id int64 1 500 | thm_name stringlengths 5 86 | thm_stmt stringlengths 30 2.63k | lean_root stringclasses 23
values | rel_path stringlengths 13 61 | imports listlengths 0 35 | used_lib_defs listlengths 1 144 | used_repo_defs listlengths 1 251 | lib_lemmas listlengths 1 172 | repo_lemmas listlengths 1 148 | used_local_defs listlengths 0 85 | used_local_lemmas listlengths 0 57 | local_ctx stringlengths 35 30.7k | target_theorem stringlengths 33 1.57k | ground_truth_proof stringlengths 6 26.5k | nesting_depth int64 1 27 | transitive_dep_count int64 1 480 | subset_aristotle bool 2
classes | category stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Binius.BinaryBasefold.fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius | theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)
[NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)
(f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)
(h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)
(i := i) (steps := ste... | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean | [
"import ArkLib.Data.MvPolynomial.Multilinear",
"import ArkLib.Data.CodingTheory.Basic",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.CodingTheory.ReedSolomon",
"import ArkLib.Data.Vector.Basic",
"import ArkLib.ProofSystem.Sumcheck.Spec.S... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "hammingDist",
"content": "notation \"Δ₀(\" u \", \" v \")\" => hammingDist u v"
},
{
"name": "distFromCode",
"content": "notation \"Δ₀(\" u \", \" C \")\" => distFromCode u C"
},
{
"name": "scoped macro_rules",
"content": "scoped macro_rules\n | `(ρ $t:term) => `(LinearCo... | [
{
"name": "Fin.is_le",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Nat.lt_of_add_right_lt",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.lt_of_le_of_lt",
"module": "Init.Prelude"
},
{
"name": "Fin.eta",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "add_ze... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "lt_add_of_pos_right_of_le",
"content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"
},
{
... | [
{
"name": "Binius.BinaryBasefold.OracleFunction",
"content": "abbrev OracleFunction (i : Fin (ℓ + 1)) : Type _ := sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n ⟩ → L"
},
{
"name": "Binius.BinaryBasefold.fiber_coeff",
"content": "noncomputable def fiber_coeff\n (i : Fin r) ... | [
{
"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R",
"content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n : i.val + steps < ℓ + 𝓡"
},
{
"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff",
"content": "lemma qMap_total_fib... | import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch
import ArkLib.Data.CodingTheory.ReedSolomon
import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.MvPolynomial.Multilinear
import ArkLib.Data.Vector.Basic
import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound
namespace Binius.BinaryBa... | theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)
[NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)
(f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)
(h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)
(i := i) (steps := ste... | := by
unfold fiberwiseClose at h_fw_dist_lt
unfold hammingClose
-- 2 * Δ₀(f, ↑(BBF_Code 𝔽q β ⟨↑i, ⋯⟩)) < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, ⋯⟩)
let d_fw := fiberwiseDistance 𝔽q β (i := i) steps h_i_add_steps f
let C_i := (BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)
let d_H := Code.distFromC... | 7 | 232 | false | Applied verif. |
2 | ConcreteBinaryTower.minPoly_of_powerBasisSucc_generator | @[simp]
theorem minPoly_of_powerBasisSucc_generator (k : ℕ) :
(minpoly (ConcreteBTField k) (powerBasisSucc k).gen) = X^2 + (Z k) • X + 1 | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | @[simp]
theorem minPoly_of_powerBasisSucc_generator (k : ℕ) :
(minpoly (ConcreteBTField k) (powerBasisSucc k).gen) = X^2 + (Z k) • X + 1 := | := by
unfold powerBasisSucc
simp only
rw [←C_mul']
letI: Fintype (ConcreteBTField k) := (getBTFResult k).instFintype
refine Eq.symm (minpoly.unique' (ConcreteBTField k) (Z (k + 1)) ?_ ?_ ?_)
· exact (definingPoly_is_monic (s:=Z (k)))
· exact aeval_definingPoly_at_Z_succ k
· intro q h_degQ_lt_deg_minPoly... | 16 | 324 | false | Applied verif. |
3 | AdditiveNTT.evaluation_poly_split_identity | theorem evaluation_poly_split_identity (i : Fin (ℓ))
(coeffs : Fin (2 ^ (ℓ - i)) → L) :
let P_i: L[X] := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs
let P_even_i_plus_1: L[X] := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs
let P_odd_i_plus_1: L[X] := oddRefinement 𝔽q β h_ℓ_add_R_rate... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Polynomial.comp_assoc",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "implies_true",
"module": "Init.SimpLemmas"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fin.foldl_succ",
"module": "Init.Data.Fin.Fol... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "getBit_eq_succ_getBit_of_mul_two_add_one",
"content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = get... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.intermedia... | [
{
"name": "AdditiveNTT.Polynomial.foldl_comp",
"content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n = (Fin.foldl (n:=n) (fun acc j => (f j).... | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | theorem evaluation_poly_split_identity (i : Fin (ℓ))
(coeffs : Fin (2 ^ (ℓ - i)) → L) :
let P_i: L[X] := | := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs
let P_even_i_plus_1: L[X] := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs
let P_odd_i_plus_1: L[X] := oddRefinement 𝔽q β h_ℓ_add_R_rate i coeffs
let q_i: L[X] := qMap 𝔽q β ⟨i, by omega⟩
P_i = (P_even_i_plus_1.comp q_i) + X * (P_odd_i_plus... | 7 | 78 | false | Applied verif. |
4 | Nat.getBit_repr | theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →
j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.Icc",
"module": "Mathlib.Order.Interval.Finset.Defs"
},
{
"name": "And",
"module": "Init.Prelude"
},
{
"name": "AddCommMonoid",
"module":... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.shiftRight_add",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "add_comm",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Finset.Icc_self",
"module": "Mathlib.Order.Interval.Finset.Basic"
},
{
"name": "Finset.mem_Icc",
"module": "Mathlib.Order.... | [
{
"name": "sum_Icc_split",
"content": "theorem sum_Icc_split {α : Type*} [AddCommMonoid α] (f : ℕ → α) (a b c : ℕ)\n (h₁ : a ≤ b) (h₂ : b ≤ c):\n ∑ i ∈ Finset.Icc a c, f i = ∑ i ∈ Finset.Icc a b, f i + ∑ i ∈ Finset.Icc (b+1) c, f i"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Nat.getBit_of_shiftRight",
"content": "lemma getBit_of_shiftRight {n p : ℕ}:\n ∀ k, getBit k (n >>> p) = getBit (k+p) n"
}
] | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →
j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k := | := by
induction ℓ with
| zero =>
-- Base case : ℓ = 0
intro j h_j
have h_j_zero : j = 0 := by exact Nat.lt_one_iff.mp h_j
subst h_j_zero
simp only [zero_tsub, Finset.Icc_self, Finset.sum_singleton, pow_zero, mul_one]
unfold getBit
rw [Nat.shiftRight_zero, Nat.and_one_is_mod]
| succ ℓ₁ ... | 2 | 24 | true | Applied verif. |
5 | Nat.getBit_of_binaryFinMapToNat | lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :
∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val
= if h_k: k < n then m ⟨k, by omega⟩ else 0 | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.univ",
"module": "Mathlib.Data.Fintype.Defs"
},
{
"name": "Ne",
"module": "Init.Core"
},
... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.and_one_is_mod",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.mod_lt",
"module": "Init.Prelude"
},
{
"name": "Nat.ofNat_pos",
"module": "Mathlib.Data.Nat.Cast.Order.Ring"
},
{
"name": "gt_iff_lt",
"module": "Init.Core"
},
{
"name": "Na... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "Nat.binaryFinMapToNat",
"content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="
}
] | [
{
"name": "Nat.getBit_lt_2",
"content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"
},
{
"name": "Nat.getBit_eq_testBit",
"content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"
},
{
"name": "Nat.getBit_zero_eq_zero",
"content": "lemma getBit... | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :
∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val
= if h_k: k < n then m ⟨k, by omega⟩ else 0 := | := by
-- We prove this by induction on `n`.
induction n with
| zero =>
intro k;
simp only [Nat.pow_zero, Fin.val_eq_zero, not_lt_zero', ↓reduceDIte]
exact getBit_zero_eq_zero
| succ n ih =>
-- Inductive step: Assume the property holds for `n`, prove it for `n+1`.
have h_lt: 2^n - 1 < 2^n := ... | 4 | 104 | true | Applied verif. |
6 | ConcreteBinaryTower.towerEquiv_commutes_left_diff | lemma towerEquiv_commutes_left_diff (i d : ℕ) : ∀ r : ConcreteBTField i,
(AlgebraTower.algebraMap i (i+d) (by omega)) ((towerEquiv i).ringEquiv r) =
(towerEquiv (i+d)).ringEquiv ((AlgebraTower.algebraMap i (i+d) (by omega)) r) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "BT... | [
{
"name": "Ne.dite_eq_left_iff",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Nat.add_one_sub_one",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.left_eq_add",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "RingHom.comp_id",
"module": "Mathlib.Algebra.Ring.Hom.Defs"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma towerEquiv_commutes_left_diff (i d : ℕ) : ∀ r : ConcreteBTField i,
(AlgebraTower.algebraMap i (i+d) (by omega)) ((towerEquiv i).ringEquiv r) =
(towerEquiv (i+d)).ringEquiv ((AlgebraTower.algebraMap i (i+d) (by omega)) r) := | := by
-- If d = 0, then this is trivial
-- For d > 0 : let j = i+d
-- lhs of goal : right => 《 0, ringMap x 》 => up => 《 algMap 0 = 0, algMap (ringMap x) 》
-- rhs of goal : up => 《 0, algMap x 》 => right => 《 ringMap 0 = 0, ringMap (algMap x) 》
-- where both `algMap (ringMap x)` and `ringMap (algMap x)`... | 10 | 306 | false | Applied verif. |
7 | AdditiveNTT.intermediateNormVpoly_comp | omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in
theorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))
(l : Fin (ℓ - (i.val + k.val) + 1)) :
intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by
simp only; omega⟩) =
(intermediateN... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Fin.cast_eq_self",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fin.coe_cast",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Fin.coe_castSucc",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
}
] | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.intermedia... | [] | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in
theorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))
(l : Fin (ℓ - (i.val + k.val) + 1)) :
intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by
simp only; omega⟩) =
(intermediateN... | := by
induction l using Fin.succRecOnSameFinType with
| zero =>
simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod, add_zero, Fin.eta, Fin.zero_eta]
have h_eq_X : intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i + ↑k, by omega⟩ 0 = X := by
simp only [intermediateNormVpoly, Fin.coe_ofNat_eq_mod, Nat.zero_mod,... | 5 | 38 | false | Applied verif. |
8 | AdditiveNTT.inductive_rec_form_W_comp | omit h_Fq_char_prime hF₂ in
lemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)
(h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])
(f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))
: ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =
((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -
... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Fact.out",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Fin.le_zero_iff'",
"module": "Mathlib.Data.Fin.SuccPred"
},
{
"name": "LinearIndependent.injective",
"module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"
},
{
"name": "Nat.not_lt_zero",
"module": "Ini... | [
{
"name": "Fin.lt_succ'",
"content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
},
{
"name": "Fin.le_succ",
"content": "lemma F... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.algEquivAevalXSubC",
... | [
{
"name": "AdditiveNTT.βᵢ_not_in_Uᵢ",
"content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n β i ∉ U 𝔽q β i"
},
{
"name": "AdditiveNTT.root_U_lift_down",
"content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n (i : Fin r) (h_i_add_1 : i + 1 < r) (a ... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | omit h_Fq_char_prime hF₂ in
lemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)
(h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])
(f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))
: ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =
((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -
... | := by
intro p
set W_i := W 𝔽q β i
set q := Fintype.card 𝔽q
set v := W_i.eval (β i)
-- First, we must prove that v is non-zero to use its inverse.
have hv_ne_zero : v ≠ 0 := by
unfold v W_i
exact Wᵢ_eval_βᵢ_neq_zero 𝔽q β i
-- Proof flow:
-- `Wᵢ₊₁(X) = ∏_{c ∈ 𝔽q} (Wᵢ ∘ (X - c • βᵢ))` -- from... | 6 | 229 | false | Applied verif. |
9 | AdditiveNTT.odd_index_intermediate_novel_basis_decomposition | lemma odd_index_intermediate_novel_basis_decomposition
(i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :
intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by
apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)
⟩ = X * (intermediateNovelBasisX 𝔽q β h_ℓ_ad... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Polynomial.comp_assoc",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "implies_true",
"module": "Init.SimpLemmas"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fin.foldl_succ",
"module": "Init.Data.Fin.Fol... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "getBit_eq_succ_getBit_of_mul_two_add_one",
"content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = get... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.intermedia... | [
{
"name": "AdditiveNTT.Polynomial.foldl_comp",
"content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n = (Fin.foldl (n:=n) (fun acc j => (f j).... | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | lemma odd_index_intermediate_novel_basis_decomposition
(i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :
intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by
apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)
⟩ = X * (intermediateNovelBasisX 𝔽q β h_ℓ_ad... | := by
unfold intermediateNovelBasisX
rw [prod_comp]
-- ∏ k ∈ Fin (ℓ - i), (Wₖ⁽ⁱ⁾(X))^((2j₊₁)ₖ)
-- = X * ∏ k ∈ Fin (ℓ - (i+1)), (Wₖ⁽ⁱ⁺¹⁾(X))^((j)ₖ) ∘ q⁽ⁱ⁾(X)
simp only [pow_comp]
conv_rhs =>
enter [2]
enter [2, x, 1]
rw [intermediateNormVpoly_comp_qmap_helper 𝔽q β h_ℓ_add_R_rate
⟨i, by om... | 5 | 50 | false | Applied verif. |
10 | AdditiveNTT.finToBinaryCoeffs_sDomainToFin | omit h_β₀_eq_1 in
lemma finToBinaryCoeffs_sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)
(x : sDomain 𝔽q β h_ℓ_add_R_rate i) :
let pointFinIdx := (sDomainToFin 𝔽q β h_ℓ_add_R_rate i h_i) x
finToBinaryCoeffs 𝔽q (i := i) (idx :=pointFinIdx) =
(sDomain_basis 𝔽q β
h_ℓ_add_R_rate i h_i).repr x | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"
},
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : ... | [
{
"name": "Fintype.card_le_one_iff_subsingleton",
"module": "Mathlib.Data.Fintype.EquivFin"
},
{
"name": "Fintype.card_units",
"module": "Mathlib.Data.Fintype.Units"
},
{
"name": "Nat.le_of_eq",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Subsingleton.elim",
"module": "... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "getBit_of_binaryFinMapToNat",
"content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n ∀ k... | [
{
"name": "AdditiveNTT.sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n AdditiveNTT.normalizedW_is_additive 𝔽q β i\n Submodule.map (polyEvalLinearMap... | [
{
"name": "AdditiveNTT.𝔽q_element_eq_zero_or_eq_one",
"content": "omit h_Fq_char_prime in\nlemma 𝔽q_element_eq_zero_or_eq_one : ∀ c: 𝔽q, c = 0 ∨ c = 1"
}
] | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | omit h_β₀_eq_1 in
lemma finToBinaryCoeffs_sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)
(x : sDomain 𝔽q β h_ℓ_add_R_rate i) :
let pointFinIdx := | := (sDomainToFin 𝔽q β h_ℓ_add_R_rate i h_i) x
finToBinaryCoeffs 𝔽q (i := i) (idx :=pointFinIdx) =
(sDomain_basis 𝔽q β
h_ℓ_add_R_rate i h_i).repr x:= by
simp only
ext j
-- Unfold the definitions to get to the core logic
dsimp [sDomainToFin, finToBinaryCoeffs, splitPointIntoCoeffs]
-- `Nat.getBit... | 5 | 84 | false | Applied verif. |
11 | AdditiveNTT.sDomain_eq_image_of_upper_span | lemma sDomain_eq_image_of_upper_span (i : Fin r) (h_i : i < ℓ + R_rate) :
let V_i := Submodule.span 𝔽q (Set.range (sBasis β h_ℓ_add_R_rate i h_i))
let W_i_map := polyEvalLinearMap (normalizedW 𝔽q β i)
(normalizedW_is_additive 𝔽q β i)
sDomain 𝔽q β h_ℓ_add_R_rate i
= Submodule.map W_i_map V_i | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import Mathlib.Tactic",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"
},
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : ... | [
{
"name": "Fin.mk_le_of_le_val",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Fin.mk_lt_of_lt_val",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Nat.lt_sub_of_add_lt",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "normalizedWᵢ_vanishing",
"content": "lemma normalizedWᵢ_vanishing (i : Fin r) :\n ∀ u ∈ U 𝔽q β i, (normalizedW 𝔽q β i).eval u = 0"
},
{... | [
{
"name": "AdditiveNTT.sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n AdditiveNTT.normalizedW_is_additive 𝔽q β i\n Submodule.map (polyEvalLinearMap... | [
{
"name": "AdditiveNTT.sBasis_range_eq",
"content": "omit [NeZero r] [Field L] [Fintype L] [DecidableEq L] [Field 𝔽q] [Algebra 𝔽q L] in\nlemma sBasis_range_eq (i : Fin r) (h_i : i < ℓ + R_rate) :\n β '' Set.Ico i ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩\n = Set.range (sBasis β h_ℓ_add_R_rate i h_i)"
}
] | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | lemma sDomain_eq_image_of_upper_span (i : Fin r) (h_i : i < ℓ + R_rate) :
let V_i := | := Submodule.span 𝔽q (Set.range (sBasis β h_ℓ_add_R_rate i h_i))
let W_i_map := polyEvalLinearMap (normalizedW 𝔽q β i)
(normalizedW_is_additive 𝔽q β i)
sDomain 𝔽q β h_ℓ_add_R_rate i
= Submodule.map W_i_map V_i :=
by
-- Proof: U_{ℓ+R} is the direct sum of Uᵢ and Vᵢ.
-- Any x in U_{ℓ+R} can be w... | 11 | 81 | false | Applied verif. |
12 | AdditiveNTT.initial_tiled_coeffs_correctness | omit [DecidableEq 𝔽q] hF₂ in
lemma initial_tiled_coeffs_correctness (h_ℓ : ℓ ≤ r) (a : Fin (2 ^ ℓ) → L) :
let b: Fin (2^(ℓ + R_rate)) → L := tileCoeffs a
additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate b a (i := ⟨ℓ, by omega⟩) | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs",
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Fintype.card_pos",
"module": "Mathlib.Data.Fintype.Card"
},
{
"name": "Polynomial.C_1",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Polynomial.C_comp",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "Polynomial.C_mul",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "W_linear_comp_decomposition",
"content": "omit hF₂ in\ntheorem W_linear_comp_decomposition (i : Fin r) (h_i_add_1 : i + 1 < r) :\n ∀ p: L[X... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.qCompositi... | [
{
"name": "AdditiveNTT.qMap_comp_normalizedW",
"content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"
},
{
"name": "AdditiveNTT.qCompositionChain_eq_foldl",
"content": "lemma qCompositionChain_eq_fol... | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | omit [DecidableEq 𝔽q] hF₂ in
lemma initial_tiled_coeffs_correctness (h_ℓ : ℓ ≤ r) (a : Fin (2 ^ ℓ) → L) :
let b: Fin (2^(ℓ + R_rate)) → L := | := tileCoeffs a
additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate b a (i := ⟨ℓ, by omega⟩) := by
unfold additiveNTTInvariant
simp only
intro j
unfold coeffsBySuffix
simp only [tileCoeffs, evaluationPointω, intermediateEvaluationPoly, Fin.eta]
have h_ℓ_sub_ℓ: 2^(ℓ - ℓ) = 1 := by norm_num
set f_r... | 14 | 134 | false | Applied verif. |
13 | MlPoly.mobius_apply_zeta_apply_eq_id | theorem mobius_apply_zeta_apply_eq_id (n : ℕ) [NeZero n] (r : Fin n) (l : Fin (r.val + 1))
(v : Vector R (2 ^ n)) : lagrangeToMono_segment n r l (monoToLagrange_segment n r l v) = v | ArkLib | ArkLib/Data/MlPoly/Basic.lean | [
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.List.Lemmas",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Vector.Basic",
"import Mathlib.RingTheory.MvPolynomial.Basic",
"import ToMathlib.General"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Vector",
"module": "Init.Data.Vector.Basic"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "BitVec.ofFin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
... | [
{
"name": "Fin.succRecOnSameFinType",
"content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n (zero : motive (0 : Fin r))\n (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n | ⟨0, _⟩ => by admit /- proof elided -/\n | ⟨Nat.succ i_val... | [
{
"name": "List.length_ofFn",
"module": "Init.Data.List.OfFn"
},
{
"name": "List.getElem_ofFn",
"module": "Init.Data.List.OfFn"
},
{
"name": "List.get_eq_getElem",
"module": "Init.Data.List.Lemmas"
},
{
"name": "Fin.is_lt",
"module": "Init.Data.Fin.Lemmas"
},
{
"n... | [
{
"name": "testBit_true_eq_getBit_eq_1",
"content": "lemma testBit_true_eq_getBit_eq_1 (k n : Nat) : n.testBit k = ((Nat.getBit k n) = 1)"
},
{
"name": "testBit_false_eq_getBit_eq_0",
"content": "lemma testBit_false_eq_getBit_eq_0 (k n : Nat) :\n (n.testBit k = false) = ((Nat.getBit k n) = 0)"
... | [
{
"name": "MlPoly",
"content": "@[reducible]\ndef MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n)"
},
{
"name": "MlPoly.monoToLagrangeLevel",
"content": "@[inline] def monoToLagrangeLevel {n : ℕ} (j : Fin n) : Vector R (2 ^ n) → Vector R (2 ^ n) :=\n fun v =>\n let stride : ℕ := 2 ^ j.val ... | [
{
"name": "MlPoly.forwardRange_length",
"content": "lemma forwardRange_length (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n (forwardRange n r l).length = r.val - l.val + 1"
},
{
"name": "MlPoly.forwardRange_eq_of_r_eq",
"content": "lemma forwardRange_eq_of_r_eq (n : ℕ) (r1 r2 : Fin n) (h_r_eq... | import ArkLib.Data.Nat.Bitwise
import Mathlib.RingTheory.MvPolynomial.Basic
import ToMathlib.General
import ArkLib.Data.Fin.BigOperators
import ArkLib.Data.List.Lemmas
import ArkLib.Data.Vector.Basic
@[reducible]
def MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n)
variable {R : Type*} {n : ℕ}
namespace MlPoly
... | theorem mobius_apply_zeta_apply_eq_id (n : ℕ) [NeZero n] (r : Fin n) (l : Fin (r.val + 1))
(v : Vector R (2 ^ n)) : lagrangeToMono_segment n r l (monoToLagrange_segment n r l v) = v := | := by
induction r using Fin.succRecOnSameFinType with
| zero =>
rw [lagrangeToMono_segment, monoToLagrange_segment, forwardRange]
simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod, Fin.val_eq_zero, tsub_self, zero_add,
List.ofFn_succ, Fin.isValue, Fin.cast_zero, Nat.mod_succ, add_zero, Fin.mk_zero',
... | 7 | 84 | false | Applied verif. |
14 | Nat.getLowBits_succ | lemma getLowBits_succ {n: ℕ} (numLowBits: ℕ) :
getLowBits (numLowBits + 1) n = getLowBits numLowBits n
+ (getBit numLowBits n) <<< numLowBits | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BEq",
"module": "Init.Prelude"
},
{
"name": "Nat.testBit",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "Nat.and_distrib_right",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Bool",
"modu... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.and_one_is_mod",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.eq_of_testBit_eq",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.mod_two_bne_zero",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "Nat.one_and_eq_mod_two",
"module": "I... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "Nat.getLowBits",
"content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"
}
] | [
{
"name": "Nat.getBit_lt_2",
"content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"
},
{
"name": "Nat.eq_iff_eq_all_getBits",
"content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"
},
{
"name": "Nat.shiftRight_and_one_distrib",
"content": "lemm... | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | lemma getLowBits_succ {n: ℕ} (numLowBits: ℕ) :
getLowBits (numLowBits + 1) n = getLowBits numLowBits n
+ (getBit numLowBits n) <<< numLowBits := | := by
apply eq_iff_eq_all_getBits.mpr;
intro k
have h_getBit_lt_numLowBits: getBit numLowBits n < 2 := by exact getBit_lt_2
interval_cases h_getBit: getBit numLowBits n
· rw [Nat.zero_shiftLeft]
simp only [add_zero]
-- ⊢ getLowBits n (numLowBits + 1) >>> k &&& 1 = getLowBits n numLowBits >>> k &&& 1
... | 4 | 103 | true | Applied verif. |
15 | rsum_eq_t1_square_aux | theorem rsum_eq_t1_square_aux
{curBTField : Type*} [Field curBTField] -- curBTField ≃ 𝔽_{2^{2^k}}
(u : curBTField) -- here u is already lifted to curBTField
(k : ℕ)
(x_pow_card : ∀ (x : curBTField), x ^ (2 ^ (2 ^ (k))) = x)
(u_ne_zero : u ≠ 0)
(trace_map_prop : TraceMapProperty curBTField u k):
∑ j ∈ Fi... | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Prelude.lean | [
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.FieldTheory.Finite.GaloisField",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.StdBasis"
] | [
{
"name": "Field",
"module": "Mathlib.Algebra.Field.Defs"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.range",
"module": "Mathlib.Data.Finset.Range"
},
{
"name": "False.elim",
"module": "Init.Prelude"
},
{
"name": "Finset.Icc",
... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.pow_le_pow_right",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.pow_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.zero_le",
"module": "Init.Prelude"
},
{
"name": "Finset.mem_Icc",
"module": "Mathlib.Order.Interval.Finset.Defs"
},
{
"... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "TraceMapProperty",
"content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"
}
] | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
}
] | import Mathlib.FieldTheory.Finite.GaloisField
import ArkLib.Data.Fin.BigOperators
import ArkLib.Data.Nat.Bitwise
import Mathlib.LinearAlgebra.StdBasis
noncomputable section Preliminaries
open Polynomial
open AdjoinRoot
open Module
notation : 10 "GF(" term : 10 ")" => GaloisField term 1
structure TraceMapProper... | theorem rsum_eq_t1_square_aux
{curBTField : Type*} [Field curBTField] -- curBTField ≃ 𝔽_{2^{2^k}}
(u : curBTField) -- here u is already lifted to curBTField
(k : ℕ)
(x_pow_card : ∀ (x : curBTField), x ^ (2 ^ (2 ^ (k))) = x)
(u_ne_zero : u ≠ 0)
(trace_map_prop : TraceMapProperty curBTField u k):
∑ j ∈ Fi... | := by
have trace_map_icc_t1 : ∑ j ∈ Finset.Icc 0 (2^(k)-1), u ^ (2^j) = 1 := by
rw [←Nat.range_succ_eq_Icc_zero (2^(k)-1), Nat.sub_add_cancel (h:=one_le_two_pow_n (k))]
exact trace_map_prop.1
have trace_map_icc_t1_inv : ∑ j ∈ Finset.Icc 0 (2^(k)-1), u⁻¹ ^ (2^j) = 1 := by
rw [←Nat.range_succ_eq_Icc_zero... | 2 | 35 | true | Applied verif. |
16 | AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C | omit h_Fq_char_prime hF₂ in
lemma rootMultiplicity_prod_W_comp_X_sub_C
(i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :
rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =
if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0 | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Finset.prod_eq_zero_iff",
"module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"
},
{
"name": "Nat.not_lt_zero",
"module": "Init.Prelude"
},
{
"name": "Polynomial.X_sub_C_ne_zero",
"module": "Mathlib.Algebra.Polynomial.Degree.Operations"
},
{
"name": "Set.Ic... | [
{
"name": "Fin.lt_succ'",
"content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
},
{
"name": "Fin.le_succ",
"content": "lemma F... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.algEquivAevalXSubC",
... | [
{
"name": "AdditiveNTT.βᵢ_not_in_Uᵢ",
"content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n β i ∉ U 𝔽q β i"
},
{
"name": "AdditiveNTT.root_U_lift_down",
"content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n (i : Fin r) (h_i_add_1 : i + 1 < r) (a ... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | omit h_Fq_char_prime hF₂ in
lemma rootMultiplicity_prod_W_comp_X_sub_C
(i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :
rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =
if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0 := | := by
rw [←Polynomial.count_roots]
set f := fun c: 𝔽q => (W 𝔽q β i).comp (X - C (c • β i)) with hf
-- ⊢ Multiset.count a (univ.prod f).roots = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0
have h_prod_ne_zero: univ.prod f ≠ 0 := Prod_W_comp_X_sub_C_ne_zero 𝔽q β i
rw [roots_prod (f := f) (s := univ (α := 𝔽q)) h_... | 4 | 157 | false | Applied verif. |
17 | Binius.BinaryBasefold.is_fiber_iff_generates_quotient_point | theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)
(h_i_add_steps : i.val + steps ≤ ℓ)
(x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :
let qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := s... | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean | [
"import ArkLib.Data.MvPolynomial.Multilinear",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.CodingTheory.ReedSolomon",
"import ArkLib.Data.Vector.Basic",
"import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound",
"import ArkLib.Data.FieldThe... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\... | [
{
"name": "Fin.eta",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "add_zero",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Module.Basis.repr_linearCombination",
"module": "Mathlib.LinearAlgebra.Basis.Defs"
},
{
"name": "Module.Basis.repr_symm_apply",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "lt_add_of_pos_right_of_le",
"content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"
},
{
... | [
{
"name": "Binius.BinaryBasefold.fiber_coeff",
"content": "noncomputable def fiber_coeff\n (i : Fin r) (steps : ℕ)\n (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n if hj : j.val < steps then\n if Nat.getBit (k := j) (n := elemen... | [
{
"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R",
"content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n : i.val + steps < ℓ + 𝓡"
},
{
"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff",
"content": "lemma qMap_total_fib... | import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch
import ArkLib.Data.CodingTheory.ReedSolomon
import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.MvPolynomial.Multilinear
import ArkLib.Data.Vector.Basic
import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound
namespace Binius.BinaryBa... | theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)
(h_i_add_steps : i.val + steps ≤ ℓ)
(x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :
let qMapFiber := | := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)
(h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y)
let k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)
(h_i_add_steps := h_i_add_steps) (x := x)
y = iteratedQuotientMap 𝔽... | 6 | 127 | false | Applied verif. |
18 | ConcreteBinaryTower.Z_square_eq | lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))
(curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :
letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps
(Z (k + 1)) ^ 2 = 《 Z (k), 1 》 | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "Al... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dcast (Eq.refl n) bv = bv"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))
(curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :
letI : Field (ConcreteBTField (k + 1)) := | := mkFieldInstance curBTFieldProps
(Z (k + 1)) ^ 2 = 《 Z (k), 1 》 := by
letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps
have hmul : ∀ (a b : ConcreteBTField (k - 1)), concrete_mul a b = a * b := fun a b => rfl
rw [pow_two]
change concrete_mul (Z (k + 1)) (Z (k + 1)) = 《 Z (k), 1 》
... | 8 | 140 | false | Applied verif. |
19 | Binius.BinaryBasefold.qMap_total_fiber_disjoint | theorem qMap_total_fiber_disjoint
(i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i + steps ≤ ℓ)
{y₁ y₂ : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩}
(hy_ne : y₁ ≠ y₂) :
Disjoint
((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)
(h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_ad... | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean | [
"import ArkLib.Data.MvPolynomial.Multilinear",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.CodingTheory.ReedSolomon",
"import ArkLib.Data.Vector.Basic",
"import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound",
"import ArkLib.Data.FieldThe... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n AdditiveNTT.normalizedW_is_additive 𝔽q β i\n Submodule.map (polyEvalLinearMap W_i_norm h_... | [
{
"name": "Fin.eta",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "add_zero",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Module.Basis.repr_linearCombination",
"module": "Mathlib.LinearAlgebra.Basis.Defs"
},
{
"name": "Module.Basis.repr_symm_apply",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "lt_add_of_pos_right_of_le",
"content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"
},
{
... | [
{
"name": "Binius.BinaryBasefold.fiber_coeff",
"content": "noncomputable def fiber_coeff\n (i : Fin r) (steps : ℕ)\n (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n if hj : j.val < steps then\n if Nat.getBit (k := j) (n := elemen... | [
{
"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R",
"content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n : i.val + steps < ℓ + 𝓡"
},
{
"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff",
"content": "lemma qMap_total_fib... | import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch
import ArkLib.Data.CodingTheory.ReedSolomon
import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.MvPolynomial.Multilinear
import ArkLib.Data.Vector.Basic
import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound
namespace Binius.BinaryBa... | theorem qMap_total_fiber_disjoint
(i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i + steps ≤ ℓ)
{y₁ y₂ : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩}
(hy_ne : y₁ ≠ y₂) :
Disjoint
((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)
(h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_ad... | := by
-- Proof by contradiction. Assume the intersection is non-empty.
rw [Finset.disjoint_iff_inter_eq_empty]
by_contra h_nonempty
-- Let `x` be an element in the intersection of the two fiber sets.
obtain ⟨x, h_x_mem_inter⟩ := Finset.nonempty_of_ne_empty h_nonempty
have hx₁ := Finset.mem_of_mem_inter_left ... | 6 | 136 | false | Applied verif. |
20 | AdditiveNTT.even_index_intermediate_novel_basis_decomposition | lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :
intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by
apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)
⟩ = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Polynomial.comp_assoc",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "implies_true",
"module": "Init.SimpLemmas"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fin.foldl_succ",
"module": "Init.Data.Fin.Fol... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "getBit_zero_of_two_mul",
"content": "lemma getBit_zero_of_two_mul {n : ℕ} : getBit 0 (2*n) = 0"
},
{
"name": "lt_two_pow_of_lt_two_pow... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.intermedia... | [
{
"name": "AdditiveNTT.Polynomial.foldl_comp",
"content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n = (Fin.foldl (n:=n) (fun acc j => (f j).... | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :
intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by
apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)
⟩ = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by... | := by
unfold intermediateNovelBasisX
rw [prod_comp]
-- ∏ k ∈ Fin (ℓ - i), (Wₖ⁽ⁱ⁾(X))^((2j)ₖ) = ∏ k ∈ Fin (ℓ - (i+1)), (Wₖ⁽ⁱ⁺¹⁾(X))^((j)ₖ) ∘ q⁽ⁱ⁾(X)
simp only [pow_comp]
conv_rhs =>
enter [2, x]
rw [intermediateNormVpoly_comp_qmap_helper 𝔽q]
-- ⊢ ∏ x, intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i,... | 5 | 50 | false | Applied verif. |
21 | ConcreteBinaryTower.split_algebraMap_eq_zero_x | lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :
letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)
split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "Ne.dite_eq_left_iff",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Nat.add_one_sub_one",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.left_eq_add",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "RingHom.comp_id",
"module": "Mathlib.Algebra.Ring.Hom.Defs"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :
letI instAlgebra := | := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)
split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x) := by
-- this one is long because of the `cast` stuff, but it should be quite straightforward
-- via def of `canonicalAlgMap` and `split_of_join`
apply Eq.symm
letI inst... | 8 | 229 | false | Applied verif. |
22 | ConcreteBinaryTower.split_bitvec_eq_iff_fromNat | theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
split h_pos x = (hi_btf, lo_btf) ↔
(hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧
lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
}
] | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"
},
{
"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2",
"content": "lemma one_le_sub_midd... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
split h_pos x = (hi_btf, lo_btf) ↔
(hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧
lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) := | := by
have lhs_lo_case := BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat (num_bits:=2 ^ (k - 1))
(n:=2 ^ k) (Nat.two_pow_pos (k - 1)) (x:=x)
have rhs_hi_case_bitvec_eq := BitVec.extractLsb_eq_shift_ofNat (n:=2 ^ k) (r:=2 ^ k - 1)
(l:=2 ^ (k - 1)) (x:=x)
constructor
· -- Forward direction : split x = (hi_b... | 4 | 40 | false | Applied verif. |
23 | AdditiveNTT.basisVectors_span | theorem basisVectors_span (ℓ : Nat) (h_ℓ : ℓ ≤ r) :
Submodule.span L (Set.range (basisVectors 𝔽q β ℓ h_ℓ)) = ⊤ | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "WithBot",
"module": "Mathlib.Order.TypeTags"
},
{
"name": "Subspace",
"module": "Mathli... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "finiteDimensional_degreeLT",
"content": "instance finiteDimensional_degreeLT {n : ℕ} (h_n_pos : 0 < n) :\n FiniteDimensional L L⦃< n⦄[X] :="
},
{
"name": "coeff.{u}",
"content": "def coeff... | [
{
"name": "Fin.card_Ico",
"module": "Mathlib.Order.Interval.Finset.Fin"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fintype.card_ofFinset",
"module": "Mathlib.Data.Fintype.Card"
},
{
"name": "LinearIndependent.injective",
"module"... | [
{
"name": "getBit_repr",
"content": "theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →\n j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k"
},
{
"name": "getBit_of_shiftRight",
"content": "lemma getBit_of_shiftRight {n p : ℕ}:\n ∀ k, getBit k (n >>> p) = getBit (k+p) n"
},
{
"name": "getBi... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.normalizedW",
"conten... | [
{
"name": "AdditiveNTT.finrank_U",
"content": "omit [Fintype L] [Fintype 𝔽q] h_Fq_char_prime in\nlemma finrank_U (i : Fin r) :\n Module.finrank 𝔽q (U 𝔽q β i) = i"
},
{
"name": "AdditiveNTT.U_card",
"content": "lemma U_card (i : Fin r) :\n Fintype.card (U 𝔽q β i) = (Fintype.card 𝔽q)^i.va... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | theorem basisVectors_span (ℓ : Nat) (h_ℓ : ℓ ≤ r) :
Submodule.span L (Set.range (basisVectors 𝔽q β ℓ h_ℓ)) = ⊤ := | := by
have h_li := basisVectors_linear_independent 𝔽q β ℓ h_ℓ
let n := 2 ^ ℓ
have h_n: n = 2 ^ ℓ := by omega
have h_n_pos: 0 < n := by
rw [h_n]
exact Nat.two_pow_pos ℓ
have h_finrank_eq_n : Module.finrank L (L⦃< n⦄[X]) = n := finrank_degreeLT_n n
-- We have `n` linearly independent vectors in an `n... | 9 | 163 | false | Applied verif. |
24 | MlPoly.coeff_of_toMvPolynomial_eq_coeff_of_MlPoly | theorem coeff_of_toMvPolynomial_eq_coeff_of_MlPoly (p : MlPoly R n) (m : Fin n →₀ ℕ) :
coeff m (toMvPolynomial p) =
if h_binary: (∀ j: Fin n, m j ≤ 1) then
let i_of_m: ℕ := Nat.binaryFinMapToNat (m:=m) (h_binary:=h_binary)
p[i_of_m]
else
0 | ArkLib | ArkLib/Data/MlPoly/Equiv.lean | [
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.MlPoly.Basic",
"import ArkLib.Data.MvPolynomial.Notation"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.univ",
"module": "Mathlib.Data.Fintype.Defs"
},
{
"name": "Finsupp",
"module": "Mathlib.Data... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "MlPoly",
"content": "@[reducible]\ndef MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n) "
},
{
"name": "binaryFinMapToNat",
"content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary... | [
{
"name": "Finsupp.onFinset_apply",
"module": "Mathlib.Data.Finsupp.Defs"
},
{
"name": "Fintype.sum_eq_zero",
"module": "Mathlib.Data.Fintype.BigOperators"
},
{
"name": "MvPolynomial.coeff_monomial",
"module": "Mathlib.Algebra.MvPolynomial.Basic"
},
{
"name": "MvPolynomial.co... | [
{
"name": "getBit_lt_2",
"content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"
},
{
"name": "eq_iff_eq_all_getBits",
"content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"
},
{
"name": "getBit_of_lt_two_pow",
"content": "lemma getBit_of_lt_two... | [
{
"name": "MlPoly.monomialOfNat",
"content": "noncomputable def monomialOfNat (i : ℕ) : (Fin n) →₀ ℕ :=\n Finsupp.onFinset (s:=Finset.univ (α:=Fin n)) (fun j => Nat.getBit j.val i) (by admit /- proof elided -/\n )"
},
{
"name": "MlPoly.toMvPolynomial",
"content": "def toMvPolynomial (p : MlP... | [
{
"name": "MlPoly.eq_monomialOfNat_iff_eq_bitRepr",
"content": "theorem eq_monomialOfNat_iff_eq_bitRepr (m : Fin n →₀ ℕ)\n (h_binary : ∀ j : Fin n, m j ≤ 1) (i: Fin (2^n)) :\n monomialOfNat i = m ↔ i = Nat.binaryFinMapToNat m h_binary"
},
{
"name": "MlPoly.toMvPolynomial_is_multilinear",
"cont... | import ArkLib.Data.MlPoly.Basic
import ArkLib.Data.MvPolynomial.Notation
open MvPolynomial
variable {R : Type*} [CommRing R] {n : ℕ}
noncomputable section
namespace MlPoly
noncomputable def monomialOfNat (i : ℕ) : (Fin n) →₀ ℕ :=
Finsupp.onFinset (s:=Finset.univ (α:=Fin n)) (fun j => Nat.getBit j.val i) (by adm... | theorem coeff_of_toMvPolynomial_eq_coeff_of_MlPoly (p : MlPoly R n) (m : Fin n →₀ ℕ) :
coeff m (toMvPolynomial p) =
if h_binary: (∀ j: Fin n, m j ≤ 1) then
let i_of_m: ℕ := | := Nat.binaryFinMapToNat (m:=m) (h_binary:=h_binary)
p[i_of_m]
else
0
:= by
if h_binary: (∀ j: Fin n, m j ≤ 1) then
unfold toMvPolynomial
simp only [h_binary, implies_true, ↓reduceDIte]
let i_of_m := Nat.binaryFinMapToNat m h_binary
have h_mono_eq : monomialOfNat i_of_m = m := by
... | 6 | 57 | false | Applied verif. |
25 | Polynomial.Bivariate.degreeX_mul | @[simp, grind _=_]
lemma degreeX_mul [IsDomain F] (f g : F[X][Y]) (hf : f ≠ 0) (hg : g ≠ 0) :
degreeX (f * g) = degreeX f + degreeX g | ArkLib | ArkLib/Data/Polynomial/Bivariate.lean | [
"import ArkLib.Data.Polynomial.Prelims"
] | [
{
"name": "Semiring",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "IsDomain",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"nam... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Finset.sum_eq_single",
"module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"
},
{
"name": "Finset.sum_union",
"module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"
},
{
"name": "Finset.sup_lt_iff",
"module": "Mathlib.Data.Finset.Lattice.Fold"
},
{
"name... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Polynomial.Bivariate.coeff",
"content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"
},
{
"name": "Polynomial.Bivariate.degreeX",
"content": "def degreeX (f : F[X][Y]) : ℕ := f.support.sup (fun n => (f.coeff n).natDegree)"
}
] | [
{
"name": "Polynomial.Bivariate.natDeg_sum_eq_of_unique",
"content": "lemma natDeg_sum_eq_of_unique {α : Type} {s : Finset α} {f : α → F[X]} {deg : ℕ}\n (mx : α) (h : mx ∈ s) :\n (f mx).natDegree = deg →\n (∀ y ∈ s, y ≠ mx → (f y).natDegree < deg ∨ f y = 0) →\n (∑ x ∈ s, f x).natDegree = deg"
},... | import ArkLib.Data.Polynomial.Prelims
open Polynomial
open Polynomial.Bivariate
namespace Polynomial.Bivariate
noncomputable section
variable {F : Type} [Semiring F]
def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i
def degreeX (f : F[X][Y]) : ℕ := f.support.sup (fun n =>... | @[simp, grind _=_]
lemma degreeX_mul [IsDomain F] (f g : F[X][Y]) (hf : f ≠ 0) (hg : g ≠ 0) :
degreeX (f * g) = degreeX f + degreeX g := | := by
letI s₁ := {n ∈ f.support | (f.coeff n).natDegree = degreeX f}
letI s₂ := {n ∈ g.support | (g.coeff n).natDegree = degreeX g}
have f_mdeg_nonempty : s₁.Nonempty := by
obtain ⟨mfx, _, _⟩ :=
Finset.exists_mem_eq_sup _ (show f.support.Nonempty by grind) fun n ↦ (f.coeff n).natDegree
use mfx
g... | 2 | 34 | false | Applied verif. |
26 | Binius.BinaryBasefold.card_qMap_total_fiber | omit [CharP L 2] [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 [NeZero ℓ] in
theorem card_qMap_total_fiber (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :
Fintype.card (Set.image (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)
... | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean | [
"import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound",
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import ArkLib.Data.MvPolynomial.Multilinear",
"import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\... | [
{
"name": "Fin.eta",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "add_zero",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Module.Basis.repr_linearCombination",
"module": "Mathlib.LinearAlgebra.Basis.Defs"
},
{
"name": "Module.Basis.repr_symm_apply",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "lt_add_of_pos_right_of_le",
"content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"
},
{
... | [
{
"name": "Binius.BinaryBasefold.fiber_coeff",
"content": "noncomputable def fiber_coeff\n (i : Fin r) (steps : ℕ)\n (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n if hj : j.val < steps then\n if Nat.getBit (k := j) (n := elemen... | [
{
"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R",
"content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n : i.val + steps < ℓ + 𝓡"
},
{
"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff",
"content": "lemma qMap_total_fib... | import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch
import ArkLib.Data.CodingTheory.ReedSolomon
import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.MvPolynomial.Multilinear
import ArkLib.Data.Vector.Basic
import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound
namespace Binius.BinaryBa... | omit [CharP L 2] [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 [NeZero ℓ] in
theorem card_qMap_total_fiber (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :
Fintype.card (Set.image (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)
... | := by
-- The cardinality of the image of a function equals the cardinality of its domain
-- if it is injective.
rw [Set.card_image_of_injective Set.univ]
-- The domain is `Fin (2 ^ steps)`, which has cardinality `2 ^ steps`.
· -- ⊢ Fintype.card ↑Set.univ = 2 ^ steps
simp only [Fintype.card_setUniv, Fintyp... | 5 | 78 | false | Applied verif. |
27 | Binius.BinaryBasefold.qMap_total_fiber_one_level_eq | lemma qMap_total_fiber_one_level_eq (i : Fin ℓ) (h_i_add_1 : i.val + 1 ≤ ℓ)
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i + 1, by omega⟩)) (k : Fin 2) :
let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)
let x : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := qMap_total_fiber �... | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean | [
"import ArkLib.Data.MvPolynomial.Multilinear",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.CodingTheory.ReedSolomon",
"import ArkLib.Data.Vector.Basic",
"import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound",
"import ArkLib.Data.FieldThe... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Ring",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "sDomain",
"content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n let W_i_norm := normalizedW 𝔽q β i\n let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\... | [
{
"name": "Fin.eta",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "add_zero",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Module.Basis.repr_linearCombination",
"module": "Mathlib.LinearAlgebra.Basis.Defs"
},
{
"name": "Module.Basis.repr_symm_apply",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "lt_add_of_pos_right_of_le",
"content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"
},
{
... | [
{
"name": "Binius.BinaryBasefold.Fin2ToF2",
"content": "def Fin2ToF2 (𝔽q : Type*) [Ring 𝔽q] (k : Fin 2) : 𝔽q :=\n if k = 0 then 0 else 1"
},
{
"name": "Binius.BinaryBasefold.fiber_coeff",
"content": "noncomputable def fiber_coeff\n (i : Fin r) (steps : ℕ)\n (j : Fin (ℓ + 𝓡 - i)) (elem... | [
{
"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R",
"content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n : i.val + steps < ℓ + 𝓡"
},
{
"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff",
"content": "lemma qMap_total_fib... | import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch
import ArkLib.Data.CodingTheory.ReedSolomon
import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.MvPolynomial.Multilinear
import ArkLib.Data.Vector.Basic
import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound
namespace Binius.BinaryBa... | lemma qMap_total_fiber_one_level_eq (i : Fin ℓ) (h_i_add_1 : i.val + 1 ≤ ℓ)
(y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i + 1, by omega⟩)) (k : Fin 2) :
let basis_x := | := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)
let x : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)
(steps := 1) (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k
let y_lifted : sDomain 𝔽q β h_ℓ_add_R_rate ⟨... | 5 | 95 | false | Applied verif. |
28 | ReedSolomonCode.minDist | theorem minDist [Field F] [DecidableEq F] (inj : Function.Injective α) [NeZero n] (h : n ≤ m) :
minDist ((ReedSolomon.code ⟨α, inj⟩ n) : Set (Fin m → F)) = m - n + 1 | ArkLib | ArkLib/Data/CodingTheory/ReedSolomon.lean | [
"import ArkLib.Data.CodingTheory.Basic",
"import Mathlib.LinearAlgebra.Lagrange",
"import ArkLib.Data.MvPolynomial.LinearMvExtension",
"import Mathlib.RingTheory.Henselian",
"import ArkLib.Data.CodingTheory.Prelims",
"import ArkLib.Data.Fin.Lift",
"import ArkLib.Data.Polynomial.Interface"
] | [
{
"name": "Fintype",
"module": "Mathlib.Data.Fintype.Defs"
},
{
"name": "Polynomial.X",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Semiring",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "toFun",
"module": "ToMathlib.Control.Monad.Hom"
},
{
"... | [
{
"name": "wt",
"content": "def wt [Zero F]\n (v : ι → F) : ℕ := #{i | v i ≠ 0}"
},
{
"name": "dim",
"content": "noncomputable def dim [Semiring F] (LC : LinearCode ι F) : ℕ :=\n Module.finrank F LC"
},
{
"name": "LinearCode.{u,",
"content": "abbrev LinearCode.{u, v} (ι : Type u) [... | [
{
"name": "Finset.image_subset_iff",
"module": "Mathlib.Data.Finset.Image"
},
{
"name": "Finset.sum_image",
"module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"
},
{
"name": "Finset.sum_le_sum",
"module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"
},
{
"name": "... | [
{
"name": "rank_eq_if_det_ne_zero",
"content": "lemma rank_eq_if_det_ne_zero {U : Matrix (Fin n) (Fin n) F} [IsDomain F] :\n Matrix.det U ≠ 0 → U.rank = n"
},
{
"name": "rank_eq_if_subUpFull_eq",
"content": "lemma rank_eq_if_subUpFull_eq (h : n ≤ m) :\n (subUpFull U (Fin.castLE h)).rank = n ... | [
{
"name": "ReedSolomon.evalOnPoints",
"content": "def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n toFun := fun p => fun x => p.eval (domain x)\n map_add' := fun x y => by admit /- proof elided -/"
},
{
"name": "ReedSolomon.code",
"content": "def code (deg : ℕ) [Semiring F]: Submodul... | [
{
"name": "Vandermonde.nonsquare_mulVecLin",
"content": "lemma nonsquare_mulVecLin [CommSemiring F] {ι' : ℕ} {α₁ : ι ↪ F} {α₂ : Fin ι' → F} {i : ι} :\n (nonsquare ι' α₁).mulVecLin α₂ i = ∑ x, α₂ x * α₁ i ^ x.1"
},
{
"name": "Vandermonde.subUpFull_of_vandermonde_is_vandermonde",
"content": "lemm... | import ArkLib.Data.MvPolynomial.LinearMvExtension
import ArkLib.Data.Polynomial.Interface
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.RingTheory.Henselian
namespace ReedSolomon
open Polynomial NNReal
variable {F : Type*} {ι : Type*} (domain : ι ↪ F)
def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) whe... | theorem minDist [Field F] [DecidableEq F] (inj : Function.Injective α) [NeZero n] (h : n ≤ m) :
minDist ((ReedSolomon.code ⟨α, inj⟩ n) : Set (Fin m → F)) = m - n + 1 := | := by
have : NeZero m := by constructor; aesop
refine le_antisymm ?p₁ ?p₂
case p₁ =>
have distUB := singletonBound (LC := ReedSolomon.code ⟨α, inj⟩ n)
rw [dim_eq_deg_of_le inj h] at distUB
simp at distUB
zify [dist_le_length] at distUB
omega
case p₂ =>
rw [dist_eq_minWtCodewords]
app... | 8 | 118 | false | Applied verif. |
29 | Vector.foldl_succ | theorem foldl_succ
{α β} {n : ℕ} [NeZero n] (f : β → α → β) (init : β) (v : Vector α n) :
v.foldl (f:=f) (b:=init) = v.tail.foldl (f:=f) (b:=f init v.head) | ArkLib | ArkLib/Data/Vector/Basic.lean | [
"import Mathlib.Data.Matrix.Mul",
"import Mathlib.Algebra.Order.Sub.Basic",
"import Mathlib.Algebra.Order.Star.Basic",
"import Mathlib.Algebra.BigOperators.Fin",
"import ToMathlib.General"
] | [
{
"name": "NeZero",
"module": "Init.Data.NeZero"
},
{
"name": "Vector",
"module": "Init.Data.Vector.Basic"
},
{
"name": "Array",
"module": "Init.Prelude"
},
{
"name": "Array.foldl",
"module": "Init.Data.Array.Basic"
},
{
"name": "List",
"module": "Init.Prelude... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Array.foldl_toList",
"module": "Init.Data.Array.Bootstrap"
},
{
"name": "Array.toList_extract",
"module": "Init.Data.Array.Lemmas"
},
{
"name": "List.drop_one",
"module": "Init.Data.List.TakeDrop"
},
{
"name": "List.extract_eq_drop_take",
"module": "Init.Data.L... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [] | [] | import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Algebra.Order.Sub.Basic
import Mathlib.Data.Matrix.Mul
import ToMathlib.General
namespace Vector | theorem foldl_succ
{α β} {n : ℕ} [NeZero n] (f : β → α → β) (init : β) (v : Vector α n) :
v.foldl (f:=f) (b:=init) = v.tail.foldl (f:=f) (b:=f init v.head) := | := by
simp_rw [Vector.foldl] -- get
simp only [size_toArray]
have hl_foldl_eq_toList_foldl := Array.foldl_toList (f:=f) (init:=init) (xs:=v.toArray)
have hl_foldl_eq: Array.foldl f init v.toArray 0 n = Array.foldl f init v.toArray := by
simp only [size_toArray]
conv_lhs =>
rw [hl_foldl_eq, hl_foldl_eq... | 1 | 29 | true | Applied verif. |
30 | ConcreteBinaryTower.join_eq_bitvec_iff_fromNat | theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
x = 《 hi_btf, lo_btf 》 ↔
(hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧
lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "AlgebraTowerEquiv.symm",
"content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dcast (Eq.refl n) bv = bv"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
x = 《 hi_btf, lo_btf 》 ↔
(hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧
lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) := | := by
-- Idea : derive from theorem join_eq_iff_dcast_extractLsb
constructor
· -- Forward direction
intro h_join
have h := join_eq_iff_dcast_extractLsb h_pos x hi_btf lo_btf
have ⟨h_hi, h_lo⟩ := h.mp h_join
have hi_eq : hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) := by
rw [h_hi]
... | 6 | 94 | false | Applied verif. |
31 | ConcreteBinaryTower.split_one | lemma split_one {k : ℕ} (h_k : k > 0) :
split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1)) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
}
] | [
{
"name": "Nat.ne_zero_of_lt",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.one_lt_two_pow_iff",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "Nat.shiftRight_eq_zero",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "BitVec.toNat_ofNat",
"module": "Init.Data.Bi... | [
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"
},
{
"name": "zero_lt_pow_n",
"content": "theorem zero_lt_pow_n (m : ℕ)... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero",
"content": "theorem BitVec.dcas... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma split_one {k : ℕ} (h_k : k > 0) :
split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1)) := | := by
rw [split]
let lo_bits := BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) (one (k:=k))
let hi_bits := BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) (one (k:=k))
apply Prod.ext
· simp only
simp only [BitVec.extractLsb, BitVec.extractLsb']
rw [one]
have one_toNat_eq := one_bitvec... | 4 | 43 | false | Applied verif. |
32 | AdditiveNTT.W_prod_comp_decomposition | lemma W_prod_comp_decomposition
(i : Fin r) (hi : i > 0) :
(W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1))) | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Polynomial.splits_X_sub_C",
"module": "Mathlib.Algebra.Polynomial.Splits"
},
{
"name": "Polynomial.splits_prod",
"module": "Mathlib.Algebra.Polynomial.Splits"
},
{
"name": "Finset.prod_eq_zero_iff",
"module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"
},
{
... | [
{
"name": "Fin.lt_succ'",
"content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
},
{
"name": "Fin.le_succ",
"content": "lemma F... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.algEquivAevalXSubC",
... | [
{
"name": "AdditiveNTT.βᵢ_not_in_Uᵢ",
"content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n β i ∉ U 𝔽q β i"
},
{
"name": "AdditiveNTT.root_U_lift_down",
"content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n (i : Fin r) (h_i_add_1 : i + 1 < r) (a ... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | lemma W_prod_comp_decomposition
(i : Fin r) (hi : i > 0) :
(W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1))) := | := by
-- ⊢ W 𝔽q β i = ∏ c, (W 𝔽q β (i - 1)).comp (X - C (c • β (i - 1)))
-- Define P and Q for clarity
set P := W 𝔽q β i
set Q := ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))
-- c : 𝔽q => univ
-- c ∈ finsetX
-- STRATEGY: Prove P = Q by showing they are monic, split, and have the same roots.
-... | 5 | 173 | false | Applied verif. |
33 | ConcreteBinaryTower.towerRingHomBackwardMap_forwardMap_eq | lemma towerRingHomBackwardMap_forwardMap_eq (k : ℕ) (x : ConcreteBTField k) :
towerRingHomBackwardMap (k:=k) (towerRingHomForwardMap (k:=k) x) = x | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "BitVec.cast_ofNat",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "BitVec.ofNat_eq_ofNat",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "BitVec.eq_zero_or_eq_one",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "congrArg",
"module": "Init.Prelude"
},
... | [
{
"name": "dcast_eq_root_cast",
"content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"
},
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : ... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.bitvec_cast_eq_dcast",
"content": "theorem BitVec.bitvec_cast_eq_dcast {n m : Nat} (h :... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma towerRingHomBackwardMap_forwardMap_eq (k : ℕ) (x : ConcreteBTField k) :
towerRingHomBackwardMap (k:=k) (towerRingHomForwardMap (k:=k) x) = x := | := by
induction k with
| zero =>
unfold towerRingHomBackwardMap towerRingHomForwardMap
simp only [↓reduceDIte, RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]
rcases concrete_eq_zero_or_eq_one (a:=x) (by omega) with x_zero | x_one
· rw [x_zero, zero_is_0]
unfold towerRingEquiv... | 15 | 299 | false | Applied verif. |
34 | AdditiveNTT.additiveNTT_correctness | theorem additiveNTT_correctness (h_ℓ : ℓ ≤ r)
(original_coeffs : Fin (2 ^ ℓ) → L)
(output_buffer : Fin (2 ^ (ℓ + R_rate)) → L)
(h_alg : output_buffer = additiveNTT 𝔽q β h_ℓ_add_R_rate original_coeffs) :
let P := polynomialFromNovelCoeffs 𝔽q β ℓ h_ℓ original_coeffs
∀ (j : Fin (2^(ℓ + R_rate))),
... | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Nat.Bitwise",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs",
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "MvPolynomial",
"module": "Mathlib.Algebra.MvPolynomial.Basic"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "MvPolynomial.op... | [
{
"name": "notation:70 s:70 \" ^^ \" t:71 => Fintype.piFinset fun (i : t)",
"content": "notation:70 s:70 \" ^^ \" t:71 => Fintype.piFinset fun (i : t) ↦ s i"
},
{
"name": "macro_rules (kind := mvEval)",
"content": "macro_rules (kind := mvEval)\n | `($p⸨$x⸩) => `(MvPolynomial.eval ($x ∘ Fin.cast... | [
{
"name": "Polynomial.comp_assoc",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "implies_true",
"module": "Init.SimpLemmas"
},
{
"name": "Fin.coe_ofNat_eq_mod",
"module": "Mathlib.Data.Fin.Basic"
},
{
"name": "Fin.foldl_succ",
"module": "Init.Data.Fin.Fol... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "getBit_eq_succ_getBit_of_mul_two_add_one",
"content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = get... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.qCompositi... | [
{
"name": "AdditiveNTT.qMap_eval_𝔽q_eq_0",
"content": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem qMap_eval_𝔽q_eq_0 (i : Fin r) :\n ∀ c: 𝔽q, (qMap 𝔽q β i).eval (algebraMap 𝔽q L c) = 0"
},
{
"name": "AdditiveNTT.qMap_comp_normalizedW",
"con... | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | theorem additiveNTT_correctness (h_ℓ : ℓ ≤ r)
(original_coeffs : Fin (2 ^ ℓ) → L)
(output_buffer : Fin (2 ^ (ℓ + R_rate)) → L)
(h_alg : output_buffer = additiveNTT 𝔽q β h_ℓ_add_R_rate original_coeffs) :
let P := | := polynomialFromNovelCoeffs 𝔽q β ℓ h_ℓ original_coeffs
∀ (j : Fin (2^(ℓ + R_rate))),
output_buffer j = P.eval (evaluationPointω 𝔽q β h_ℓ_add_R_rate ⟨0, by omega⟩ j) :=
by
simp only [Fin.zero_eta]
intro j
simp only [h_alg]
unfold additiveNTT
set output_foldl := Fin.foldl ℓ (fun current_b i ↦ NTT... | 14 | 317 | false | Applied verif. |
35 | InductiveMerkleTree.functional_completeness | theorem functional_completeness (α : Type) {s : Skeleton}
(idx : SkeletonLeafIndex s)
(leaf_data_tree : LeafData α s)
(hash : α → α → α) :
(getPutativeRoot_with_hash
idx
(leaf_data_tree.get idx)
(generateProof
(buildMerkleTree_with_hash leaf_data_tree hash) idx)
(hash)) =
(buildMerkleTre... | ArkLib | ArkLib/CommitmentScheme/InductiveMerkleTree.lean | [
"import ArkLib.ToMathlib.Data.IndexedBinaryTree.Basic",
"import Mathlib.Data.Vector.Snoc",
"import ArkLib.CommitmentScheme.Basic",
"import VCVio",
"import ArkLib.ToVCVio.Oracle"
] | [
{
"name": "Repr",
"module": "Init.Data.Repr"
},
{
"name": "List",
"module": "Init.Prelude"
}
] | [
{
"name": "FullData.leftSubtree",
"content": "def FullData.leftSubtree {α : Type} {s_left s_right : Skeleton}\n (tree : FullData α (Skeleton.internal s_left s_right)) :\n FullData α s_left :=\n match tree with\n | FullData.internal _ left _right =>\n left"
},
{
"name": "Skeleton",
"co... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "LeafData.rightSubtree_internal",
"content": "@[simp]\ntheorem LeafData.rightSubtree_internal {α} {s_left s_right : Skeleton}\n (left : LeafData α s_left) (right : LeafData α s_right) :\n (LeafData.internal left right).rightSubtree = right"
},
{
"name": "LeafData.leftSubtree_internal... | [
{
"name": "InductiveMerkleTree.buildMerkleTree_with_hash",
"content": "def buildMerkleTree_with_hash {s} (leaf_tree : LeafData α s) (hashFn : α → α → α) :\n (FullData α s) :=\n match leaf_tree with\n | LeafData.leaf a => FullData.leaf a\n | LeafData.internal left right =>\n let leftTree := buildMer... | [
{
"name": "InductiveMerkleTree.generateProof_ofLeft",
"content": "@[simp]\ntheorem generateProof_ofLeft {sleft sright : Skeleton}\n (cache_tree : FullData α (Skeleton.internal sleft sright))\n (idxLeft : SkeletonLeafIndex sleft) :\n generateProof cache_tree (BinaryTree.SkeletonLeafIndex.ofLeft idxL... | import VCVio
import ArkLib.ToMathlib.Data.IndexedBinaryTree.Basic
import ArkLib.CommitmentScheme.Basic
import Mathlib.Data.Vector.Snoc
import ArkLib.ToVCVio.Oracle
namespace InductiveMerkleTree
open List OracleSpec OracleComp BinaryTree
section spec
variable (α : Type)
end spec
variable {α : Type}
def buildM... | theorem functional_completeness (α : Type) {s : Skeleton}
(idx : SkeletonLeafIndex s)
(leaf_data_tree : LeafData α s)
(hash : α → α → α) :
(getPutativeRoot_with_hash
idx
(leaf_data_tree.get idx)
(generateProof
(buildMerkleTree_with_hash leaf_data_tree hash) idx)
(hash)) =
(buildMerkleTre... | := by
induction s with
| leaf =>
match leaf_data_tree with
| LeafData.leaf a =>
cases idx with
| ofLeaf =>
simp [buildMerkleTree_with_hash, getPutativeRoot_with_hash]
| internal s_left s_right left_ih right_ih =>
match leaf_data_tree with
| LeafData.internal left right =>
... | 4 | 31 | false | Applied verif. |
36 | ConcreteBinaryTower.aeval_definingPoly_at_Z_succ | lemma aeval_definingPoly_at_Z_succ (k : ℕ) :
(aeval (Z (k + 1))) (definingPoly (s:=Z (k))) = 0 | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dcast (Eq.refl n) bv = bv"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma aeval_definingPoly_at_Z_succ (k : ℕ) :
(aeval (Z (k + 1))) (definingPoly (s:=Z (k))) = 0 := | := by
rw [aeval_def]
set f := algebraMap (ConcreteBTField k) (ConcreteBTField (k + 1))
have h_f_is_canonical_embedding :
f = concreteTowerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) := by rfl
rw [definingPoly, eval₂_add, eval₂_add] -- break down into sum of terms
rw [eval₂_X_pow]
rw [C_mul']
-- ⊢ Z (k... | 10 | 257 | false | Applied verif. |
37 | AdditiveNTT.inductive_linear_map_W | omit hF₂ in
lemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)
(h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))
: IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p) | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Fact.out",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Fin.le_zero_iff'",
"module": "Mathlib.Data.Fin.SuccPred"
},
{
"name": "LinearIndependent.injective",
"module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"
},
{
"name": "Nat.not_lt_zero",
"module": "Ini... | [
{
"name": "Fin.lt_succ'",
"content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
},
{
"name": "Fin.le_succ",
"content": "lemma F... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.algEquivAevalXSubC",
... | [
{
"name": "AdditiveNTT.βᵢ_not_in_Uᵢ",
"content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n β i ∉ U 𝔽q β i"
},
{
"name": "AdditiveNTT.root_U_lift_down",
"content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n (i : Fin r) (h_i_add_1 : i + 1 < r) (a ... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | omit hF₂ in
lemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)
(h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))
: IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p) := | := by
have h_rec_form := inductive_rec_form_W_comp
(hβ_lin_indep := hβ_lin_indep) (h_prev_linear_map := h_prev_linear_map) (i :=i)
set q := Fintype.card 𝔽q
set v := (W 𝔽q β i).eval (β i)
-- `∀ f(X), f(X) ∈ L[X]`:
constructor
· intro f g
-- 1. Proof flow
-- `Wᵢ₊₁(f(X)+g(X)) = Wᵢ(f(X)+g(X))² ... | 7 | 238 | false | Applied verif. |
38 | ConcreteBinaryTower.join_eq_join_via_add_smul | @[simp]
theorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | @[simp]
theorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)
(hi_btf lo_btf : ConcreteBTField (k - 1)) :
《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf := | := by
unfold join_via_add_smul
set instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)
set hi_lifted := instAlgebra.2 hi_btf with h_hi_lifted
-- First, show `hi_btf • Z k` corresponds to `join h_pos hi_btf 0`.
have h_hi_term : hi_btf • Z k = 《 hi_btf, 0 》 := by
apply join_of_split
e... | 14 | 250 | false | Applied verif. |
39 | AdditiveNTT.W_linearity | theorem W_linearity (i : Fin r)
: IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p) | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean | [
"import Mathlib.Algebra.Polynomial.Degree.Definitions",
"import ArkLib.Data.Fin.BigOperators",
"import ArkLib.Data.Polynomial.MonomialBasis",
"import Mathlib.LinearAlgebra.StdBasis",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.Polynomial.Frobenius"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "Fin.succRecOnSameFinType",
"content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n (zero : motive (0 : Fin r))\n (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i... | [
{
"name": "Fact.out",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Fin.le_zero_iff'",
"module": "Mathlib.Data.Fin.SuccPred"
},
{
"name": "LinearIndependent.injective",
"module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"
},
{
"name": "Nat.not_lt_zero",
"module": "Ini... | [
{
"name": "Fin.lt_succ'",
"content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"
},
{
"name": "Fin.val_add_one'",
"content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"
},
{
"name": "Fin.le_succ",
"content": "lemma F... | [
{
"name": "AdditiveNTT.U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "AdditiveNTT.W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "AdditiveNTT.algEquivAevalXSubC",
... | [
{
"name": "AdditiveNTT.βᵢ_not_in_Uᵢ",
"content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n β i ∉ U 𝔽q β i"
},
{
"name": "AdditiveNTT.root_U_lift_down",
"content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n (i : Fin r) (h_i_add_1 : i + 1 < r) (a ... | import ArkLib.Data.Nat.Bitwise
import ArkLib.Data.Polynomial.Frobenius
import ArkLib.Data.Polynomial.MonomialBasis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Algebra.Polynomial.Degree.Definitions
open Polynomial FiniteDimensional Finset Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable... | theorem W_linearity (i : Fin r)
: IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p) := | := by
induction i using Fin.succRecOnSameFinType with
| zero =>
-- Base Case: i = 0 => Prove W₀ is linear.
unfold W
have h_U0 : (univ : Finset (U 𝔽q β 0)) = {0} := by
ext u -- u : ↥(U 𝔽q β 0)
simp only [mem_univ, true_iff, mem_singleton]
-- ⊢ u = 0
by_contra h
have h_u :=... | 8 | 257 | false | Applied verif. |
40 | MvPolynomial.finSuccEquivNth_coeff_coeff | theorem finSuccEquivNth_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquivNth R p f) i) = coeff (m.insertNth p i) f | ArkLib | ArkLib/ToMathlib/MvPolynomial/Equiv.lean | [
"import Mathlib.Algebra.MvPolynomial.Equiv",
"import ArkLib.ToMathlib.Finsupp.Fin"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "MvPolynomial",
"module": "Mathlib.Algebra.MvPolynomial.Basic"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "MvPolynomial.optionEquivLeft",
"module": "Mathlib.Algebra.MvPolynomi... | [
{
"name": "insertNth",
"content": "def insertNth (p : Fin (n + 1)) (y : M) (s : Fin n →₀ M) : Fin (n + 1) →₀ M :=\n Finsupp.equivFunOnFinite.symm (Fin.insertNth p y s : Fin (n + 1) → M)"
},
{
"name": "removeNth",
"content": "def removeNth (p : Fin (n + 1)) (s : Fin (n + 1) →₀ M) : Fin n →₀ M :=... | [
{
"name": "AlgEquiv.coe_trans",
"module": "Mathlib.Algebra.Algebra.Equiv"
},
{
"name": "Function.comp_apply",
"module": "Init.Core"
},
{
"name": "MvPolynomial.aeval_C",
"module": "Mathlib.Algebra.MvPolynomial.Eval"
},
{
"name": "MvPolynomial.coe_eval₂Hom",
"module": "Math... | [
{
"name": "insertNth_self_removeNth",
"content": "theorem insertNth_self_removeNth : insertNth p (t p) (removeNth p t) = t"
},
{
"name": "insertNth_apply_succAbove",
"content": "@[simp]\ntheorem insertNth_apply_succAbove : insertNth p y s (p.succAbove i) = s i"
},
{
"name": "removeNth_ap... | [
{
"name": "MvPolynomial.finSuccEquivNth",
"content": "def finSuccEquivNth : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] Polynomial (MvPolynomial (Fin n) R) :=\n (renameEquiv R (_root_.finSuccEquiv' p)).trans (optionEquivLeft R (Fin n))"
}
] | [
{
"name": "MvPolynomial.finSuccEquivNth_eq",
"content": "theorem finSuccEquivNth_eq :\n (finSuccEquivNth R p : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =\n eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))\n (Fin.insertNth p Polynomial.X (Polynomial.... | import Mathlib.Algebra.MvPolynomial.Equiv
import ArkLib.ToMathlib.Finsupp.Fin
namespace MvPolynomial
open Function Finsupp Polynomial
noncomputable section
section FinSuccEquivNth
variable {n : ℕ} {σ : Type*} (R : Type*) [CommSemiring R] (p : Fin (n + 1))
def finSuccEquivNth : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] ... | theorem finSuccEquivNth_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquivNth R p f) i) = coeff (m.insertNth p i) f := | := by
induction' f using MvPolynomial.induction_on' with u a p q hp hq generalizing i m
· simp only [finSuccEquivNth_apply, coe_eval₂Hom, eval₂_monomial, RingHom.coe_comp, comp_apply,
prod_pow, Fin.prod_univ_succAbove _ p, Fin.insertNth_apply_same,
Fin.insertNth_apply_succAbove, Polynomial.coeff_C_mul, ... | 3 | 70 | false | Applied verif. |
41 | ReedSolomonCode.genMatIsVandermonde | lemma genMatIsVandermonde [Fintype ι] [Field F] [DecidableEq F] [inst : NeZero m] {α : ι ↪ F} :
fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m | ArkLib | ArkLib/Data/CodingTheory/ReedSolomon.lean | [
"import Mathlib.LinearAlgebra.Lagrange",
"import ArkLib.Data.MvPolynomial.LinearMvExtension",
"import Mathlib.RingTheory.Henselian",
"import ArkLib.Data.Fin.Lift",
"import ArkLib.Data.Polynomial.Interface"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Matrix",
"module": "Mathlib.LinearAlgebra.Matrix.Defs"
},
{
"name": "Semiring",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "Matrix.of",
"module": "Mathlib.LinearAlgebra.Matrix.Defs"
},
{
"name": "Poly... | [
{
"name": "polynomialOfCoeffs",
"content": "def polynomialOfCoeffs (coeffs : Fin deg → F) : F[X] :=\n ⟨\n Finset.map ⟨Fin.val, Fin.val_injective⟩ {i | coeffs i ≠ 0},\n fun i ↦ if h : i < deg then coeffs ⟨i, h⟩ else 0,\n fun a ↦ by admit /- proof elided -/\n ⟩"
},
{
"name": "liftF'",
"... | [
{
"name": "Polynomial.mem_degreeLT",
"module": "Mathlib.RingTheory.Polynomial.Basic"
},
{
"name": "Polynomial.natDegree_lt_iff_degree_lt",
"module": "Mathlib.Algebra.Polynomial.Degree.Definitions"
},
{
"name": "Matrix.mulVecLin_apply",
"module": "Mathlib.LinearAlgebra.Matrix.ToLin"
... | [
{
"name": "liftF'_p_coeff",
"content": "@[simp]\nlemma liftF'_p_coeff {p : F[X]} {k : ℕ} {i : Fin k} : liftF' p.coeff i = p.coeff i"
},
{
"name": "coeff_polynomialOfCoeffs_eq_coeffs",
"content": "@[simp]\nlemma coeff_polynomialOfCoeffs_eq_coeffs :\n Fin.liftF' (polynomialOfCoeffs coeffs).coeff ... | [
{
"name": "ReedSolomon.evalOnPoints",
"content": "def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n toFun := fun p => fun x => p.eval (domain x)\n map_add' := fun x y => by admit /- proof elided -/"
},
{
"name": "ReedSolomon.code",
"content": "def code (deg : ℕ) [Semiring F]: Submodul... | [
{
"name": "Vandermonde.nonsquare_mulVecLin",
"content": "lemma nonsquare_mulVecLin [CommSemiring F] {ι' : ℕ} {α₁ : ι ↪ F} {α₂ : Fin ι' → F} {i : ι} :\n (nonsquare ι' α₁).mulVecLin α₂ i = ∑ x, α₂ x * α₁ i ^ x.1"
},
{
"name": "Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials",
... | import ArkLib.Data.MvPolynomial.LinearMvExtension
import ArkLib.Data.Polynomial.Interface
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.RingTheory.Henselian
namespace ReedSolomon
open Polynomial NNReal
variable {F : Type*} {ι : Type*} (domain : ι ↪ F)
def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) whe... | lemma genMatIsVandermonde [Fintype ι] [Field F] [DecidableEq F] [inst : NeZero m] {α : ι ↪ F} :
fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m := | := by
unfold fromColGenMat ReedSolomon.code
ext x; rw [LinearMap.mem_range, Submodule.mem_map]
refine ⟨
fun ⟨coeffs, h⟩ ↦ ⟨polynomialOfCoeffs coeffs, h.symm ▸ ?p₁⟩,
fun ⟨p, h⟩ ↦ ⟨Fin.liftF' p.coeff, ?p₂⟩
⟩
· rw [
←coeff_polynomialOfCoeffs_eq_coeffs (coeffs := coeffs),
Vandermonde.mulVecLin... | 3 | 47 | false | Applied verif. |
42 | UniPoly.toImpl_toPoly_of_canonical | lemma toImpl_toPoly_of_canonical [LawfulBEq R] (p : UniPolyC R) : p.toPoly.toImpl = p | ArkLib | ArkLib/Data/UniPoly/Basic.lean | [
"import Mathlib.Algebra.Tropical.Basic",
"import ArkLib.Data.Array.Lemmas",
"import Mathlib.RingTheory.Polynomial.Basic"
] | [
{
"name": "inline",
"module": "Init.Core"
},
{
"name": "Array",
"module": "Init.Prelude"
},
{
"name": "Bool",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Option",
... | [
{
"name": "findIdxRev?",
"content": "def findIdxRev? (cond : α → Bool) (as : Array α) : Option (Fin as.size) :=\n find ⟨ as.size, Nat.lt_succ_self _ ⟩\nwhere\n find : Fin (as.size + 1) → Option (Fin as.size)\n | 0 => none\n | ⟨ i+1, h ⟩ =>\n if (cond as[i]) then\n some ⟨ i, Nat.lt_of_suc... | [
{
"name": "Nat.lt_succ_self",
"module": "Init.Prelude"
},
{
"name": "Array.foldl_induction",
"module": "Init.Data.Array.Lemmas"
},
{
"name": "Array.getD_eq_getD_getElem?",
"module": "Init.Data.Array.Lemmas"
},
{
"name": "Array.getElem?_eq_none",
"module": "Init.Data.Array... | [
{
"name": "findIdxRev?_eq_some",
"content": "theorem findIdxRev?_eq_some {cond} {as : Array α} (h : ∃ i, ∃ hi : i < as.size, cond as[i]) :\n ∃ k : Fin as.size, findIdxRev? cond as = some k"
},
{
"name": "findIdxRev?_eq_none",
"content": "theorem findIdxRev?_eq_none {cond} {as : Array α} (h : ∀ ... | [
{
"name": "UniPoly",
"content": "@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R"
},
{
"name": "Polynomial.toImpl",
"content": "def Polynomial.toImpl {R : Type*} [Semiring R] (p : R[X]) : UniPoly R :=\n match p.degree with\n | ⊥ => #[]\n | some d => .ofFn (fun i : Fin (d... | [
{
"name": "UniPoly.Trim.last_nonzero_none",
"content": "theorem last_nonzero_none [LawfulBEq R] {p : UniPoly R} :\n (∀ i, (hi : i < p.size) → p[i] = 0) → p.last_nonzero = none"
},
{
"name": "UniPoly.Trim.last_nonzero_some",
"content": "theorem last_nonzero_some [LawfulBEq R] {p : UniPoly R} {i}... | import Mathlib.Algebra.Tropical.Basic
import Mathlib.RingTheory.Polynomial.Basic
import ArkLib.Data.Array.Lemmas
open Polynomial
@[reducible, inline, specialize]
def UniPoly (R : Type*) := Array R
def Polynomial.toImpl {R : Type*} [Semiring R] (p : R[X]) : UniPoly R :=
match p.degree with
| ⊥ => #[]
| some d... | lemma toImpl_toPoly_of_canonical [LawfulBEq R] (p : UniPolyC R) : p.toPoly.toImpl = p := | := by
-- we will change something slightly more general: `toPoly` is injective on canonical polynomials
suffices h_inj : ∀ q : UniPolyC R, p.toPoly = q.toPoly → p = q by
have : p.toPoly = p.toPoly.toImpl.toPoly := by rw [toPoly_toImpl]
exact h_inj ⟨ p.toPoly.toImpl, trim_toImpl p.toPoly ⟩ this |> congrArg S... | 8 | 128 | false | Applied verif. |
43 | ConcreteBinaryTower.split_sum_eq_sum_split | theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)
(hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))
(h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))
(h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :
split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "AlgebraTowerEquiv.symm",
"content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv... | [
{
"name": "BitVec.ofNat_xor",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "BitVec.xor_eq",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dcast (Eq.refl n) bv = bv"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)
(hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))
(h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))
(h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :
split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁) := | := by
have h_x₀ := join_of_split h_pos x₀ hi₀ lo₀ h_split_x₀
have h_x₁ := join_of_split h_pos x₁ hi₁ lo₁ h_split_x₁
-- Approach : convert equation to Nat realm for simple proof
have h₀ := (split_bitvec_eq_iff_fromNat (k:=k) (h_pos:=h_pos) x₀ hi₀ lo₀).mp h_split_x₀
have h₁ := (split_bitvec_eq_iff_fromNat (k:=k... | 8 | 106 | false | Applied verif. |
44 | ConcreteBinaryTower.concrete_eq_zero_or_eq_one | theorem concrete_eq_zero_or_eq_one {k : ℕ} {a : ConcreteBTField k} (h_k_zero : k = 0)
: a = zero ∨ a = one | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "AlgebraTowerEquiv.symm",
"content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv... | [
{
"name": "BitVec.cast_ofNat",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "BitVec.ofNat_eq_ofNat",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "BitVec.eq_zero_or_eq_one",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "congrArg",
"module": "Init.Prelude"
},
... | [
{
"name": "dcast_eq_root_cast",
"content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"
}
] | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.bitvec_cast_eq_dcast",
"content": "theorem BitVec.bitvec_cast_eq_dcast {n m : Nat} (h : n = m) (bv : BitVec n) :\n BitVec.cast h bv = DCast.dcast h bv"
},
{
"name": "ConcreteBinaryTower.BitVec.cast_one",
"content": "@[simp] theorem BitVec.cast_one {n m : ℕ}... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | theorem concrete_eq_zero_or_eq_one {k : ℕ} {a : ConcreteBTField k} (h_k_zero : k = 0)
: a = zero ∨ a = one := | := by
if h_k_zero : k = 0 then
have h_2_pow_k_eq_1 : 2 ^ k = 1 := by rw [h_k_zero]; norm_num
let a0 : ConcreteBTField 0 := Eq.mp (congrArg ConcreteBTField h_k_zero) a
have a0_is_eq_mp_a : a0 = Eq.mp (congrArg ConcreteBTField h_k_zero) a := by rfl
-- Approach : convert to BitVec.cast and derive equalit... | 4 | 32 | false | Applied verif. |
45 | ConcreteBinaryTower.concrete_mul_left_distrib0 | lemma concrete_mul_left_distrib0 (a b c : ConcreteBTField 0) :
concrete_mul a (b + c) = concrete_mul a b + concrete_mul a c | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "Al... | [
{
"name": "BitVec.xor_self",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "BitVec.eq_zero_or_eq_one",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "BitVec.xor_eq_zero_iff",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "if_neg",
"module": "Init.Core"
},
{
... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.add_self_cancel",
"content": "lemma add_self_cancel {k : ℕ} (a : ConcreteBTField k) : a + a = 0"
},
{
"name": "ConcreteBinaryTower.add_eq_zero_iff_eq",
"content": "lemma add_eq_zero_iff_eq {k : ℕ} (a b : ConcreteBTField k) : a + b = 0 ↔ a = b"
},
{
"name": ... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma concrete_mul_left_distrib0 (a b c : ConcreteBTField 0) :
concrete_mul a (b + c) = concrete_mul a b + concrete_mul a c := | := by
rcases eq_zero_or_eq_one (a := a) with (ha | ha)
· simp [ha, concrete_mul, zero_is_0] -- a = zero
· simp [ha, concrete_mul, zero_is_0, one_is_1];
rcases eq_zero_or_eq_one (a := b + c) with (hb_add_c | hb_add_c)
· simp [hb_add_c, zero_is_0];
rw [zero_is_0] at hb_add_c
have b_eq_c : b = c... | 5 | 32 | false | Applied verif. |
46 | coeffs_of_comp_minus_x | theorem coeffs_of_comp_minus_x {f : Polynomial F} {n : ℕ} :
(f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n | ArkLib | ArkLib/Data/FieldTheory/NonBinaryField/Basic.lean | [
"import Mathlib.Tactic.FieldSimp",
"import Mathlib.Algebra.Polynomial.FieldDivision",
"import Mathlib.Tactic.LinearCombination"
] | [
{
"name": "Semiring",
"module": "Mathlib.Algebra.Ring.Defs"
},
{
"name": "Even",
"module": "Mathlib.Algebra.Group.Even"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Polynomial.X",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
... | [
{
"name": "coeff.{u}",
"content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"
}
] | [
{
"name": "Nat.even_add_one",
"module": "Mathlib.Algebra.Group.Nat.Even"
},
{
"name": "Nat.even_iff",
"module": "Mathlib.Algebra.Group.Nat.Even"
},
{
"name": "Polynomial.coeff_X",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Polynomial.degree_pos_induction_on",
... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [] | [
{
"name": "coeffs_of_comp_minus_x_pos_degree",
"content": "private lemma coeffs_of_comp_minus_x_pos_degree {f : Polynomial F} {n : ℕ} (h : 0 < f.degree) :\n (f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n"
}
] | import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
section NonBinaryField
variable {F : Type*} [NonBinaryField F]
end NonBinaryField
section
variable {F : Type*} [Field F]
open Polynomial | theorem coeffs_of_comp_minus_x {f : Polynomial F} {n : ℕ} :
(f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n := | := by
by_cases hpos : 0 < f.degree
· rw [coeffs_of_comp_minus_x_pos_degree hpos]
· have : f.natDegree = 0 := by aesop (add simp natDegree_pos_iff_degree_pos.symm)
cases n <;> aesop (add simp natDegree_eq_zero) | 2 | 12 | false | Applied verif. |
47 | UniPoly.Trim.eq_degree_of_equiv | lemma eq_degree_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.degree = q.degree | ArkLib | ArkLib/Data/UniPoly/Basic.lean | [
"import Mathlib.Algebra.Tropical.Basic",
"import ArkLib.Data.Array.Lemmas",
"import Mathlib.RingTheory.Polynomial.Basic"
] | [
{
"name": "inline",
"module": "Init.Core"
},
{
"name": "Array",
"module": "Init.Prelude"
},
{
"name": "Bool",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Option",
... | [
{
"name": "findIdxRev?",
"content": "def findIdxRev? (cond : α → Bool) (as : Array α) : Option (Fin as.size) :=\n find ⟨ as.size, Nat.lt_succ_self _ ⟩\nwhere\n find : Fin (as.size + 1) → Option (Fin as.size)\n | 0 => none\n | ⟨ i+1, h ⟩ =>\n if (cond as[i]) then\n some ⟨ i, Nat.lt_of_suc... | [
{
"name": "Nat.lt_succ_self",
"module": "Init.Prelude"
},
{
"name": "Bool.false_eq_true",
"module": "Init.Data.Bool"
},
{
"name": "bne_iff_ne",
"module": "Init.SimpLemmas"
},
{
"name": "bne_self_eq_false",
"module": "Init.SimpLemmas"
},
{
"name": "ne_eq",
"mod... | [
{
"name": "findIdxRev?_eq_some",
"content": "theorem findIdxRev?_eq_some {cond} {as : Array α} (h : ∃ i, ∃ hi : i < as.size, cond as[i]) :\n ∃ k : Fin as.size, findIdxRev? cond as = some k"
},
{
"name": "findIdxRev?_eq_none",
"content": "theorem findIdxRev?_eq_none {cond} {as : Array α} (h : ∀ ... | [
{
"name": "UniPoly",
"content": "@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R"
},
{
"name": "UniPoly.coeff",
"content": "@[reducible]\ndef coeff (p : UniPoly Q) (i : ℕ) : Q := p.getD i 0"
},
{
"name": "UniPoly.last_nonzero",
"content": "def last_nonzero (p : U... | [
{
"name": "UniPoly.Trim.last_nonzero_none",
"content": "theorem last_nonzero_none [LawfulBEq R] {p : UniPoly R} :\n (∀ i, (hi : i < p.size) → p[i] = 0) → p.last_nonzero = none"
},
{
"name": "UniPoly.Trim.last_nonzero_some",
"content": "theorem last_nonzero_some [LawfulBEq R] {p : UniPoly R} {i}... | import Mathlib.Algebra.Tropical.Basic
import Mathlib.RingTheory.Polynomial.Basic
import ArkLib.Data.Array.Lemmas
open Polynomial
@[reducible, inline, specialize]
def UniPoly (R : Type*) := Array R
namespace UniPoly
variable {R : Type*} [Ring R] [BEq R]
variable {Q : Type*} [Ring Q]
@[reducible]
def coeff (p : U... | lemma eq_degree_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.degree = q.degree := | := by
unfold equiv degree
intro h_equiv
induction p using last_nonzero_induct with
| case1 p h_none_p h_all_zero =>
have h_zero_p : ∀ i, p.coeff i = 0 := coeff_eq_zero.mp h_all_zero
have h_zero_q : ∀ i, q.coeff i = 0 := by intro i; rw [← h_equiv, h_zero_p]
have h_none_q : q.last_nonzero = none := la... | 3 | 36 | false | Applied verif. |
48 | ConcreteBinaryTower.towerRingHomForwardMap_Z | lemma towerRingHomForwardMap_Z (k : ℕ) :
towerRingHomForwardMap k (Z k) = BinaryTower.Z k | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "BT... | [
{
"name": "BitVec.extractLsb_ofNat",
"module": "Init.Data.BitVec.Lemmas"
},
{
"name": "BitVec.zero_eq",
"module": "Init.Data.BitVec.Basic"
},
{
"name": "Nat.shiftRight_zero",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Bas... | [
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "join_via_add_smul_zero",
"content": "lemma join_via_add_smul_zero {k : ℕ} (h_pos : k > 0) :\n ⋘ 0, 0 ⋙ = 0"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dcast (Eq.refl n) bv = bv"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq",
"content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma towerRingHomForwardMap_Z (k : ℕ) :
towerRingHomForwardMap k (Z k) = BinaryTower.Z k := | := by
induction k with
| zero =>
unfold towerRingHomForwardMap
simp only [RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, ↓reduceDIte,
towerRingEquivFromConcrete0]
rfl
| succ k ih =>
unfold towerRingHomForwardMap
simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduc... | 9 | 196 | false | Applied verif. |
49 | Nat.num_eq_highBits_add_lowBits | lemma num_eq_highBits_add_lowBits {n: ℕ} (numLowBits: ℕ) :
n = getHighBits numLowBits n + getLowBits numLowBits n | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Nat.and_distrib_right",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.testBit",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "Bool",
"module": "Init.Prelude"
},
{
"name": "Nat.binaryRec",... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.shiftRight_add",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "add_comm",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Nat.and_one_is_mod",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.eq_of_testBit_eq",
"module": "Init.Data.Na... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "Nat.getLowBits",
"content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"
},
{
"name": "Nat.getHighBits_no_shl",
"content": "def getHighBits_no_shl (numLow... | [
{
"name": "Nat.getBit_lt_2",
"content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"
},
{
"name": "Nat.eq_iff_eq_all_getBits",
"content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"
},
{
"name": "Nat.shiftRight_and_one_distrib",
"content": "lemm... | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | lemma num_eq_highBits_add_lowBits {n: ℕ} (numLowBits: ℕ) :
n = getHighBits numLowBits n + getLowBits numLowBits n := | := by
apply eq_iff_eq_all_getBits.mpr; unfold getBit
intro k
--- use 2 getBit extractions to get the condition for getLowBits of ((n >>> numLowBits) <<<
-- numLowBits)
set highBits_no_shl := n >>> numLowBits
have h_getBit_highBits_shl := getBit_of_shiftLeft (n := highBits_no_shl) (p := numLowBits)
have h... | 4 | 103 | false | Applied verif. |
50 | BerlekampWelch.elocPolyF_deg | @[simp]
lemma elocPolyF_deg {ωs f : Fin n → F} : (ElocPolyF ωs f p).natDegree = Δ₀(f, p.eval ∘ ωs) | ArkLib | ArkLib/Data/CodingTheory/BerlekampWelch/ElocPoly.lean | [
"import ArkLib.Data.CodingTheory.Basic",
"import Init.Data.List.FinRange",
"import ArkLib.Data.Fin.Lift",
"import Mathlib.Data.Finset.Insert",
"import Mathlib.Data.Fintype.Card",
"import Mathlib.Algebra.Polynomial.FieldDivision",
"import Mathlib.Data.Matrix.Mul",
"import Mathlib.Algebra.Field.Basic",
... | [
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "List",
"module": "Init.Prelude"
},
{
"name": "List.prod",
"module": "Batteries.Data.List.Basic"
},
{
"name": "List.range",
"module": "Init.Data.List.Basic"
},
{
"name": "Polynomial.... | [
{
"name": "hammingDist",
"content": "notation \"Δ₀(\" u \", \" v \")\" => hammingDist u v"
},
{
"name": "distFromCode",
"content": "notation \"Δ₀(\" u \", \" C \")\" => distFromCode u C"
},
{
"name": "scoped macro_rules",
"content": "scoped macro_rules\n | `(ρ $t:term) => `(LinearCo... | [
{
"name": "List.mem_range",
"module": "Init.Data.List.Nat.Range"
},
{
"name": "List.pmap_eq_map",
"module": "Init.Data.List.Attach"
},
{
"name": "List.pmap_eq_map_attach",
"module": "Init.Data.List.Attach"
},
{
"name": "Polynomial.ext",
"module": "Mathlib.Algebra.Polynomi... | [
{
"name": "contract_eq_liftF_of_lt",
"content": "lemma contract_eq_liftF_of_lt {k : ℕ} (h₁ : k < m) :\n contract m f' k = liftF f' k"
},
{
"name": "liftF_succ",
"content": "@[simp]\nlemma liftF_succ {f : Fin (n + 1) → α} : liftF f n = f ⟨n, Nat.lt_add_one _⟩"
}
] | [
{
"name": "BerlekampWelch.ElocPoly",
"content": "protected noncomputable def ElocPoly (n : ℕ) (ωs f : ℕ → F) (p : Polynomial F) : Polynomial F :=\n List.prod <| (List.range n).map fun i =>\n if f i = p.eval (ωs i)\n then 1\n else X - C (ωs i)"
},
{
"name": "BerlekampWelch.ElocPolyF",
"... | [
{
"name": "BerlekampWelch.elocPoly_zero",
"content": "@[simp]\nprotected lemma elocPoly_zero : ElocPoly 0 ωs f p = 1"
},
{
"name": "BerlekampWelch.elocPoly_succ",
"content": "@[simp]\nprotected lemma elocPoly_succ :\n ElocPoly (n + 1) ωs f p =\n ElocPoly n ωs f p *\n if f n = p.eval (ωs n)\... | import Init.Data.List.FinRange
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Matrix.Mul
import ... | @[simp]
lemma elocPolyF_deg {ωs f : Fin n → F} : (ElocPolyF ωs f p).natDegree = Δ₀(f, p.eval ∘ ωs) := | := by
rw [elocPolyF_eq_elocPoly']
induction' n with n ih
· simp only [elocPoly_zero, natDegree_one, hamming_zero_eq_dist]
exact funext_iff.2 (Fin.elim0 ·)
· rw [
elocPoly_succ,
natDegree_mul (by simp)
(by aesop (erase simp liftF_succ)
(add simp [... | 4 | 42 | false | Applied verif. |
51 | Fin.zero_dappend | @[simp]
theorem zero_dappend {motive : Fin (0 + n) → Sort u} {u : (i : Fin 0) → motive (castAdd n i)}
(v : (i : Fin n) → motive (natAdd 0 i)) :
dappend (motive := motive) u v = fun i => cast (by simp) (v (i.cast (by omega))) | ArkLib | ArkLib/Data/Fin/Tuple/Lemmas.lean | [
"import ArkLib.Data.Fin.Tuple.Notation"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin.last",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Fin.castAdd",
"module": "Init.Data.Fin.Basic"... | [
{
"name": "dappend",
"content": "@[elab_as_elim]\ndef dappend {m n : ℕ} {motive : Fin (m + n) → Sort u}\n (u : (i : Fin m) → motive (Fin.castAdd n i))\n (v : (i : Fin n) → motive (Fin.natAdd m i))\n (i : Fin (m + n)) : motive i :=\n match n with\n | 0 => u i\n | k + 1 => dconcat (dappend u (fun ... | [
{
"name": "Fin.ext",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Fin.snoc_castSucc",
"module": "Mathlib.Data.Fin.Tuple.Basic"
},
{
"name": "Fin.snoc_last",
"module": "Mathlib.Data.Fin.Tuple.Basic"
},
{
"name": "Fin.forall_fin_zero_pi",
"module": "Mathlib.Data.Fin.Tuple... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [] | [
{
"name": "Fin.dconcat_last",
"content": "@[simp]\ntheorem dconcat_last {motive : Fin (n + 1) → Sort u} (v : (i : Fin n) → motive (castSucc i))\n (a : motive (last n)) : (v :+ᵈ⟨motive⟩ a) (last n) = a"
},
{
"name": "Fin.dconcat_castSucc",
"content": "@[simp]\ntheorem dconcat_castSucc {motive ... | import ArkLib.Data.Fin.Tuple.Notation
namespace Fin
variable {m n : ℕ} {α : Sort u} | @[simp]
theorem zero_dappend {motive : Fin (0 + n) → Sort u} {u : (i : Fin 0) → motive (castAdd n i)}
(v : (i : Fin n) → motive (natAdd 0 i)) :
dappend (motive := motive) u v = fun i => cast (by simp) (v (i.cast (by omega))) := | := by
induction n with
| zero => ext i; exact Fin.elim0 i
| succ n ih =>
simp [dappend, ih, dconcat_eq_snoc, Fin.cast, last]
ext i
by_cases h : i.val < n
· have : i = Fin.castSucc ⟨i.val, by simp [h]⟩ := by ext; simp
rw [this, snoc_castSucc]
simp
· have : i.val = n := by omega
... | 5 | 27 | false | Applied verif. |
52 | BerlekampWelch.solutionToQ_zero | @[simp]
lemma solutionToQ_zero {v : Fin (2 * 0 + 0) → F} :
solutionToQ (F := F) 0 0 v = 0 := rfl | ArkLib | ArkLib/Data/CodingTheory/BerlekampWelch/Condition.lean | [
"import Mathlib.Data.Matrix.Reflection",
"import ArkLib.Data.CodingTheory.Basic",
"import ArkLib.Data.CodingTheory.BerlekampWelch.Sorries",
"import Init.Data.List.FinRange",
"import Mathlib.Data.Finset.Insert",
"import ArkLib.Data.Polynomial.Interface",
"import Mathlib.Data.Fintype.Card",
"import Math... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Polynomial",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.add",
"module": "Mathlib.Algebra.Group.Pointwise.Finset.Basic"
},
{
"... | [
{
"name": "liftF",
"content": "def liftF (f : Fin n → α) : ℕ → α :=\n fun m ↦ if h : m < n then f ⟨m, h⟩ else 0"
}
] | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "BerlekampWelch.solutionToQ",
"content": "def solutionToQ (e k : ℕ) (v : Fin (2 * e + k) → F) : Polynomial F :=\n ⟨\n (Finset.range (e + k)).filter (fun x => liftF v (e + x) ≠ 0),\n fun i => if i < e + k then liftF v (e + i) else 0,\n by admit /- proof elided -/\n ⟩"
}
] | [] | import Init.Data.List.FinRange
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Matrix.Mul
import ... | @[simp]
lemma solutionToQ_zero {v : Fin (2 * 0 + 0) → F} :
solutionToQ (F := F) 0 0 v = 0 := | := rfl | 2 | 6 | false | Applied verif. |
53 | BinaryTower.eq_join_via_add_smul_eq_iff_split | theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)
(x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :
x = ⋘ hi_btf, lo_btf ⋙ ↔
split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf) | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Basic.lean | [
"import Mathlib.Tactic.DepRewrite",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.RingTheory.AlgebraTower"
] | [
{
"name": "Field",
"module": "Mathlib.Algebra.Field.Defs"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.range",
"module": "Mathlib.Data.Finset.Range"
},
{
"name": "Polynomial.C",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "TraceMapProperty",
"content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n inverse_t... | [
{
"name": "Ne.dite_eq_left_iff",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Nat.add_one_sub_one",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.left_eq_add",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "RingHom.comp_id",
"module": "Mathlib.Algebra.Ring.Hom.Defs"
... | [
{
"name": "degree_definingPoly",
"content": "lemma degree_definingPoly {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n (definingPoly s).degree = 2"
},
{
"name": "degree_s_smul_X_add_1",
"content": "lemma degree_s_smul_X_add_1 {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n (C... | [
{
"name": "BinaryTower.BinaryTowerResult",
"content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n vec : (List.Vector F (k + 1))\n instField : (Field F)\n instFintype : Fintype F\n specialElement : F\n specialElementNeZero : NeZero specialElement\n firstElementOfVecIsSpecialElement [Inhab... | [
{
"name": "BinaryTower.poly_natDegree_eq_2",
"content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"
},
{
"name": "BinaryTower.BTField.cast_BTField_eq",
"content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n BTField k = BTField m"
},
{
"name": "Binar... | import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude
import ArkLib.Data.RingTheory.AlgebraTower
import Mathlib.Tactic.DepRewrite
namespace BinaryTower
noncomputable section
open Polynomial AdjoinRoot Module
section BTFieldDefs
structure BinaryTowerResult (F : Type _) (k : ℕ) where
vec : (List.Vector F (k +... | theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)
(x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :
x = ⋘ hi_btf, lo_btf ⋙ ↔
split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf) := | := by
have h_k_sub_1_add_1_eq_k : k - 1 + 1 = k := by omega
have h_BTField_eq := BTField.cast_BTField_eq (k:=k) (m:=k-1+1) (h_eq:=by omega)
set p := unique_linear_decomposition_succ (k:=(k-1)) (x:=(Eq.mp (h:=h_BTField_eq) x)) with hp
-- -- ⊢ x = join_via_add_smul k h_pos hi lo
have h_p_satisfy := p.choose_spe... | 6 | 98 | false | Applied verif. |
54 | BinaryTower.algebraMap_eq_zero_x | lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : BTField i) :
letI instAlgebra := binaryAlgebraTower (l:=i) (r:=j) (h_le:=by omega)
letI instAlgebraPred := binaryAlgebraTower (l:=i) (r:=j-1) (h_le:=by omega)
algebraMap (BTField i) (BTField j) x
= ⋘ 0, algebraMap (BTField i) (BTField (j-1)) x ⋙ | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Basic.lean | [
"import Mathlib.Tactic.DepRewrite",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.RingTheory.AlgebraTower"
] | [
{
"name": "Field",
"module": "Mathlib.Algebra.Field.Defs"
},
{
"name": "Finset",
"module": "Mathlib.Data.Finset.Defs"
},
{
"name": "Finset.range",
"module": "Mathlib.Data.Finset.Range"
},
{
"name": "Polynomial.C",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "TraceMapProperty",
"content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n inverse_t... | [
{
"name": "Ne.dite_eq_left_iff",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Nat.add_one_sub_one",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.left_eq_add",
"module": "Init.Data.Nat.Lemmas"
},
{
"name": "RingHom.comp_id",
"module": "Mathlib.Algebra.Ring.Hom.Defs"
... | [
{
"name": "degree_definingPoly",
"content": "lemma degree_definingPoly {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n (definingPoly s).degree = 2"
},
{
"name": "degree_s_smul_X_add_1",
"content": "lemma degree_s_smul_X_add_1 {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n (C... | [
{
"name": "BinaryTower.BinaryTowerResult",
"content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n vec : (List.Vector F (k + 1))\n instField : (Field F)\n instFintype : Fintype F\n specialElement : F\n specialElementNeZero : NeZero specialElement\n firstElementOfVecIsSpecialElement [Inhab... | [
{
"name": "BinaryTower.poly_natDegree_eq_2",
"content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"
},
{
"name": "BinaryTower.BTField.cast_BTField_eq",
"content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n BTField k = BTField m"
},
{
"name": "Binar... | import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude
import ArkLib.Data.RingTheory.AlgebraTower
import Mathlib.Tactic.DepRewrite
namespace BinaryTower
noncomputable section
open Polynomial AdjoinRoot Module
section BTFieldDefs
structure BinaryTowerResult (F : Type _) (k : ℕ) where
vec : (List.Vector F (k +... | lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : BTField i) :
letI instAlgebra := | := binaryAlgebraTower (l:=i) (r:=j) (h_le:=by omega)
letI instAlgebraPred := binaryAlgebraTower (l:=i) (r:=j-1) (h_le:=by omega)
algebraMap (BTField i) (BTField j) x
= ⋘ 0, algebraMap (BTField i) (BTField (j-1)) x ⋙ := by
set d := j - i with d_eq
induction hd : d with
| zero =>
have h_i_eq_j : i... | 8 | 114 | false | Applied verif. |
55 | Nat.getBit_of_sub_two_pow_of_bit_1 | lemma getBit_of_sub_two_pow_of_bit_1 {n i j: ℕ} (h_getBit_eq_1: getBit i n = 1) :
getBit j (n - 2^i) = (if j = i then 0 else getBit j n) | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BEq",
"module": "Init.Prelude"
},
{
"name": "Bool",
"module": "Init.Prelude"
},
{
"name": "Nat.testBit",
"module": "Init.Data.Nat.Bitwise.Basic"
},
{
"name": "Nat.binaryRec",
"module": "Mathlib.Data.Nat.B... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Bool.toNat_true",
"module": "Init.Data.Bool"
},
{
"name": "Nat.and_one_is_mod",
"module": "Init.Data.Nat.Bitwise.Lemmas"
},
{
"name": "Nat.and_two_pow",
"module": "Mathlib.Data.Nat.Bitwise"
},
{
"name": "Nat.mod_two_bne_zero",
"module": "Init.Data.Nat.Lemmas"
... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
}
] | [
{
"name": "Nat.testBit_true_eq_getBit_eq_1",
"content": "lemma testBit_true_eq_getBit_eq_1 (k n : Nat) : n.testBit k = ((Nat.getBit k n) = 1)"
},
{
"name": "Nat.getBit_two_pow",
"content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"
},
{
"name": "Nat.and... | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | lemma getBit_of_sub_two_pow_of_bit_1 {n i j: ℕ} (h_getBit_eq_1: getBit i n = 1) :
getBit j (n - 2^i) = (if j = i then 0 else getBit j n) := | := by
have h_2_pow_i_lt_n: 2^i ≤ n := by
apply Nat.ge_two_pow_of_testBit
rw [Nat.testBit_true_eq_getBit_eq_1]
exact h_getBit_eq_1
have h_xor_eq_sub := (Nat.xor_eq_sub_iff_submask (n:=n) (m:=2^i) (h_2_pow_i_lt_n)).mpr (by
exact and_two_pow_eq_two_pow_of_getBit_1 h_getBit_eq_1)
rw [h_xor_eq_sub.symm... | 4 | 78 | false | Applied verif. |
56 | Binius.BinaryBasefold.toOutCodewordsCount_succ_eq | lemma toOutCodewordsCount_succ_eq (i : Fin ℓ) :
(toOutCodewordsCount ℓ ϑ i.succ) =
if isCommitmentRound ℓ ϑ i then (toOutCodewordsCount ℓ ϑ i.castSucc) + 1
else (toOutCodewordsCount ℓ ϑ i.castSucc) | ArkLib | ArkLib/ProofSystem/Binius/BinaryBasefold/Basic.lean | [
"import ArkLib.ProofSystem.Binius.BinaryBasefold.Prelude"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Decidable",
"module": "Init.Prelude"
},
{
"name": "False.elim",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "Ne",... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.succ_div_of_dvd",
"module": "Init.Data.Nat.Div.Lemmas"
},
{
"name": "Nat.succ_div_of_not_dvd",
"module": "Init.Data.Nat.Div.Lemmas"
},
{
"name": "Fin.coe_castSucc",
"module": "Init.Data.Fin.Lemmas"
},
{
"name": "Fin.val_pos_iff",
"module": "Mathlib.Data.Fin... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Binius.BinaryBasefold.toOutCodewordsCount",
"content": "def toOutCodewordsCount (i : Fin (ℓ + 1)) : ℕ :="
},
{
"name": "Binius.BinaryBasefold.isCommitmentRound",
"content": "def isCommitmentRound (i : Fin ℓ) : Prop :=\n ϑ ∣ i.val + 1 ∧ i.val + 1 ≠ ℓ"
}
] | [
{
"name": "Binius.BinaryBasefold.div_add_one_eq_if_dvd",
"content": "lemma div_add_one_eq_if_dvd (i ϑ : ℕ) [NeZero ϑ] :\n (i + 1) / ϑ = if ϑ ∣ i + 1 then i / ϑ + 1 else i / ϑ"
},
{
"name": "Binius.BinaryBasefold.toOutCodewordsCount_succ_eq_add_one_iff",
"content": "omit hdiv in\nlemma toOutCo... | import ArkLib.ProofSystem.Binius.BinaryBasefold.Prelude
noncomputable section
namespace Binius.BinaryBasefold
open OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial
Binius.BinaryBasefold
open scoped NNReal
open ReedSolomon Code BerlekampWelch
open Finset AdditiveNTT Polynomial MvPoly... | lemma toOutCodewordsCount_succ_eq (i : Fin ℓ) :
(toOutCodewordsCount ℓ ϑ i.succ) =
if isCommitmentRound ℓ ϑ i then (toOutCodewordsCount ℓ ϑ i.castSucc) + 1
else (toOutCodewordsCount ℓ ϑ i.castSucc) := | := by
have h_succ_val: i.succ.val = i.val + 1 := rfl
by_cases hv: ϑ ∣ i.val + 1 ∧ i.val + 1 ≠ ℓ
· have h_succ := (toOutCodewordsCount_succ_eq_add_one_iff ℓ ϑ i).mp hv
rw [←h_succ];
simp only [left_eq_ite_iff, Nat.add_eq_left, one_ne_zero, imp_false, Decidable.not_not]
exact hv
· rw [isCommitmentRoun... | 3 | 53 | false | Applied verif. |
57 | AdditiveNTT.evalWAt_eq_W | theorem evalWAt_eq_W (i : Fin r) (x : L) :
evalWAt (β := β) (ℓ := ℓ) (R_rate := R_rate) (i := i) x =
(W (𝔽q := 𝔽q) (β := β) (i := i)).eval x | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Impl",
"import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT",
"import ArkLib.Data.Nat.Bitwise",
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "split",
"content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n let lo_bits : BitVec (2 ^ (k - 1) - 1 -... | [
{
"name": "Bool.false_eq_true",
"module": "Init.Data.Bool"
},
{
"name": "Fact.out",
"module": "Mathlib.Logic.Basic"
},
{
"name": "Finset.mem_univ",
"module": "Mathlib.Data.Fintype.Defs"
},
{
"name": "Finset.sum_congr",
"module": "Mathlib.Algebra.BigOperators.Group.Finset.... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "eq_iff_eq_all_getBits",
"content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"
},
{
"name": "getBit... | [
{
"name": "AdditiveNTT.bitsToU",
"content": "def bitsToU (i : Fin r) (k : Fin (2 ^ i.val)) :\n AdditiveNTT.U (L := L) (𝔽q := 𝔽q) (β := β) i :=\n let val := (Finset.univ : Finset (Fin i)).sum fun j =>\n if (Nat.getBit (n := k.val) (k := j.val) == 1) then\n β ⟨j, by admit /- proof elided -/\n ... | [
{
"name": "AdditiveNTT.List.prod_finRange_eq_finset_prod",
"content": "lemma List.prod_finRange_eq_finset_prod {M : Type*} [CommMonoid M] {n : ℕ} (f : Fin n → M) :\n ((List.finRange n).map f).prod = ∏ i : Fin n, f i"
},
{
"name": "AdditiveNTT.bitsToU_bijective",
"content": "theorem bitsToU_bi... | import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT
import ArkLib.Data.FieldTheory.BinaryField.Tower.Impl
namespace AdditiveNTT
open ConcreteBinaryTower
section HelperFunctions
end HelperFunctions
variable {r : ℕ} [NeZero r]
variable {L : Type} [Field L] [Fintype L] [DecidableEq L]
variable {𝔽q : Type} [Fie... | theorem evalWAt_eq_W (i : Fin r) (x : L) :
evalWAt (β := β) (ℓ := ℓ) (R_rate := R_rate) (i := i) x =
(W (𝔽q := 𝔽q) (β := β) (i := i)).eval x := | := by
-- 1. Convert implementation to mathematical product over Fin(2^i)
unfold evalWAt getUElements
rw [List.map_map]
rw [List.prod_finRange_eq_finset_prod] -- Now the pattern matches!
-- 2. Prepare RHS
rw [AdditiveNTT.W, Polynomial.eval_prod]
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial... | 4 | 73 | false | Applied verif. |
58 | AdditiveNTT.normalizedW_eq_qMap_composition | lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :
normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i | ArkLib | ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean | [
"import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis",
"import Mathlib.Data.Finsupp.Defs",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Tactic",
"import ArkLib.Data.Polynomial.Frobenius",
"import Mathlib.LinearAlgebra.LinearIndependent.Defs"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Subspace",
"module": "Mathlib.Algebra.Module.Submodule.Basic"
},
{
"name": "Set",
"module": "Mathlib.Data.Set.Defs"
},
{
"name": "Set.Ico",
"module": "Mathlib.Order.Interval.Set.Defs"
},
{
"name": "Submodule"... | [
{
"name": "W",
"content": "noncomputable def W (i : Fin r) : L[X] :=\n ∏ u : U 𝔽q β i, (X - C u.val)"
},
{
"name": "U",
"content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"
},
{
"name": "normalizedW",
"content": "noncomputable def normalizedW (... | [
{
"name": "Fintype.card_pos",
"module": "Mathlib.Data.Fintype.Card"
},
{
"name": "Polynomial.C_1",
"module": "Mathlib.Algebra.Polynomial.Basic"
},
{
"name": "Polynomial.C_comp",
"module": "Mathlib.Algebra.Polynomial.Eval.Defs"
},
{
"name": "Polynomial.C_mul",
"module": "M... | [
{
"name": "Xⱼ_zero_eq_one",
"content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"
},
{
"name": "W_linear_comp_decomposition",
"content": "omit hF₂ in\ntheorem W_linear_comp_decomposition (i : Fin r) (h_i_add_1 : i + 1 < r) :\n ∀ p: L[X... | [
{
"name": "AdditiveNTT.qMap",
"content": "noncomputable def qMap (i : Fin r) : L[X] :=\n let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"
},
{
"name": "AdditiveNTT.qCompositi... | [
{
"name": "AdditiveNTT.qMap_comp_normalizedW",
"content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"
}
] | import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis
import Mathlib.Tactic
import Mathlib.Data.Finsupp.Defs
import Mathlib.LinearAlgebra.LinearIndependent.Defs
open Polynomial AdditiveNTT Module
namespace AdditiveNTT
variable {r : ℕ} [NeZero r]
variable {L : Type u} [Field L] [Fintype L] [DecidableEq ... | lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :
normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i := | :=
by
-- We proceed by induction on i.
induction i using Fin.succRecOnSameFinType with
| zero =>
-- Base case: i = 0
-- We need to show `normalizedW ... 0 = qCompositionChain 0`.
-- The RHS is `X` by definition of the chain.
rw [qCompositionChain.eq_def]
-- The LHS is `C (1 / eval (β 0) (W ...... | 11 | 86 | false | Applied verif. |
59 | Nat.getHighBits_no_shl_joinBits | lemma getHighBits_no_shl_joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) :
getHighBits_no_shl n (joinBits low high).val = high.val | ArkLib | ArkLib/Data/Nat/Bitwise.lean | [
"import Mathlib.Algebra.Order.BigOperators.Group.Finset",
"import ArkLib.Data.Fin.BigOperators",
"import Mathlib.Algebra.BigOperators.Ring.Finset",
"import Mathlib.Data.Nat.Bitwise",
"import Mathlib.Data.Finsupp.Basic",
"import Mathlib.Algebra.Order.Ring.Star",
"import Mathlib.Data.Nat.Digits.Defs",
"... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Bool",
"module": "Init.Prelude"
},
{
"name": "Nat.binaryRec",
"module": "Mathlib.Data.Nat.BinaryRec"
},
{
"name": "Nat.bit",
"module": "Mathlib.Data.Nat.Binary... | [
{
"name": "...",
"content": "..."
}
] | [
{
"name": "Nat.add_mul_div_left",
"module": "Init.Data.Nat.Div.Basic"
},
{
"name": "add_comm",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "mul_comm",
"module": "Mathlib.Algebra.Group.Defs"
},
{
"name": "Nat.and_div_two_pow",
"module": "Init.Data.Nat.Bitwise.Lemma... | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Nat.getBit",
"content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"
},
{
"name": "Nat.getHighBits_no_shl",
"content": "def getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits"
},
{
"name": "Nat.joinBits",
"content": "def joinBits {n m : ℕ} (low : Fin ... | [
{
"name": "Nat.getBit_lt_2",
"content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"
},
{
"name": "Nat.getBit_eq_testBit",
"content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"
},
{
"name": "Nat.eq_iff_eq_all_getBits",
"content": "lemma eq_i... | import ArkLib.Data.Fin.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Data.Finsupp.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperato... | lemma getHighBits_no_shl_joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) :
getHighBits_no_shl n (joinBits low high).val = high.val := | := by
unfold joinBits getHighBits_no_shl
dsimp
have h_and_zero := and_shl_eq_zero_of_lt_two_pow (a := high.val) (b := low.val) (hb := low.isLt)
rw [←Nat.sum_of_and_eq_zero_is_or h_and_zero]
rw [Nat.add_shiftRight_distrib h_and_zero]
rw [Nat.shiftLeft_shiftRight]
rw [Nat.shiftRight_eq_div_pow]
have h: lo... | 4 | 97 | false | Applied verif. |
60 | ConcreteBinaryTower.towerRingHomForwardMap_backwardMap_eq | lemma towerRingHomForwardMap_backwardMap_eq (k : ℕ) (x : BTField k) :
towerRingHomForwardMap (k:=k) (towerRingHomBackwardMap (k:=k) x) = x | ArkLib | ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean | [
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude",
"import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic",
"import ArkLib.Data.Classes.DCast"
] | [
{
"name": "Eq",
"module": "Init.Prelude"
},
{
"name": "id",
"module": "Init.Prelude"
},
{
"name": "BitVec",
"module": "Init.Prelude"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "BitVec.cast",
"module": "Init.Data.BitVec.Basic"
},
{
"name"... | [
{
"name": "GaloisField",
"content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"
},
{
"name": "DCast",
"content": "class DCast (α : Sort*) (β : α → Sort*) where\n dcast : ∀ {a a' : α}, a = a' → β a → β a'\n dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"
},
{
"name": "su... | [
{
"name": "Nat.sub_add_cancel",
"module": "Init.Data.Nat.Basic"
},
{
"name": "Nat.sub_zero",
"module": "Init.Data.Nat.Basic"
},
{
"name": "BitVec.ofNat_toNat",
"module": "Init.Data.BitVec.Bootstrap"
},
{
"name": "BitVec.setWidth_eq",
"module": "Init.Data.BitVec.Lemmas"
... | [
{
"name": "one_le_two_pow_n",
"content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"
},
{
"name": "dcast_eq",
"content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"
},
{
"name": "one_le_sub_consecutive_two_pow",
"content": "theorem one_le_sub_consecutive_two_pow (n : ℕ):... | [
{
"name": "ConcreteBinaryTower.ConcreteBTField",
"content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"
},
{
"name": "ConcreteBinaryTower.BitVec",
"content": "instance BitVec.instDCast : DCast Nat BitVec where\n dcast h := BitVec.cast h\n dcast_id := by admit /- proof elided -/... | [
{
"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq",
"content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n ConcreteBTField k = ConcreteBTField m"
},
{
"name": "ConcreteBinaryTower.BitVec.dcast_id",
"content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n DCast.dca... | import ArkLib.Data.Classes.DCast
import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic
namespace ConcreteBinaryTower
open Polynomial
def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)
section BitVecDCast
instance BitVec.instDCast : DCast Nat BitVec where
dcast h := BitVec.cast h
dcast_id := by admit /... | lemma towerRingHomForwardMap_backwardMap_eq (k : ℕ) (x : BTField k) :
towerRingHomForwardMap (k:=k) (towerRingHomBackwardMap (k:=k) x) = x := | := by
induction k with
| zero =>
unfold towerRingHomForwardMap towerRingHomBackwardMap
simp only [↓reduceDIte, RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]
rcases GF_2_value_eq_zero_or_one x with x_zero | x_one
· rw [x_zero];
unfold towerRingEquivFromConcrete0 -- ⊢ towerRin... | 15 | 285 | false | Applied verif. |
61 | Capless.preservation | theorem preservation
(hr : Reduce state state')
(ht : TypedState state Γ E) :
Preserve Γ E state' | capless-lean | Capless/Soundness/Preservation.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Subst.Type.Typing",
"import Capless.Renaming.Capture.Typing",
"import Capless.Weakening.TypedCont.Term",
"import Capless.Basic",
"import Capless.Typing.Basic",
"import Capless.CaptureSet",
"import Capless.Store",
"import Capless.Narrowing.Typing",... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
}
] | [
{
"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t",
"content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"
},
{
"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u",
"content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"
},
{
"name": "notat... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Subcapt.refl",
"content": "theorem Subcapt.refl :\n Subcapt Γ C C"
},
{
"name": "ESubtyp.cweaken",
"content": "theorem ESubtyp.cweaken\n (h : ESubtyp Γ E1 E2) :\n ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"
},
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken... | [
{
"name": "Capless.Preserve",
"content": "inductive Preserve : Context n m k -> EType n m k -> State n' m' k' -> Prop where\n| mk :\n TypedState state Γ E ->\n Preserve Γ E state\n| mk_weaken :\n TypedState state (Γ.var P) E.weaken ->\n Preserve Γ E state\n| mk_tweaken :\n TypedState state (Γ.tvar b) E... | [
{
"name": "Capless.value_typing_widen",
"content": "theorem value_typing_widen\n (hv : Typed Γ v (EType.type (S^C)) Cv)\n (hs : Γ ⊢ (S^C1) <: (S'^C2)) :\n Typed Γ v (S'^C) Cv"
},
{
"name": "Capless.EType.weaken_cweaken_helper",
"content": "theorem EType.weaken_cweaken_helper {S : SType n m k}... | import Capless.Store
import Capless.Type
import Capless.Reduction
import Capless.Inversion.Typing
import Capless.Inversion.Lookup
import Capless.Renaming.Term.Subtyping
import Capless.Renaming.Type.Subtyping
import Capless.Renaming.Capture.Subtyping
import Capless.Subst.Term.Typing
import Capless.Subst.Type.Ty... | theorem preservation
(hr : Reduce state state')
(ht : TypedState state Γ E) :
Preserve Γ E state' := | := by
cases hr
case apply hl =>
cases ht
case mk hs hsc ht hc =>
have hg := TypedStore.is_tight hs
have ⟨T0, Cf, F0, E0, hx, hy, he1, hs1⟩:= Typed.app_inv ht
have ⟨Sv, Cv, Cv0, hv, hbx, hvs⟩ := Store.lookup_inv_typing hl hs hx
have hv' := value_typing_widen hv hvs
have ⟨hcfs, h... | 7 | 334 | false | Type systems |
62 | Capless.Typed.rename | theorem Typed.rename
{Γ : Context n m k} {Δ : Context n' m k}
(h : Typed Γ t E Ct)
(ρ : VarMap Γ f Δ) :
Typed Δ (t.rename f) (E.rename f) (Ct.rename f) | capless-lean | Capless/Renaming/Term/Typing.lean | [
"import Capless.Renaming.Term.Subtyping",
"import Capless.Renaming.Term.Subcapturing",
"import Capless.Typing",
"import Capless.Type.Basic",
"import Capless.CaptureSet",
"import Capless.Renaming.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.Subset.rename",
"content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n (h : C1 ⊆ C2) :\n C1.rename f ⊆ C2.rename f"
},
{
"name": "Subcapt.rename",
"content": "theorem Subcapt.rename\n (h : Subcapt Γ C1 C2)\n (ρ : VarMap Γ f Δ) :\n Subcapt Δ (C1.rename ... | [] | [] | import Capless.Typing
import Capless.Renaming.Basic
import Capless.Renaming.Term.Subtyping
namespace Capless | theorem Typed.rename
{Γ : Context n m k} {Δ : Context n' m k}
(h : Typed Γ t E Ct)
(ρ : VarMap Γ f Δ) :
Typed Δ (t.rename f) (E.rename f) (Ct.rename f) := | := by
induction h generalizing n'
case var hb =>
simp [Term.rename, EType.rename, CType.rename]
apply Typed.var
have hb1 := ρ.map _ _ hb
simp [CType.rename] at hb1
trivial
case pack ih =>
simp [Term.rename, EType.rename]
apply Typed.pack
have ih := ih (ρ.cext _)
simp [Term.rena... | 4 | 111 | false | Type systems |
63 | Capless.Typed.subst | theorem Typed.subst
{Γ : Context n m k} {Δ : Context n' m k}
(h : Typed Γ t E Ct)
(σ : VarSubst Γ f Δ) :
Typed Δ (t.rename f) (E.rename f) (Ct.rename f) | capless-lean | Capless/Subst/Term/Typing.lean | [
"import Capless.Renaming.Capture.Typing",
"import Capless.Renaming.Term.Typing",
"import Capless.Renaming.Term.Subtyping",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Subst.Basic",
"import Capless.Subst.Term.Subcapturing",
"import Capless.Typing.Basic",
"import Capless.Renaming.Term.S... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
}
] | [
{
"name": "macro \"easy\" : tactic => `(tactic| assumption)",
"content": "macro \"easy\" : tactic => `(tactic| assumption)"
},
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:ma... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.crename",
"content": "theorem CSubtyp.crename\n (h : CSubtyp Γ C1 C2)\n (ρ : CVarMap Γ f Δ) :\n CSubtyp Δ (C1.crename f) (C2.crename f)"
},
{
"name": "SSubtyp.crename",
"content": "theorem SSubtyp.crename\n (h : SSubtyp Γ S1 S2)\n (ρ : CVarMap Γ f Δ) :\n SSubtyp Δ (S1.cr... | [] | [] | import Capless.Typing
import Capless.Subst.Basic
import Capless.Subst.Term.Subtyping
import Capless.Renaming.Term.Typing
namespace Capless | theorem Typed.subst
{Γ : Context n m k} {Δ : Context n' m k}
(h : Typed Γ t E Ct)
(σ : VarSubst Γ f Δ) :
Typed Δ (t.rename f) (E.rename f) (Ct.rename f) := | := by
induction h generalizing n'
case var hb =>
simp [Term.rename, EType.rename, CType.rename]
have hb1 := σ.map _ _ hb
simp [CType.rename] at hb1
apply Typed.precise_capture
trivial
case pack ih =>
simp [Term.rename, EType.rename]
apply pack
have ih := ih σ.cext
simp [EType.r... | 5 | 190 | false | Type systems |
64 | Capless.Typed.csubst | theorem Typed.csubst
{Γ : Context n m k} {Δ : Context n m k'}
(h : Typed Γ t E Ct)
(σ : CVarSubst Γ f Δ) :
Typed Δ (t.crename f) (E.crename f) (Ct.crename f) | capless-lean | Capless/Subst/Capture/Typing.lean | [
"import Capless.Renaming.Capture.Typing",
"import Capless.Renaming.Term.Typing",
"import Capless.Renaming.Term.Subtyping",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Subst.Basic",
"import Capless.Renaming.Term.Subcapturing",
"import Capless.CaptureSet",
"import Capless.Subst.Capture.... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.crename",
"content": "theorem CSubtyp.crename\n (h : CSubtyp Γ C1 C2)\n (ρ : CVarMap Γ f Δ) :\n CSubtyp Δ (C1.crename f) (C2.crename f)"
},
{
"name": "SSubtyp.crename",
"content": "theorem SSubtyp.crename\n (h : SSubtyp Γ S1 S2)\n (ρ : CVarMap Γ f Δ) :\n SSubtyp Δ (S1.cr... | [] | [] | import Capless.Subst.Basic
import Capless.Subst.Capture.Subtyping
import Capless.Typing
namespace Capless | theorem Typed.csubst
{Γ : Context n m k} {Δ : Context n m k'}
(h : Typed Γ t E Ct)
(σ : CVarSubst Γ f Δ) :
Typed Δ (t.crename f) (E.crename f) (Ct.crename f) := | := by
induction h generalizing k'
case var hb =>
simp [Term.crename, EType.crename, CType.crename]
have hb1 := σ.map _ _ hb
simp [CType.crename] at hb1
apply Typed.var; trivial
case pack ih =>
simp [Term.crename, EType.crename]
apply pack
have ih := ih σ.cext
... | 5 | 195 | false | Type systems |
65 | Capless.Typed.crename | theorem Typed.crename
{Γ : Context n m k} {Δ : Context n m k'}
(h : Typed Γ t E Ct)
(ρ : CVarMap Γ f Δ) :
Typed Δ (t.crename f) (E.crename f) (Ct.crename f) | capless-lean | Capless/Renaming/Capture/Typing.lean | [
"import Capless.Typing",
"import Capless.Renaming.Capture.Subtyping",
"import Capless.Type.Basic",
"import Capless.CaptureSet",
"import Capless.Renaming.Capture.Subcapturing",
"import Capless.Renaming.Basic"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "SType.cweaken_crename",
"content": "theorem SType.cweaken_crename {S : SType n m k} :\n (S.crename f).cweaken = S.cweaken.crename f.ext"
},
{
"name": "SType.crename_crename",
"content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n (S.cre... | [] | [] | import Capless.Typing
import Capless.Renaming.Basic
import Capless.Renaming.Capture.Subtyping
namespace Capless | theorem Typed.crename
{Γ : Context n m k} {Δ : Context n m k'}
(h : Typed Γ t E Ct)
(ρ : CVarMap Γ f Δ) :
Typed Δ (t.crename f) (E.crename f) (Ct.crename f) := | := by
induction h generalizing k'
case var hb =>
simp [Term.crename, EType.crename, CType.crename]
apply var
have hb1 := ρ.map _ _ hb
simp [CType.crename] at hb1
exact hb1
case pack ih =>
simp [Term.crename, EType.crename]
apply pack
have ih := ih (ρ.cext _)
simp [Term.crename,... | 3 | 119 | false | Type systems |
66 | Capless.Typed.trename | theorem Typed.trename
{Γ : Context n m k} {Δ : Context n m' k}
(h : Typed Γ t E Ct)
(ρ : TVarMap Γ f Δ) :
Typed Δ (t.trename f) (E.trename f) Ct | capless-lean | Capless/Renaming/Type/Typing.lean | [
"import Capless.Renaming.Type.Subtyping",
"import Capless.Typing",
"import Capless.Type.Basic",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Renaming.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "EType.trename_topen",
"content": "theorem EType.trename_topen {E : EType n (m+1) k} :\n (E.topen X).trename f = (E.trename f.ext).topen (f X)"
},
{
"name": "EType.trename_trename",
"content": "theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n ... | [] | [] | import Capless.Typing
import Capless.Renaming.Basic
import Capless.Renaming.Type.Subtyping
namespace Capless | theorem Typed.trename
{Γ : Context n m k} {Δ : Context n m' k}
(h : Typed Γ t E Ct)
(ρ : TVarMap Γ f Δ) :
Typed Δ (t.trename f) (E.trename f) Ct := | := by
induction h generalizing m'
case var =>
simp [Term.trename, EType.trename, CType.trename]
apply var
rename_i hb
have hb1 := ρ.map _ _ hb
simp [CType.trename] at hb1
trivial
case pack ih =>
simp [Term.trename, EType.trename]
apply pack
have ih := ih (ρ.cext _)
simp [Te... | 3 | 111 | false | Type systems |
67 | Capless.Typed.tsubst | theorem Typed.tsubst
{Γ : Context n m k} {Δ : Context n m' k}
(h : Typed Γ t E Ct)
(σ : TVarSubst Γ f Δ) :
Typed Δ (t.trename f) (E.trename f) Ct | capless-lean | Capless/Subst/Type/Typing.lean | [
"import Capless.Renaming.Type.Subtyping",
"import Capless.Renaming.Term.Typing",
"import Capless.Typing",
"import Capless.Renaming.Type.Typing",
"import Capless.Renaming.Capture.Typing",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Renaming.Term.Subtyping",
"import Capless.Subst.Type.S... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "macro \"easy\" : tactic => `(tactic| assumption)",
"content": "macro \"easy\" : tactic => `(tactic| assumption)"
},
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notatio... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.crename",
"content": "theorem CSubtyp.crename\n (h : CSubtyp Γ C1 C2)\n (ρ : CVarMap Γ f Δ) :\n CSubtyp Δ (C1.crename f) (C2.crename f)"
},
{
"name": "SSubtyp.crename",
"content": "theorem SSubtyp.crename\n (h : SSubtyp Γ S1 S2)\n (ρ : CVarMap Γ f Δ) :\n SSubtyp Δ (S1.cr... | [] | [] | import Capless.Subst.Basic
import Capless.Subst.Type.Subtyping
import Capless.Typing
namespace Capless | theorem Typed.tsubst
{Γ : Context n m k} {Δ : Context n m' k}
(h : Typed Γ t E Ct)
(σ : TVarSubst Γ f Δ) :
Typed Δ (t.trename f) (E.trename f) Ct := | := by
induction h generalizing m'
case var hb =>
simp [Term.trename, EType.trename, CType.trename]
have hb1 := σ.map _ _ hb
simp [CType.trename] at hb1
apply Typed.var; trivial
case pack ih =>
simp [Term.trename, EType.trename]
apply pack
have ih := ih σ.cext
... | 5 | 189 | false | Type systems |
68 | Capless.SSubtyp.rename | theorem SSubtyp.rename
(h : SSubtyp Γ S1 S2)
(ρ : VarMap Γ f Δ) :
SSubtyp Δ (S1.rename f) (S2.rename f) | capless-lean | Capless/Renaming/Term/Subtyping.lean | [
"import Capless.Renaming.Term.Subcapturing",
"import Capless.Subtyping",
"import Capless.Renaming.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Subcapt.rename",
"content": "theorem Subcapt.rename\n (h : Subcapt Γ C1 C2)\n (ρ : VarMap Γ f Δ) :\n Subcapt Δ (C1.rename f) (C2.rename f)"
},
{
"name": "CaptureSet.Subset.rename",
"content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n (h : C1 ⊆ C2) :\n C1.rename... | [
{
"name": "Capless.SSubtyp.rename_motive1",
"content": "def SSubtyp.rename_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n ESubtyp Δ (E1.rename f) (E2.rename f)"
},
{
"name": "Capless.SSub... | [
{
"name": "Capless.Subbound.rename",
"content": "theorem Subbound.rename\n (h : Subbound Γ B1 B2)\n (ρ : VarMap Γ f Δ) :\n Subbound Δ (B1.rename f) (B2.rename f)"
}
] | import Capless.Subtyping
import Capless.Renaming.Basic
import Capless.Renaming.Term.Subcapturing
namespace Capless
def SSubtyp.rename_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),
ESubtyp Δ (E1.rename f) (E2.... | theorem SSubtyp.rename
(h : SSubtyp Γ S1 S2)
(ρ : VarMap Γ f Δ) :
SSubtyp Δ (S1.rename f) (S2.rename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.rename_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.rename_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.rename_motive3 Γ S1 S2)
(t := h) (ρ := ρ)
case exist ih =>
unfold SSubtyp.rename_motive1 SSubtyp.rename_motive... | 4 | 49 | false | Type systems |
69 | Capless.SSubtyp.subst | theorem SSubtyp.subst
(h : SSubtyp Γ S1 S2)
(σ : VarSubst Γ f Δ) :
SSubtyp Δ (S1.rename f) (S2.rename f) | capless-lean | Capless/Subst/Term/Subtyping.lean | [
"import Capless.Subst.Term.Subcapturing",
"import Capless.Renaming.Term.Subtyping",
"import Capless.Renaming.Term.Typing",
"import Capless.Renaming.Term.Subcapturing",
"import Capless.Subtyping",
"import Capless.Subst.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
}
] | [
{
"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t",
"content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"
},
{
"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u",
"content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"
},
{
"name": "notat... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.rename",
"content": "theorem CSubtyp.rename\n (h : CSubtyp Γ T1 T2)\n (ρ : VarMap Γ f Δ) :\n CSubtyp Δ (T1.rename f) (T2.rename f)"
},
{
"name": "SSubtyp.rename",
"content": "theorem SSubtyp.rename\n (h : SSubtyp Γ S1 S2)\n (ρ : VarMap Γ f Δ) :\n SSubtyp Δ (S1.rename f) ... | [
{
"name": "Capless.SSubtyp.subst_motive1",
"content": "def SSubtyp.subst_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n ESubtyp Δ (E1.rename f) (E2.rename f)"
},
{
"name": "Capless.SSub... | [
{
"name": "Capless.Subbound.subst",
"content": "theorem Subbound.subst\n (h : Subbound Γ B1 B2)\n (σ : VarSubst Γ f Δ) :\n Subbound Δ (B1.rename f) (B2.rename f)"
}
] | import Capless.Subst.Basic
import Capless.Subtyping
import Capless.Subst.Term.Subcapturing
namespace Capless
def SSubtyp.subst_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),
ESubtyp Δ (E1.rename f) (E2.renam... | theorem SSubtyp.subst
(h : SSubtyp Γ S1 S2)
(σ : VarSubst Γ f Δ) :
SSubtyp Δ (S1.rename f) (S2.rename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.subst_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.subst_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.subst_motive3 Γ S1 S2)
(t := h) (ρ := σ)
case exist =>
unfold subst_motive1 subst_motive2
repeat intro
s... | 6 | 122 | false | Type systems |
70 | Capless.SSubtyp.csubst | theorem SSubtyp.csubst
(h : SSubtyp Γ S1 S2)
(σ : CVarSubst Γ f Δ) :
SSubtyp Δ (S1.crename f) (S2.crename f) | capless-lean | Capless/Subst/Capture/Subtyping.lean | [
"import Capless.Renaming.Capture.Typing",
"import Capless.Subst.Basic",
"import Capless.Renaming.Capture.Subtyping",
"import Capless.Subst.Capture.Subcapturing",
"import Capless.Context",
"import Capless.Subtyping",
"import Capless.Renaming.Capture.Subcapturing"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.crename",
"content": "theorem CSubtyp.crename\n (h : CSubtyp Γ C1 C2)\n (ρ : CVarMap Γ f Δ) :\n CSubtyp Δ (C1.crename f) (C2.crename f)"
},
{
"name": "SSubtyp.crename",
"content": "theorem SSubtyp.crename\n (h : SSubtyp Γ S1 S2)\n (ρ : CVarMap Γ f Δ) :\n SSubtyp Δ (S1.cr... | [
{
"name": "Capless.SSubtyp.csubst_motive1",
"content": "def SSubtyp.csubst_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n ESubtyp Δ (E1.crename f) (E2.crename f)"
},
{
"name": "Capless... | [
{
"name": "Capless.Subbound.csubst",
"content": "theorem Subbound.csubst\n (h : Subbound Γ B1 B2)\n (σ : CVarSubst Γ f Δ) :\n Subbound Δ (B1.crename f) (B2.crename f)"
}
] | import Capless.Subtyping
import Capless.Subst.Basic
import Capless.Subst.Capture.Subcapturing
namespace Capless
def SSubtyp.csubst_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),
ESubtyp Δ (E1.crename f) (E2... | theorem SSubtyp.csubst
(h : SSubtyp Γ S1 S2)
(σ : CVarSubst Γ f Δ) :
SSubtyp Δ (S1.crename f) (S2.crename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 _ => SSubtyp.csubst_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 _ => SSubtyp.csubst_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 _ => SSubtyp.csubst_motive3 Γ S1 S2)
(t := h) (ρ := σ)
case exist =>
unfold csubst_motive1 csubst_motive2
... | 6 | 112 | false | Type systems |
71 | Capless.SSubtyp.crename | theorem SSubtyp.crename
(h : SSubtyp Γ S1 S2)
(ρ : CVarMap Γ f Δ) :
SSubtyp Δ (S1.crename f) (S2.crename f) | capless-lean | Capless/Renaming/Capture/Subtyping.lean | [
"import Capless.Tactics",
"import Capless.Subtyping",
"import Capless.Renaming.Capture.Subcapturing",
"import Capless.Renaming.Basic"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Subcapt.crename",
"content": "theorem Subcapt.crename\n (h : Subcapt Γ C1 C2)\n (ρ : CVarMap Γ f Δ) :\n Subcapt Δ (C1.crename f) (C2.crename f)"
},
{
"name": "CaptureSet.Subset.crename",
"content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n (h : C1 ⊆ C2) :\n C1... | [
{
"name": "Capless.SSubtyp.crename_motive1",
"content": "def SSubtyp.crename_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n ESubtyp Δ (E1.crename f) (E2.crename f)"
},
{
"name": "Capless... | [
{
"name": "Capless.Subbound.crename",
"content": "theorem Subbound.crename\n (h : Subbound Γ B1 B2)\n (ρ : CVarMap Γ f Δ) :\n Subbound Δ (B1.crename f) (B2.crename f)"
}
] | import Capless.Tactics
import Capless.Subtyping
import Capless.Renaming.Basic
import Capless.Renaming.Capture.Subcapturing
namespace Capless
def SSubtyp.crename_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),
... | theorem SSubtyp.crename
(h : SSubtyp Γ S1 S2)
(ρ : CVarMap Γ f Δ) :
SSubtyp Δ (S1.crename f) (S2.crename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.crename_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.crename_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.crename_motive3 Γ S1 S2)
(t := h) (ρ := ρ)
case exist =>
unfold SSubtyp.crename_motive1 SSubtyp.crename_moti... | 6 | 60 | false | Type systems |
72 | Capless.SSubtyp.tsubst | theorem SSubtyp.tsubst
(h : SSubtyp Γ S1 S2)
(σ : TVarSubst Γ f Δ) :
SSubtyp Δ (S1.trename f) (S2.trename f) | capless-lean | Capless/Subst/Type/Subtyping.lean | [
"import Capless.Renaming.Type.Subtyping",
"import Capless.Renaming.Type.Typing",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Subst.Type.Subcapturing",
"import Capless.Subtyping",
"import Capless.Subst.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.trename",
"content": "theorem CSubtyp.trename\n (h : CSubtyp Γ T1 T2)\n (ρ : TVarMap Γ f Δ) :\n CSubtyp Δ (T1.trename f) (T2.trename f)"
},
{
"name": "SSubtyp.trename",
"content": "theorem SSubtyp.trename\n (h : SSubtyp Γ S1 S2)\n (ρ : TVarMap Γ f Δ) :\n SSubtyp Δ (S1.tr... | [
{
"name": "Capless.SSubtyp.tsubst_motive1",
"content": "def SSubtyp.tsubst_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n ESubtyp Δ (E1.trename f) (E2.trename f)"
},
{
"name": "Capless... | [
{
"name": "Capless.Subbound.tsubst",
"content": "theorem Subbound.tsubst\n (h : Subbound Γ B1 B2)\n (σ : TVarSubst Γ f Δ) :\n Subbound Δ B1 B2"
}
] | import Capless.Subst.Basic
import Capless.Subtyping
import Capless.Subst.Type.Subcapturing
namespace Capless
def SSubtyp.tsubst_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),
ESubtyp Δ (E1.trename f) (E2.tr... | theorem SSubtyp.tsubst
(h : SSubtyp Γ S1 S2)
(σ : TVarSubst Γ f Δ) :
SSubtyp Δ (S1.trename f) (S2.trename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 _ => SSubtyp.tsubst_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 _ => SSubtyp.tsubst_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 _ => SSubtyp.tsubst_motive3 Γ S1 S2)
(t := h) (ρ := σ)
case exist =>
unfold tsubst_motive1 tsubst_motive2
... | 5 | 121 | false | Type systems |
73 | Capless.SSubtyp.trename | theorem SSubtyp.trename
(h : SSubtyp Γ S1 S2)
(ρ : TVarMap Γ f Δ) :
SSubtyp Δ (S1.trename f) (S2.trename f) | capless-lean | Capless/Renaming/Type/Subtyping.lean | [
"import Capless.Tactics",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Subtyping",
"import Capless.Renaming.Basic"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Subcapt.trename",
"content": "theorem Subcapt.trename\n (h : Subcapt Γ C1 C2)\n (ρ : TVarMap Γ f Δ) :\n Subcapt Δ C1 C2"
}
] | [
{
"name": "Capless.SSubtyp.trename_motive1",
"content": "def SSubtyp.trename_motive1\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop :=\n ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n ESubtyp Δ (E1.trename f) (E2.trename f)"
},
{
"name": "Capless... | [
{
"name": "Capless.Subbound.trename",
"content": "theorem Subbound.trename\n (h : Subbound Γ T1 T2)\n (ρ : TVarMap Γ f Δ) :\n Subbound Δ T1 T2"
}
] | import Capless.Tactics
import Capless.Subtyping
import Capless.Renaming.Basic
import Capless.Renaming.Type.Subcapturing
namespace Capless
def SSubtyp.trename_motive1
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop :=
∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),
ES... | theorem SSubtyp.trename
(h : SSubtyp Γ S1 S2)
(ρ : TVarMap Γ f Δ) :
SSubtyp Δ (S1.trename f) (S2.trename f) := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.trename_motive1 Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.trename_motive2 Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.trename_motive3 Γ S1 S2)
(t := h) (ρ := ρ)
case exist =>
unfold trename_motive1 trename_motive2
repeat i... | 6 | 45 | false | Type systems |
74 | Capless.SSubtyp.sub_dealias_cforall_inv | theorem SSubtyp.sub_dealias_cforall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))
(h2 : SType.Dealias Γ S2 (SType.cforall B2 E2))
(hs : SSubtyp Γ S1 S2) :
Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2 | capless-lean | Capless/Inversion/Subtyping.lean | [
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Inversion.Basic",
"import Capless.Narrowing",
"import Capless.Subtyping",
"import Capless.Store",
"import Capless.Inversion.Context"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "And",
"module": "Init.Prelude"
},
{
"name": "Exists",
"module": "Init.Core"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "refl",
"module": "Mathlib.Order.Defs.Unbundled"
}
] | [
{
"name": "Context.tbound_inj",
"content": "theorem Context.tbound_inj\n (h1 : Context.TBound Γ X b1)\n (h2 : Context.TBound Γ X b2) : b1 = b2"
},
{
"name": "Context.tvar_tbound_succ_inv",
"content": "theorem Context.tvar_tbound_succ_inv\n (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n ... | [
{
"name": "Capless.SSubtyp.dealias_right_cforall.emotive",
"content": "def SSubtyp.dealias_right_cforall.emotive\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop := True"
},
{
"name": "Capless.SSubtyp.dealias_right_cforall.cmotive",
"content": "def SSubtyp.dealias_rig... | [
{
"name": "Capless.SSubtyp.dealias_right_cforall",
"content": "theorem SSubtyp.dealias_right_cforall\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"
},
{
"name": "Capless.SType.dealias_cforall_inj'",... | import Capless.Subtyping
import Capless.Store
import Capless.Inversion.Basic
import Capless.Inversion.Context
import Capless.Subtyping.Basic
import Capless.Narrowing
namespace Capless
def SSubtyp.dealias_right_cforall.emotive
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop := True
def S... | theorem SSubtyp.sub_dealias_cforall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))
(h2 : SType.Dealias Γ S2 (SType.cforall B2 E2))
(hs : SSubtyp Γ S1 S2) :
Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2 := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_cforall_inv.emotive Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_cforall_inv.cmotive Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_cforall_inv.smotive Γ S1 S2)
(t := hs) (h1 := h1) (h2 := h2) (ht := ht)
case exi... | 7 | 120 | false | Type systems |
75 | Capless.SSubtyp.sub_dealias_forall_inv | theorem SSubtyp.sub_dealias_forall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.forall T1 E1))
(h2 : SType.Dealias Γ S2 (SType.forall T2 E2))
(hs : SSubtyp Γ S1 S2) :
CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2 | capless-lean | Capless/Inversion/Subtyping.lean | [
"import Capless.Narrowing.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Narrowing.TypedCont",
"import Capless.Inversion.Basic",
"import Capless.Narrowing",
"import Capless.Subtyping",
"import Capless.Store",
"import Capless.Inversion.Context"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "refl",
"module": "Mathlib.Order.Defs.Unbundled"
}
] | [
{
"name": "Context.tbound_inj",
"content": "theorem Context.tbound_inj\n (h1 : Context.TBound Γ X b1)\n (h2 : Context.TBound Γ X b2) : b1 = b2"
},
{
"name": "Context.tvar_tbound_succ_inv",
"content": "theorem Context.tvar_tbound_succ_inv\n (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n ... | [
{
"name": "Capless.SSubtyp.dealias_right_forall.emotive",
"content": "def SSubtyp.dealias_right_forall.emotive\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop := True"
},
{
"name": "Capless.SSubtyp.dealias_right_forall.cmotive",
"content": "def SSubtyp.dealias_right_... | [
{
"name": "Capless.SSubtyp.dealias_right_forall",
"content": "theorem SSubtyp.dealias_right_forall\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"
},
{
"name": "Capless.SType.dealias_forall_inj'",
... | import Capless.Subtyping
import Capless.Store
import Capless.Inversion.Basic
import Capless.Inversion.Context
import Capless.Subtyping.Basic
import Capless.Narrowing
namespace Capless
def SSubtyp.dealias_right_forall.emotive
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop := True
def SS... | theorem SSubtyp.sub_dealias_forall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.forall T1 E1))
(h2 : SType.Dealias Γ S2 (SType.forall T2 E2))
(hs : SSubtyp Γ S1 S2) :
CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2 := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_forall_inv.emotive Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_forall_inv.cmotive Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_forall_inv.smotive Γ S1 S2)
(t := hs) (h1 := h1) (h2 := h2) (ht := ht)
case exist ... | 5 | 128 | false | Type systems |
76 | Capless.SSubtyp.sub_dealias_tforall_inv | theorem SSubtyp.sub_dealias_tforall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))
(h2 : SType.Dealias Γ S2 (SType.tforall T2 E2))
(hs : SSubtyp Γ S1 S2) :
SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2 | capless-lean | Capless/Inversion/Subtyping.lean | [
"import Capless.Narrowing.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Inversion.Basic",
"import Capless.Narrowing",
"import Capless.Subtyping",
"import Capless.Store",
"import Capless.Inversion.Context"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "And",
"module": "Init.Prelude"
},
{
"name": "Exists",
"module": "Init.Core"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "refl",
"module": "Mathlib.Order.Defs.Unbundled"
}
] | [
{
"name": "Context.tbound_inj",
"content": "theorem Context.tbound_inj\n (h1 : Context.TBound Γ X b1)\n (h2 : Context.TBound Γ X b2) : b1 = b2"
},
{
"name": "Context.tvar_tbound_succ_inv",
"content": "theorem Context.tvar_tbound_succ_inv\n (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n ... | [
{
"name": "Capless.SSubtyp.dealias_right_tforall.emotive",
"content": "def SSubtyp.dealias_right_tforall.emotive\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop := True"
},
{
"name": "Capless.SSubtyp.dealias_right_tforall.cmotive",
"content": "def SSubtyp.dealias_rig... | [
{
"name": "Capless.SSubtyp.dealias_right_tforall",
"content": "theorem SSubtyp.dealias_right_tforall\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"
},
{
"name": "Capless.SType.dealias_tforall_inj'",... | import Capless.Subtyping
import Capless.Store
import Capless.Inversion.Basic
import Capless.Inversion.Context
import Capless.Subtyping.Basic
import Capless.Narrowing
namespace Capless
def SSubtyp.dealias_right_tforall.emotive
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop := True
def S... | theorem SSubtyp.sub_dealias_tforall_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))
(h2 : SType.Dealias Γ S2 (SType.tforall T2 E2))
(hs : SSubtyp Γ S1 S2) :
SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2 := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_tforall_inv.emotive Γ E1 E2)
(motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_tforall_inv.cmotive Γ C1 C2)
(motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_tforall_inv.smotive Γ S1 S2)
(t := hs) (h1 := h1) (h2 := h2) (ht := ht)
case exi... | 5 | 121 | false | Type systems |
77 | Capless.SSubtyp.sub_dealias_boxed_inv | theorem SSubtyp.sub_dealias_boxed_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.box T1))
(h2 : SType.Dealias Γ S2 (SType.box T2))
(hs : SSubtyp Γ S1 S2) :
CSubtyp Γ T1 T2 | capless-lean | Capless/Inversion/Subtyping.lean | [
"import Capless.Inversion.Basic",
"import Capless.Narrowing",
"import Capless.Subtyping.Basic",
"import Capless.Subtyping",
"import Capless.Store",
"import Capless.Inversion.Context"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "refl",
"module": "Mathlib.Order.Defs.Unbundled"
}
] | [
{
"name": "Context.tbound_inj",
"content": "theorem Context.tbound_inj\n (h1 : Context.TBound Γ X b1)\n (h2 : Context.TBound Γ X b2) : b1 = b2"
},
{
"name": "Context.tvar_tbound_succ_inv",
"content": "theorem Context.tvar_tbound_succ_inv\n (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n ... | [
{
"name": "Capless.SSubtyp.dealias_right_boxed.emotive",
"content": "def SSubtyp.dealias_right_boxed.emotive\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop := True"
},
{
"name": "Capless.SSubtyp.dealias_right_boxed.cmotive",
"content": "def SSubtyp.dealias_right_box... | [
{
"name": "Capless.SSubtyp.dealias_right_boxed",
"content": "theorem SSubtyp.dealias_right_boxed\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.box T2)) :\n ∃ T1, SType.Dealias Γ S1 (SType.box T1)"
},
{
"name": "Capless.SType.dealias_boxed_inj'",
"content": "theore... | import Capless.Subtyping
import Capless.Store
import Capless.Inversion.Basic
import Capless.Inversion.Context
import Capless.Subtyping.Basic
import Capless.Narrowing
namespace Capless
def SSubtyp.dealias_right_boxed.emotive
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop := True
def SSu... | theorem SSubtyp.sub_dealias_boxed_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.box T1))
(h2 : SType.Dealias Γ S2 (SType.box T2))
(hs : SSubtyp Γ S1 S2) :
CSubtyp Γ T1 T2 := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 _ => SSubtyp.dealias_boxed_inv.emotive Γ E1 E2)
(motive_2 := fun Γ C1 C2 _ => SSubtyp.dealias_boxed_inv.cmotive Γ C1 C2)
(motive_3 := fun Γ S1 S2 _ => SSubtyp.dealias_boxed_inv.smotive Γ S1 S2)
(t := hs) (h1 := h1) (h2 := h2) (ht := ht)
case exist => ... | 7 | 109 | false | Type systems |
78 | Capless.SSubtyp.sub_dealias_label_inv | theorem SSubtyp.sub_dealias_label_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.label T1))
(h2 : SType.Dealias Γ S2 (SType.label T2))
(hs : SSubtyp Γ S1 S2) :
SSubtyp Γ T2 T1 | capless-lean | Capless/Inversion/Subtyping.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Subtyping.Basic",
"import Capless.Inversion.Basic",
"import Capless.Narrowing",
"import Capless.Subtyping",
"import Capless.Store",
"import Capless.Inversion.Context"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Exists",
"module": "Init.Core"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "refl",
"module": "Mathlib.Order.Defs.Unbundled"
}
] | [
{
"name": "ESubtyp.refl",
"content": "theorem ESubtyp.refl :\n ESubtyp Γ E E"
},
{
"name": "CSubtyp.refl",
"content": "theorem CSubtyp.refl :\n CSubtyp Γ T T"
},
{
"name": "Context.tbound_inj",
"content": "theorem Context.tbound_inj\n (h1 : Context.TBound Γ X b1)\n (h2 : Context.... | [
{
"name": "Capless.SSubtyp.dealias_right_label.emotive",
"content": "def SSubtyp.dealias_right_label.emotive\n (Γ : Context n m k)\n (E1 : EType n m k)\n (E2 : EType n m k)\n : Prop := True"
},
{
"name": "Capless.SSubtyp.dealias_right_label.cmotive",
"content": "def SSubtyp.dealias_right_lab... | [
{
"name": "Capless.SSubtyp.dealias_right_label",
"content": "theorem SSubtyp.dealias_right_label\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.label T2)) :\n ∃ T1, SType.Dealias Γ S1 (SType.label T1)"
},
{
"name": "Capless.SType.dealias_label_inj'",
"content": "th... | import Capless.Subtyping
import Capless.Store
import Capless.Inversion.Basic
import Capless.Inversion.Context
import Capless.Subtyping.Basic
import Capless.Narrowing
namespace Capless
def SSubtyp.dealias_right_label.emotive
(Γ : Context n m k)
(E1 : EType n m k)
(E2 : EType n m k)
: Prop := True
def SSu... | theorem SSubtyp.sub_dealias_label_inv
(ht : Γ.IsTight)
(h1 : SType.Dealias Γ S1 (SType.label T1))
(h2 : SType.Dealias Γ S2 (SType.label T2))
(hs : SSubtyp Γ S1 S2) :
SSubtyp Γ T2 T1 := | := by
apply SSubtyp.rec
(motive_1 := fun Γ E1 E2 _ => SSubtyp.dealias_label_inv.emotive Γ E1 E2)
(motive_2 := fun Γ C1 C2 _ => SSubtyp.dealias_label_inv.cmotive Γ C1 C2)
(motive_3 := fun Γ S1 S2 _ => SSubtyp.dealias_label_inv.smotive Γ S1 S2)
(t := hs) (h1 := h1) (h2 := h2) (ht := ht)
case exist => ... | 5 | 112 | false | Type systems |
79 | Capless.progress | theorem progress
(ht : TypedState state Γ E) :
Progress state | capless-lean | Capless/Soundness/Progress.lean | [
"import Capless.Inversion.Context",
"import Capless.Weakening.IsValue",
"import Mathlib.Data.Fin.Basic",
"import Capless.WellScoped.Basic",
"import Capless.Inversion.Subtyping",
"import Capless.Inversion.Lookup",
"import Capless.Inversion.Typing",
"import Capless.Store",
"import Capless.Reduction",
... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
},
{
"name": "Fin.elim0",
"module": "Init.... | [
{
"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t",
"content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"
},
{
"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u",
"content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"
},
{
"name": "notat... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Term.IsValue.weaken",
"content": "theorem Term.IsValue.weaken\n (hv : Term.IsValue t) :\n Term.IsValue t.weaken"
},
{
"name": "Term.IsValue.tweaken",
"content": "theorem Term.IsValue.tweaken\n (hv : Term.IsValue t) :\n Term.IsValue t.tweaken"
},
{
"name": "Term.IsValue.cwe... | [
{
"name": "Capless.Progress",
"content": "inductive Progress : State n m k -> Prop where\n| halt_var :\n Progress ⟨σ, Cont.none, Term.var x⟩\n| halt_value {t : Term n m k} :\n t.IsValue ->\n Progress ⟨σ, Cont.none, t⟩\n| step :\n Reduce state state' ->\n Progress state"
}
] | [
{
"name": "Capless.Store.lookup_exists",
"content": "theorem Store.lookup_exists {σ : Store n m k} {x : Fin n} :\n (∃ v, Store.Bound σ x v ∧ v.IsValue) ∨ (∃ S, Store.LBound σ x S)"
},
{
"name": "Capless.Store.val_lookup_exists",
"content": "theorem Store.val_lookup_exists {σ : Store n m k} {x :... | import Mathlib.Data.Fin.Basic
import Capless.Reduction
import Capless.Narrowing.TypedCont
import Capless.Inversion.Lookup
import Capless.Inversion.Typing
import Capless.Weakening.IsValue
import Capless.WellScoped.Basic
namespace Capless
inductive Progress : State n m k -> Prop where
| halt_var :
Progress ⟨σ, ... | theorem progress
(ht : TypedState state Γ E) :
Progress state := | := by
cases ht
case mk hs ht hsc hc =>
induction ht
case var =>
cases hc <;> aesop
case label =>
cases hc <;> aesop
case pack =>
cases hc <;> aesop
case sub hsub ih _ _ _ =>
apply ih <;> try easy
apply WellScoped.subcapt; easy; easy
apply! TypedCont.narrow
... | 8 | 164 | false | Type systems |
80 | Capless.TypedCont.lweaken | theorem TypedCont.lweaken
(h : TypedCont Γ E cont E' Ct) :
TypedCont (Γ.label S) E.weaken cont.weaken E'.weaken Ct.weaken | capless-lean | Capless/Weakening/TypedCont/Term.lean | [
"import Capless.Type.Basic",
"import Capless.Basic",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subtyping",
"import Capless.CaptureSet",
"import Capless.Weakening.Subcapturing",
"import Capless.Store"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "SSubtyp.weaken",
"content": "theorem SSubtyp.weaken\n (h : SSubtyp Γ S1 S2) :\n ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"
},
{
"name": "Su... | [] | [
{
"name": "Capless.EType.weaken1_weaken",
"content": "theorem EType.weaken1_weaken (E : EType n m k) :\n E.weaken.weaken1 = E.weaken.weaken"
},
{
"name": "Capless.CaptureSet.weaken1_weaken",
"content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n C.weaken.weaken1 = C.weaken.weak... | import Capless.Store
import Capless.Weakening.Typing
import Capless.Weakening.Subtyping
import Capless.Weakening.Subcapturing
namespace Capless | theorem TypedCont.lweaken
(h : TypedCont Γ E cont E' Ct) :
TypedCont (Γ.label S) E.weaken cont.weaken E'.weaken Ct.weaken := | := by
induction h
case none =>
simp [Cont.weaken]
apply none
apply? ESubtyp.lweaken
case cons ih =>
simp [Cont.weaken]
have heq : ∀ {n m k} {T0 : CType n m k}, (EType.type T0).weaken = EType.type T0.weaken := by
intro T0
simp [EType.weaken, EType.rename, CType.weaken]
-- rw [he... | 7 | 140 | false | Type systems |
81 | Capless.TypedCont.weaken | theorem TypedCont.weaken
(h : TypedCont Γ E t E' C0) :
TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken | capless-lean | Capless/Weakening/TypedCont/Term.lean | [
"import Capless.Type.Basic",
"import Capless.Basic",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subtyping",
"import Capless.CaptureSet",
"import Capless.Weakening.Subcapturing",
"import Capless.Store"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "SSubtyp.weaken",
"content": "theorem SSubtyp.weaken\n (h : SSubtyp Γ S1 S2) :\n ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"
},
{
"name": "Su... | [] | [
{
"name": "Capless.EType.weaken1_weaken",
"content": "theorem EType.weaken1_weaken (E : EType n m k) :\n E.weaken.weaken1 = E.weaken.weaken"
},
{
"name": "Capless.CaptureSet.weaken1_weaken",
"content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n C.weaken.weaken1 = C.weaken.weak... | import Capless.Store
import Capless.Weakening.Typing
import Capless.Weakening.Subtyping
import Capless.Weakening.Subcapturing
namespace Capless | theorem TypedCont.weaken
(h : TypedCont Γ E t E' C0) :
TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken := | := by
induction h
case none =>
simp [Cont.weaken]
apply none
apply? ESubtyp.weaken
case cons ih =>
simp [Cont.weaken]
have heq : ∀ {n m k} {T0 : CType n m k}, (EType.type T0).weaken = EType.type T0.weaken := by
intro T0
simp [EType.weaken, EType.rename, CType.weaken]
-- rw [heq... | 5 | 128 | false | Type systems |
82 | Capless.TypedCont.cweaken | theorem TypedCont.cweaken
(h : TypedCont Γ E t E' Ct) :
TypedCont (Γ.cvar b) E.cweaken t.cweaken E'.cweaken Ct.cweaken | capless-lean | Capless/Weakening/TypedCont/Capture.lean | [
"import Capless.Type.Basic",
"import Capless.Basic",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subtyping",
"import Capless.CaptureSet",
"import Capless.Weakening.Subcapturing",
"import Capless.Store"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "SSubtyp.weaken",
"content": "theorem SSubtyp.weaken\n (h : SSubtyp Γ S1 S2) :\n ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"
},
{
"name": "Su... | [] | [
{
"name": "Capless.EType.cweaken_ex",
"content": "theorem EType.cweaken_ex (T : CType n m (k+1)) :\n (EType.ex T).cweaken = EType.ex T.cweaken1"
},
{
"name": "Capless.EType.cweaken_weaken",
"content": "theorem EType.cweaken_weaken (E : EType n m k) :\n E.weaken.cweaken = E.cweaken.weaken"
},... | import Capless.Store
import Capless.Weakening.Typing
import Capless.Weakening.Subtyping
import Capless.Weakening.Subcapturing
namespace Capless | theorem TypedCont.cweaken
(h : TypedCont Γ E t E' Ct) :
TypedCont (Γ.cvar b) E.cweaken t.cweaken E'.cweaken Ct.cweaken := | := by
induction h
case none =>
simp [Cont.cweaken]
apply none
apply? ESubtyp.cweaken
case cons ht hs _ ih =>
simp [Cont.cweaken, EType.cweaken_type]
apply cons
{ have ht1 := ht.cweaken_ext (b := b)
rw [EType.cweaken_weaken] at ht1
rw [CaptureSet.weaken_crename]
exact ht1 ... | 5 | 118 | false | Type systems |
83 | Capless.Subcapt.rename | theorem Subcapt.rename
(h : Subcapt Γ C1 C2)
(ρ : VarMap Γ f Δ) :
Subcapt Δ (C1.rename f) (C2.rename f) | capless-lean | Capless/Renaming/Term/Subcapturing.lean | [
"import Capless.Subcapturing",
"import Mathlib.Data.Finset.Image",
"import Capless.Renaming.Basic",
"import Capless.CaptureSet"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.rename_union",
"content": "theorem CaptureSet.rename_union {C1 C2 : CaptureSet n k} {f : FinFun n n'} :\n (C1 ∪ C2).rename f = C1.rename f ∪ C2.rename f"
},
{
"name": "CaptureSet.rename_singleton",
"content": "theorem CaptureSet.rename_singleton {x : Fin n} {f : FinFun n n... | [] | [
{
"name": "Capless.CaptureSet.Subset.rename",
"content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n (h : C1 ⊆ C2) :\n C1.rename f ⊆ C2.rename f"
}
] | import Capless.Subcapturing
import Capless.Renaming.Basic
import Mathlib.Data.Finset.Image
namespace Capless | theorem Subcapt.rename
(h : Subcapt Γ C1 C2)
(ρ : VarMap Γ f Δ) :
Subcapt Δ (C1.rename f) (C2.rename f) := | := by
induction h
case trans ih1 ih2 => apply trans <;> aesop
case subset hsub =>
apply subset
apply CaptureSet.Subset.rename; trivial
case union ih1 ih2 =>
simp [CaptureSet.rename_union]
apply union <;> aesop
case var hb =>
simp [CaptureSet.rename_singleton]
apply var
have hb1 := ... | 3 | 36 | false | Type systems |
84 | Capless.Store.val_lookup_exists | theorem Store.val_lookup_exists {σ : Store n m k} {x : Fin n}
(hs : TypedStore σ Γ) (hx : Typed Γ (Term.var x) (EType.type T) Cx)
(hvt : T.IsValue) :
∃ v, Store.Bound σ x v ∧ v.IsValue | capless-lean | Capless/Soundness/Progress.lean | [
"import Capless.Inversion.Context",
"import Capless.Weakening.IsValue",
"import Mathlib.Data.Fin.Basic",
"import Capless.WellScoped.Basic",
"import Capless.Inversion.Subtyping",
"import Capless.Inversion.Lookup",
"import Capless.Inversion.Typing",
"import Capless.Store",
"import Capless.Reduction",
... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
},
{
"name": "Fin.elim0",
"module": "Init.... | [
{
"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U",
"content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"
},
{
"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T",
"content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"
},
{
"name": "notation:50... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Term.IsValue.weaken",
"content": "theorem Term.IsValue.weaken\n (hv : Term.IsValue t) :\n Term.IsValue t.weaken"
},
{
"name": "Term.IsValue.tweaken",
"content": "theorem Term.IsValue.tweaken\n (hv : Term.IsValue t) :\n Term.IsValue t.tweaken"
},
{
"name": "Term.IsValue.cwe... | [] | [
{
"name": "Capless.Store.lookup_exists",
"content": "theorem Store.lookup_exists {σ : Store n m k} {x : Fin n} :\n (∃ v, Store.Bound σ x v ∧ v.IsValue) ∨ (∃ S, Store.LBound σ x S)"
}
] | import Mathlib.Data.Fin.Basic
import Capless.Reduction
import Capless.Narrowing.TypedCont
import Capless.Inversion.Lookup
import Capless.Inversion.Typing
import Capless.Weakening.IsValue
import Capless.WellScoped.Basic
namespace Capless | theorem Store.val_lookup_exists {σ : Store n m k} {x : Fin n}
(hs : TypedStore σ Γ) (hx : Typed Γ (Term.var x) (EType.type T) Cx)
(hvt : T.IsValue) :
∃ v, Store.Bound σ x v ∧ v.IsValue := | := by
have hg := TypedStore.is_tight hs
have h := Store.lookup_exists (σ := σ) (x := x)
cases h
case inl h => easy
case inr h =>
have ⟨S, hl⟩ := h
have hb := Store.bound_label hl hs
have ⟨S0, hb0, hsub⟩ := Typed.label_inv hx hb
have h := Context.lbound_inj hb hb0
subst_vars
cases hvt
... | 4 | 102 | false | Type systems |
85 | Capless.Typed.canonical_form_tlam' | theorem Typed.canonical_form_tlam'
(ht : Γ.IsTight)
(hd : SType.Dealias Γ S0 (SType.tforall S' E))
(he1 : t0 = Term.tlam S t)
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
SSubtyp Γ S' S ∧
Typed (Γ.tvar (TBinding.bound S')) t E Cf | capless-lean | Capless/Inversion/Typing.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Narrowing.Typing",
"import Capless.Tactics",
"import Capless.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Weakening.Subtyping",
"import Capless.Inversion.Subtyping",
"import Capless.Narrowing",
... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "And",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t",
"content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"
},
{
"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u",
"content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"
},
{
"name": "notat... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Typed.tnarrow",
"content": "theorem Typed.tnarrow\n (h : Typed (Γ,X<: S) t E Ct)\n (hs : SSubtyp Γ S' S) :\n Typed (Γ,X<: S') t E Ct"
},
{
"name": "SSubtyp.sub_dealias_tforall_inv",
"content": "theorem SSubtyp.sub_dealias_tforall_inv\n (ht : Γ.IsTight)\n (h1 : SType.Dealias Γ S1 ... | [] | [] | import Capless.Tactics
import Capless.Typing
import Capless.Subtyping.Basic
import Capless.Subcapturing.Basic
import Capless.Narrowing
import Capless.Weakening.Subcapturing
import Capless.Inversion.Context
import Capless.Inversion.Subtyping
namespace Capless | theorem Typed.canonical_form_tlam'
(ht : Γ.IsTight)
(hd : SType.Dealias Γ S0 (SType.tforall S' E))
(he1 : t0 = Term.tlam S t)
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
SSubtyp Γ S' S ∧
Typed (Γ.tvar (TBinding.bound S')) t E Cf := | := by
induction h <;> try (solve | cases he1 | cases he2)
case tabs =>
cases he1; cases he2
cases hd
constructor
apply SSubtyp.refl
trivial
case sub hs ih =>
subst he2
cases hs
rename_i hs
cases hs
rename_i hsc hs
have ⟨S1, E1, hd3⟩ := SSubtyp.dealias_right_tforall hs h... | 5 | 68 | false | Type systems |
86 | Capless.Subcapt.crename | theorem Subcapt.crename
(h : Subcapt Γ C1 C2)
(ρ : CVarMap Γ f Δ) :
Subcapt Δ (C1.crename f) (C2.crename f) | capless-lean | Capless/Renaming/Capture/Subcapturing.lean | [
"import Capless.Subcapturing",
"import Mathlib.Data.Finset.Image",
"import Capless.Renaming.Basic",
"import Capless.CaptureSet"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.crename_csingleton",
"content": "theorem CaptureSet.crename_csingleton {x : Fin k} {f : FinFun k k'} :\n ({c=x} : CaptureSet n k).crename f = {c=f x}"
},
{
"name": "CaptureSet.crename_union",
"content": "theorem CaptureSet.crename_union {C1 C2 : CaptureSet n k} {f : FinFun... | [] | [
{
"name": "Capless.CaptureSet.Subset.crename",
"content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n (h : C1 ⊆ C2) :\n C1.crename f ⊆ C2.crename f"
}
] | import Capless.Subcapturing
import Capless.Renaming.Basic
import Mathlib.Data.Finset.Image
namespace Capless | theorem Subcapt.crename
(h : Subcapt Γ C1 C2)
(ρ : CVarMap Γ f Δ) :
Subcapt Δ (C1.crename f) (C2.crename f) := | := by
induction h
case trans ih1 ih2 => apply trans <;> aesop
case subset hsub =>
apply subset
apply CaptureSet.Subset.crename; trivial
case union ih1 ih2 =>
simp [CaptureSet.crename_union]
apply union <;> aesop
case var hb =>
simp [CaptureSet.crename_singleton]
apply var
have hb1 ... | 3 | 43 | false | Type systems |
87 | Capless.Typed.boundary_body_typing | theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}
(ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :
Typed ((Γ.label S),c:={x=0}) t E Ct | capless-lean | Capless/Typing/Boundary.lean | [
"import Capless.Renaming.Capture.Typing",
"import Capless.Renaming.Term.Subtyping",
"import Capless.Renaming.Term.Typing",
"import Capless.Renaming.Type.Subcapturing",
"import Capless.Basic",
"import Capless.Subst.Term.Subcapturing",
"import Capless.Renaming.Term.Subcapturing",
"import Capless.Capture... | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "abs",
"module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.cweaken",
"content": "theorem CSubtyp.cweaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"
},
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "ES... | [
{
"name": "Capless.VarRename.boundary",
"content": "def VarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n VarMap\n ((Γ,c<:*),x:(Label[S.cweaken])^{c=0})\n FinFun.weaken.ext\n (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0}) :="
},
{
"name": "Capless.CVarRename.boundary",
... | [
{
"name": "Capless.Term.copen_cweaken_ext",
"content": "theorem Term.copen_cweaken_ext {t : Term n m (k+1)} :\n (t.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = t"
},
{
"name": "Capless.EType.copen_cweaken_ext",
"content": "theorem EType.copen_cweaken_ext {E : EType n m (k+1)} :\n (E.... | import Capless.Typing
import Capless.Weakening.Typing
import Capless.Narrowing.Typing
namespace Capless
def VarRename.boundary {Γ : Context n m k} {S : SType n m k} :
VarMap
((Γ,c<:*),x:(Label[S.cweaken])^{c=0})
FinFun.weaken.ext
(((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0}) :=
def CVarRenam... | theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}
(ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :
Typed ((Γ.label S),c:={x=0}) t E Ct := | := by
have h := ht.rename VarRename.boundary
have h := h.crename CVarRename.boundary
have h := h.csubst CVarSubst.boundary
simp [Term.copen_cweaken_ext, EType.copen_cweaken_ext, CaptureSet.copen_cweaken_ext] at h
have h := h.subst VarSubst.boundary
simp [Term.open_weaken_ext, EType.open_weaken_ext, CaptureS... | 5 | 220 | false | Type systems |
88 | Capless.Typed.canonical_form_lam' | theorem Typed.canonical_form_lam'
(ht : Γ.IsTight)
(he1 : t0 = Term.lam T t) (hd2 : SType.Dealias Γ S0 (SType.forall T' E))
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
CSubtyp Γ T' T ∧
Typed (Γ.var T') t E (Cf.weaken ∪ {x=0}) | capless-lean | Capless/Inversion/Typing.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Narrowing.Typing",
"import Capless.Tactics",
"import Capless.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subtyping",
"import Capless.Narrowing.TypedC... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "And",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "Typed.narrow",
"content": "theorem Typed.narrow\n (h : Typed (Γ,x: T) t E Ct)\n (hs : CSubtyp Γ T' T) :\n Typed (Γ,x: T') t E Ct"
},
{
"name": "CSubtyp.trans",
"content": "theorem CSubtyp.trans\n (h1 : CSubtyp Γ T1 T2)\n (h2 : CSubtyp Γ T2 T3) :\n CSubtyp Γ T1 T3"
},
{
"... | [] | [] | import Capless.Tactics
import Capless.Typing
import Capless.Subtyping.Basic
import Capless.Subcapturing.Basic
import Capless.Narrowing
import Capless.Weakening.Subcapturing
import Capless.Inversion.Context
import Capless.Inversion.Subtyping
namespace Capless | theorem Typed.canonical_form_lam'
(ht : Γ.IsTight)
(he1 : t0 = Term.lam T t) (hd2 : SType.Dealias Γ S0 (SType.forall T' E))
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
CSubtyp Γ T' T ∧
Typed (Γ.var T') t E (Cf.weaken ∪ {x=0}) := | := by
induction h <;> try (solve | cases he1 | cases he2)
case abs =>
cases he1; cases he2
cases hd2
constructor
{ apply CSubtyp.refl }
{ aesop }
case sub hs ih =>
subst he2
cases hs
rename_i hs
cases hs
rename_i hsc hs
have ⟨T1, E1, hd3⟩ := SSubtyp.dealias_right_forall... | 4 | 106 | false | Type systems |
89 | Capless.Typed.canonical_form_clam' | theorem Typed.canonical_form_clam'
(ht : Γ.IsTight)
(hd : SType.Dealias Γ S0 (SType.cforall B' E))
(he1 : t0 = Term.clam B t)
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
Subbound Γ B' B ∧ Typed (Γ.cvar (CBinding.bound B')) t E Cf.cweaken | capless-lean | Capless/Inversion/Typing.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Narrowing.Typing",
"import Capless.Tactics",
"import Capless.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Narrowing.Subtyping",
"import Capless.Inversion.Subtyping",
"import Capless.Narrowing",
"import Capless.Weakening.Subcapturing"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "And",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2",
"content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"
},
{
"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2",
"content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"
},
{
... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "SSubtyp.dealias_right_cforall",
"content": "theorem SSubtyp.dealias_right_cforall\n (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"
},
{
"name": "Typed.cnarrow",
"content": "theorem Type... | [] | [] | import Capless.Tactics
import Capless.Typing
import Capless.Subtyping.Basic
import Capless.Subcapturing.Basic
import Capless.Narrowing
import Capless.Weakening.Subcapturing
import Capless.Inversion.Context
import Capless.Inversion.Subtyping
namespace Capless | theorem Typed.canonical_form_clam'
(ht : Γ.IsTight)
(hd : SType.Dealias Γ S0 (SType.cforall B' E))
(he1 : t0 = Term.clam B t)
(he2 : E0 = EType.type (CType.capt Cf S0))
(h : Typed Γ t0 E0 Ct0) :
Subbound Γ B' B ∧ Typed (Γ.cvar (CBinding.bound B')) t E Cf.cweaken := | := by
induction h <;> try (solve | cases he1 | cases he2)
case cabs =>
cases he1; cases he2
cases hd
apply And.intro
{ apply Subbound.refl }
{ trivial }
case sub hs ih =>
subst he2
cases hs
rename_i hs
cases hs
rename_i hsc hs
have ⟨B1, E1, hd3⟩ := SSubtyp.dealias_right... | 4 | 56 | false | Type systems |
90 | Capless.TypedCont.tweaken | theorem TypedCont.tweaken
(h : TypedCont Γ E t E' C0) :
TypedCont (Γ.tvar S) E.tweaken t.tweaken E'.tweaken C0 | capless-lean | Capless/Weakening/TypedCont/Type.lean | [
"import Capless.Type.Basic",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subcapturing",
"import Capless.Store",
"import Capless.Weakening.Subtyping"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t",
"content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"
},
{
"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u",
"content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"
},
{
"name": "notat... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "Subbound.tweaken",
"content": "theorem Subbound.tweaken\n (h : Subbound Γ B1 B2) :\n Subbound (Γ.tvar b) B1 B2"
},
{
"name": "Subbound.weake... | [] | [
{
"name": "Capless.EType.tweaken_ex",
"content": "theorem EType.tweaken_ex (T : CType n m (k+1)) :\n (EType.ex T).tweaken = EType.ex T.tweaken"
},
{
"name": "Capless.EType.tweaken_weaken",
"content": "theorem EType.tweaken_weaken (E : EType n m k) :\n E.weaken.tweaken = E.tweaken.weaken"
},
... | import Capless.Store
import Capless.Weakening.Typing
import Capless.Weakening.Subtyping
import Capless.Weakening.Subcapturing
namespace Capless | theorem TypedCont.tweaken
(h : TypedCont Γ E t E' C0) :
TypedCont (Γ.tvar S) E.tweaken t.tweaken E'.tweaken C0 := | := by
induction h
case none =>
simp [Cont.tweaken]
apply none
apply? ESubtyp.tweaken
case cons ht hs _ ih =>
simp [Cont.tweaken]
-- simp [EType.tweaken_type]
apply cons
{ have ht1 := ht.tweaken_ext (b := S)
rw [EType.tweaken_weaken] at ht1
exact ht1 }
{ apply hs.tweaken... | 5 | 125 | false | Type systems |
91 | Capless.SType.crename_rename_comm | theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :
(S.rename f).crename g = (S.crename g).rename f :=
match S with
| SType.top => by simp [SType.rename, SType.crename]
| SType.tvar X => by simp [SType.rename, SType.crename]
| SType.forall E1 E2 => by
have ih1 := CTyp... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Type.Renaming",
"import Capless.Type.Core",
"import Capless.CaptureSet"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.crename_rename_comm",
"content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n (C.rename f).crename g = (C.crename g).rename f"
}
] | [] | [
{
"name": "Capless.CBound.crename_rename_comm",
"content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n (b.crename f).rename g = (b.rename g).crename f"
},
{
"name": "Capless.EType.crename_rename_comm",
"content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n'... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless | theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :
(S.rename f).crename g = (S.crename g).rename f := | :=
match S with
| SType.top => by simp [SType.rename, SType.crename]
| SType.tvar X => by simp [SType.rename, SType.crename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_rename_comm E1 f g
have ih2 := EType.crename_rename_comm E2 f.ext g
simp [SType.rename, SType.crename, ih1, ih2]
| SType... | 5 | 24 | false | Type systems |
92 | Capless.SType.rename_rename | theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :
(S.rename f).rename g = S.rename (g ∘ f) :=
match S with
| SType.top => by simp [SType.rename]
| SType.tvar X => by simp [SType.rename]
| SType.forall E1 E2 => by
have ih1 := CType.rename_rename E1 f g
have ih2 := ET... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Basic",
"import Capless.Type.Renaming",
"import Capless.Type.Core",
"import Capless.CaptureSet"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.rename_rename",
"content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n (C.rename f).rename g = C.rename (g ∘ f)"
},
{
"name": "FinFun.ext_comp_ext",
"content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n g.ext ∘ f.ext = FinFun.ext... | [] | [
{
"name": "Capless.CBound.rename_rename",
"content": "theorem CBound.rename_rename {b : CBound n k} :\n (b.rename f).rename g = b.rename (g ∘ f)"
},
{
"name": "Capless.EType.rename_rename",
"content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n (E.r... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end | theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :
(S.rename f).rename g = S.rename (g ∘ f) := | :=
match S with
| SType.top => by simp [SType.rename]
| SType.tvar X => by simp [SType.rename]
| SType.forall E1 E2 => by
have ih1 := CType.rename_rename E1 f g
have ih2 := EType.rename_rename E2 f.ext g.ext
simp [SType.rename, ih1, ih2, FinFun.ext_comp_ext]
| SType.tforall S E => by
have ih1 ... | 4 | 20 | false | Type systems |
93 | Capless.SType.trename_rename_comm | theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :
(S.trename g).rename f = (S.rename f).trename g :=
match S with
| SType.top => by simp [SType.trename, SType.rename]
| SType.tvar X => by simp [SType.trename, SType.rename]
| SType.forall E1 E2 => by
have ih1 := CTyp... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Type.Renaming",
"import Capless.Type.Core"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [] | [
{
"name": "Capless.EType.trename_rename_comm",
"content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n (E.trename g).rename f = (E.rename f).trename g"
},
{
"name": "Capless.CType.trename_rename_comm",
"content": "theorem CType.trename_rename_comm... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end | theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :
(S.trename g).rename f = (S.rename f).trename g := | :=
match S with
| SType.top => by simp [SType.trename, SType.rename]
| SType.tvar X => by simp [SType.trename, SType.rename]
| SType.forall E1 E2 => by
have ih1 := CType.trename_rename_comm E1 f g
have ih2 := EType.trename_rename_comm E2 f.ext g
simp [SType.trename, SType.rename, ih1, ih2]
| SType... | 4 | 20 | false | Type systems |
94 | Capless.SType.rename_id | theorem SType.rename_id {S : SType n m k} :
S.rename FinFun.id = S :=
match S with
| SType.top => by simp [SType.rename]
| SType.tvar X => by simp [SType.rename]
| SType.forall E1 E2 => by
have ih1 := CType.rename_id (T := E1)
have ih2 := EType.rename_id (E := E2)
simp [SType.rename, FinFun.id_ext... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Basic",
"import Capless.Type.Renaming",
"import Capless.Type.Core",
"import Capless.CaptureSet"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.rename_id",
"content": "theorem CaptureSet.rename_id {C : CaptureSet n k} :\n C.rename FinFun.id = C"
},
{
"name": "FinFun.id_ext",
"content": "theorem FinFun.id_ext :\n (FinFun.ext (n := n) id) = id"
}
] | [] | [
{
"name": "Capless.CBound.rename_id",
"content": "theorem CBound.rename_id {b : CBound n k} :\n b.rename FinFun.id = b"
},
{
"name": "Capless.EType.rename_id",
"content": "theorem EType.rename_id {E : EType n m k} :\n E.rename FinFun.id = E"
},
{
"name": "Capless.CType.rename_id",
... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end
end
end
end
end | theorem SType.rename_id {S : SType n m k} :
S.rename FinFun.id = S := | :=
match S with
| SType.top => by simp [SType.rename]
| SType.tvar X => by simp [SType.rename]
| SType.forall E1 E2 => by
have ih1 := CType.rename_id (T := E1)
have ih2 := EType.rename_id (E := E2)
simp [SType.rename, FinFun.id_ext, ih1, ih2]
| SType.tforall S E => by
have ih1 := SType.rename_... | 4 | 21 | false | Type systems |
95 | Capless.Context.cvar_bound_cvar_inst_inv' | theorem Context.cvar_bound_cvar_inst_inv' {Γ : Context n m k}
(he1 : Γ' = Context.cvar Γ (CBinding.bound b0))
(he2 : b' = CBinding.inst C)
(hb : Context.CBound Γ' c b') :
∃ c0 C0, c = c0.succ ∧ C = C0.cweaken ∧ Context.CBound Γ c0 (CBinding.inst C0) | capless-lean | Capless/Context.lean | [
"import Capless.Type",
"import Capless.CaptureSet"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Eq",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [
{
"name": "Capless.TBinding",
"content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"
},
{
"name": "Capless.CBinding",
"content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBi... | [
{
"name": "Capless.CBinding.eq_inst_cweaken_inv",
"content": "theorem CBinding.eq_inst_cweaken_inv {b : CBinding n k}\n (h : CBinding.inst C = b.cweaken) :\n ∃ C0, b = CBinding.inst C0"
}
] | import Capless.Type
import Capless.CaptureSet
namespace Capless
inductive TBinding : Nat -> Nat -> Nat -> Type where
| bound : SType n m k -> TBinding n m k
| inst : SType n m k -> TBinding n m k
inductive CBinding : Nat -> Nat -> Type where
| bound : CBound n k -> CBinding n k
| inst : CaptureSet n k -> CBinding n... | theorem Context.cvar_bound_cvar_inst_inv' {Γ : Context n m k}
(he1 : Γ' = Context.cvar Γ (CBinding.bound b0))
(he2 : b' = CBinding.inst C)
(hb : Context.CBound Γ' c b') :
∃ c0 C0, c = c0.succ ∧ C = C0.cweaken ∧ Context.CBound Γ c0 (CBinding.inst C0) := | := by
cases hb <;> try (solve | cases he1)
case here =>
have h := CBinding.eq_inst_cweaken_inv (Eq.symm he2)
have ⟨C0, h⟩ := h
subst h; cases he1
case there_cvar =>
have ⟨C0, h⟩ := CBinding.eq_inst_cweaken_inv (Eq.symm he2)
subst h; simp [CBinding.cweaken, CBinding.crename] at he2
rename_i... | 3 | 32 | false | Type systems |
96 | Capless.SType.crename_crename | theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :
(S.crename f).crename g = S.crename (g ∘ f) :=
match S with
| SType.top => by simp [SType.crename]
| SType.tvar X => by simp [SType.crename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_crename E1 f g
have ... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Basic",
"import Capless.Type.Renaming",
"import Capless.Type.Core",
"import Capless.CaptureSet"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.crename_crename",
"content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n (C.crename f).crename g = C.crename (g ∘ f)"
},
{
"name": "FinFun.ext_comp_ext",
"content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n g.ext ∘ f.ext = Fin... | [] | [
{
"name": "Capless.CBound.crename_crename",
"content": "theorem CBound.crename_crename {b : CBound n k} :\n (b.crename f).crename g = b.crename (g ∘ f)"
},
{
"name": "Capless.EType.crename_crename",
"content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k''... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end
end | theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :
(S.crename f).crename g = S.crename (g ∘ f) := | :=
match S with
| SType.top => by simp [SType.crename]
| SType.tvar X => by simp [SType.crename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_crename E1 f g
have ih2 := EType.crename_crename E2 f g
simp [SType.crename, ih1, ih2]
| SType.tforall S E => by
have ih1 := SType.crename_crena... | 4 | 20 | false | Type systems |
97 | Capless.SType.crename_trename_comm | theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :
(S.crename f).trename g = (S.trename g).crename f :=
match S with
| SType.top => by simp [SType.crename, SType.trename]
| SType.tvar X => by simp [SType.crename, SType.trename]
| SType.forall E1 E2 => by
have ih1 :=... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Type.Renaming",
"import Capless.Type.Core"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "List.getElem_append_left{α",
"content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"
}
] | [] | [
{
"name": "Capless.EType.crename_trename_comm",
"content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n (E.crename f).trename g = (E.trename g).crename f"
},
{
"name": "Capless.CType.crename_trename_comm",
"content": "theorem CType.crename_trenam... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end
end
end | theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :
(S.crename f).trename g = (S.trename g).crename f := | :=
match S with
| SType.top => by simp [SType.crename, SType.trename]
| SType.tvar X => by simp [SType.crename, SType.trename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_trename_comm E1 f g
have ih2 := EType.crename_trename_comm E2 f g
simp [SType.crename, SType.trename, ih1, ih2]
| STyp... | 3 | 20 | false | Type systems |
98 | Capless.SType.crename_id | theorem SType.crename_id {S : SType n m k} :
S.crename FinFun.id = S :=
match S with
| SType.top => by simp [SType.crename]
| SType.tvar X => by simp [SType.crename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_id (T := E1)
have ih2 := EType.crename_id (E := E2)
simp [SType.crename, ih1, i... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Basic",
"import Capless.Type.Renaming",
"import Capless.Type.Core",
"import Capless.CaptureSet"
] | [
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Nat",
"module": "Init.Prelude"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "CaptureSet.crename_id",
"content": "theorem CaptureSet.crename_id {C : CaptureSet n k} :\n C.crename FinFun.id = C"
},
{
"name": "FinFun.id_ext",
"content": "theorem FinFun.id_ext :\n (FinFun.ext (n := n) id) = id"
}
] | [] | [
{
"name": "Capless.CBound.crename_id",
"content": "theorem CBound.crename_id {b : CBound n k} :\n b.crename FinFun.id = b"
},
{
"name": "Capless.EType.crename_id",
"content": "theorem EType.crename_id {E : EType n m k} :\n E.crename FinFun.id = E"
},
{
"name": "Capless.CType.crename_id... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end
end
end
end
end
end
end | theorem SType.crename_id {S : SType n m k} :
S.crename FinFun.id = S := | :=
match S with
| SType.top => by simp [SType.crename]
| SType.tvar X => by simp [SType.crename]
| SType.forall E1 E2 => by
have ih1 := CType.crename_id (T := E1)
have ih2 := EType.crename_id (E := E2)
simp [SType.crename, ih1, ih2]
| SType.tforall S E => by
have ih1 := SType.crename_id (S := ... | 5 | 21 | false | Type systems |
99 | Capless.SType.trename_trename | theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :
(S.trename f).trename g = S.trename (g ∘ f) :=
match S with
| SType.top => by simp [SType.trename]
| SType.tvar X => by simp [SType.trename]
| SType.forall E1 E2 => by
have ih1 := CType.trename_trename E1 f g
have ... | capless-lean | Capless/Type/Basic.lean | [
"import Capless.Basic",
"import Capless.Type.Renaming",
"import Capless.Type.Core"
] | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
}
] | [
{
"name": "SType",
"content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "FinFun.ext_comp_ext",
"content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n g.ext ∘ f.ext = FinFun.ext (g ∘ f)"
}
] | [] | [
{
"name": "Capless.EType.trename_trename",
"content": "theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n (E.trename f).trename g = E.trename (g ∘ f)"
},
{
"name": "Capless.CType.trename_trename",
"content": "theorem CType.trename_trename (T : CType n m k)... | import Capless.Type.Core
import Capless.Type.Renaming
namespace Capless
end
end
end
end
end | theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :
(S.trename f).trename g = S.trename (g ∘ f) := | :=
match S with
| SType.top => by simp [SType.trename]
| SType.tvar X => by simp [SType.trename]
| SType.forall E1 E2 => by
have ih1 := CType.trename_trename E1 f g
have ih2 := EType.trename_trename E2 f g
simp [SType.trename, ih1, ih2]
| SType.tforall S E => by
have ih1 := SType.trename_trena... | 4 | 15 | false | Type systems |
100 | Capless.Typed.letex_inv' | theorem Typed.letex_inv' {Γ : Context n m k}
(he : t0 = Term.letex t u)
(h : Typed Γ t0 E Ct0) :
∃ T E0,
Typed Γ t (EType.ex T) Ct0 ∧
Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) u E0.cweaken.weaken Ct0.cweaken.weaken ∧
ESubtyp Γ E0 E | capless-lean | Capless/Inversion/Typing.lean | [
"import Capless.Subcapturing.Basic",
"import Capless.Tactics",
"import Capless.Typing",
"import Capless.Subtyping.Basic",
"import Capless.Weakening.Typing",
"import Capless.Weakening.Subtyping",
"import Capless.Inversion.Subtyping",
"import Capless.Narrowing",
"import Capless.Weakening.Subcapturing"... | [
{
"name": "Nat",
"module": "Init.Prelude"
},
{
"name": "Fin",
"module": "Init.Prelude"
},
{
"name": "Fin.succ",
"module": "Init.Data.Fin.Basic"
},
{
"name": "Exists",
"module": "Init.Core"
}
] | [
{
"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x",
"content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"
},
{
"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c",
"content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"
},
{
"name"... | [
{
"name": "...",
"module": ""
}
] | [
{
"name": "ESubtyp.cweaken",
"content": "theorem ESubtyp.cweaken\n (h : ESubtyp Γ E1 E2) :\n ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"
},
{
"name": "CSubtyp.weaken",
"content": "theorem CSubtyp.weaken\n (h : CSubtyp Γ E1 E2) :\n CSubtyp (Γ.var T) E1.weaken E2.weaken"
},
{
"name": "SS... | [] | [] | import Capless.Tactics
import Capless.Typing
import Capless.Subtyping.Basic
import Capless.Subcapturing.Basic
import Capless.Narrowing
import Capless.Weakening.Subcapturing
import Capless.Inversion.Context
import Capless.Inversion.Subtyping
namespace Capless | theorem Typed.letex_inv' {Γ : Context n m k}
(he : t0 = Term.letex t u)
(h : Typed Γ t0 E Ct0) :
∃ T E0,
Typed Γ t (EType.ex T) Ct0 ∧
Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) u E0.cweaken.weaken Ct0.cweaken.weaken ∧
ESubtyp Γ E0 E := | := by
induction h <;> try (solve | cases he)
case letex =>
cases he
repeat apply Exists.intro
constructor; trivial
constructor; trivial
apply ESubtyp.refl
case sub hs ih =>
have ih := ih he
obtain ⟨T, E0, ht, hu, hs0⟩ := ih
have hs1 := ESubtyp.trans hs0 hs
repeat apply Exists.i... | 3 | 92 | false | Type systems |
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