fact stringlengths 6 1.53k | type stringclasses 16
values | library stringclasses 5
values | imports listlengths 1 79 | filename stringclasses 205
values | symbolic_name stringlengths 1 55 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
lit_is_unboxed(l: base_lit) : Prop :=
match l with | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | lit_is_unboxed | |
val_is_unboxed(v : val) : Prop :=
match v with
| LitV l => lit_is_unboxed l
| InjLV (LitV l) => lit_is_unboxed l
| InjRV (LitV l) => lit_is_unboxed l
| _ => False
end. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | val_is_unboxed | |
Instancelit_is_unboxed_dec l : Decision (lit_is_unboxed l).
Proof. destruct l; simpl; exact (decide _). Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceval_is_unboxed_dec v : Decision (val_is_unboxed v).
Proof. destruct v as [ | | | [] | [] ]; simpl; exact (decide _). Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
vals_compare_safe(vl v1 : val) : Prop :=
val_is_unboxed vl ∨ val_is_unboxed v1. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | vals_compare_safe | |
Argumentsvals_compare_safe !_ !_ /. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Arguments | |
state: Type := {
heap: gmap loc (option val);
used_proph_id: gset proph_id;
}. | Record | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | state | |
to_of_valv : to_val (of_val v) = Some v.
Proof. by destruct v. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | to_of_val | |
of_to_vale v : to_val e = Some v → of_val v = e.
Proof. destruct e=>//=. by intros [= <-]. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | of_to_val | |
Instanceof_val_inj : Inj (=) (=) of_val.
Proof. intros ??. congruence. Qed. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instancebase_lit_eq_dec : EqDecision base_lit.
Proof. solve_decision. Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceun_op_eq_dec : EqDecision un_op.
Proof. solve_decision. Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instancebin_op_eq_dec : EqDecision bin_op.
Proof. solve_decision. Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceexpr_eq_dec : EqDecision expr.
Proof.
refine (
fix go (e1 e2 : expr) {struct e1} : Decision (e1 = e2) :=
match e1, e2 with
| Val v, Val v' => cast_if (decide (v = v'))
| Var x, Var x' => cast_if (decide (x = x'))
| Rec f x e, Rec f' x' e' =>
cast_if_and3 (decide (f = f')) (decid... | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceval_eq_dec : EqDecision val.
Proof. solve_decision. Defined. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instancebase_lit_countable : Countable base_lit.
Proof.
refine (inj_countable' (λ l, match l with
| LitInt n => (inl (inl n), None)
| LitBool b => (inl (inr b), None)
| LitUnit => (inr (inl false), None)
| LitPoison => (inr (inl true), None)
| LitLoc l => (inr (inr l), None)
| LitProphecy p => (inr (inl fa... | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceun_op_finite : Countable un_op.
Proof.
refine (inj_countable' (λ op, match op with NegOp => 0 | MinusUnOp => 1 end)
(λ n, match n with 0 => NegOp | _ => MinusUnOp end) _); by intros [].
Qed. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instancebin_op_countable : Countable bin_op.
Proof.
refine (inj_countable' (λ op, match op with
| PlusOp => 0 | MinusOp => 1 | MultOp => 2 | QuotOp => 3 | RemOp => 4
| AndOp => 5 | OrOp => 6 | XorOp => 7 | ShiftLOp => 8 | ShiftROp => 9
| LeOp => 10 | LtOp => 11 | EqOp => 12 | OffsetOp => 13
end) (λ n, match n ... | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceexpr_countable : Countable expr.
Proof.
set (enc :=
fix go e :=
match e with
| Val v => GenNode 0 [gov v]
| Var x => GenLeaf (inl (inl x))
| Rec f x e => GenNode 1 [GenLeaf (inl (inr f)); GenLeaf (inl (inr x)); go e]
| App e1 e2 => GenNode 2 [go e1; go e2]
| UnOp op e => GenNod... | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceval_countable : Countable val.
Proof. refine (inj_countable of_val to_val _); auto using to_of_val. Qed. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instancestate_inhabited : Inhabited state :=
populate {| heap := inhabitant; used_proph_id := inhabitant |}. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceval_inhabited : Inhabited val := populate (LitV LitUnit). | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
Instanceexpr_inhabited : Inhabited expr := populate (Val inhabitant).
Canonical Structure stateO := leibnizO state.
Canonical Structure locO := leibnizO loc.
Canonical Structure valO := leibnizO val.
Canonical Structure exprO := leibnizO expr. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
ectx_item:=
| AppLCtx (v2 : val)
| AppRCtx (e1 : expr)
| UnOpCtx (op : un_op)
| BinOpLCtx (op : bin_op) (v2 : val)
| BinOpRCtx (op : bin_op) (e1 : expr)
| IfCtx (e1 e2 : expr)
| PairLCtx (v2 : val)
| PairRCtx (e1 : expr)
| FstCtx
| SndCtx
| InjLCtx
| InjRCtx
| CaseCtx (e1 : expr) (e2 : expr)
... | Inductive | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | ectx_item | |
fill_item(Ki : ectx_item) (e : expr) : expr :=
match Ki with
| AppLCtx v2 => App e (of_val v2)
| AppRCtx e1 => App e1 e
| UnOpCtx op => UnOp op e
| BinOpLCtx op v2 => BinOp op e (Val v2)
| BinOpRCtx op e1 => BinOp op e1 e
| IfCtx e1 e2 => If e e1 e2
| PairLCtx v2 => Pair e (Val v2)
| PairRCtx e1 => Pa... | Fixpoint | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | fill_item | |
subst(x : string) (v : val) (e : expr) : expr :=
match e with
| Val _ => e
| Var y => if decide (x = y) then Val v else Var y
| Rec f y e =>
Rec f y $ if decide (BNamed x ≠ f ∧ BNamed x ≠ y) then subst x v e else e
| App e1 e2 => App (subst x v e1) (subst x v e2)
| UnOp op e => UnOp op (subst x v e)
... | Fixpoint | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | subst | |
subst'(mx : binder) (v : val) : expr → expr :=
match mx with BNamed x => subst x v | BAnon => id end. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | subst' | |
un_op_eval(op : un_op) (v : val) : option val :=
match op, v with
| NegOp, LitV (LitBool b) => Some $ LitV $ LitBool (negb b)
| NegOp, LitV (LitInt n) => Some $ LitV $ LitInt (Z.lnot n)
| MinusUnOp, LitV (LitInt n) => Some $ LitV $ LitInt (- n)
| _, _ => None
end. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | un_op_eval | |
bin_op_eval_int(op : bin_op) (n1 n2 : Z) : option base_lit :=
match op with
| PlusOp => Some $ LitInt (n1 + n2)
| MinusOp => Some $ LitInt (n1 - n2)
| MultOp => Some $ LitInt (n1 * n2)
| QuotOp => Some $ LitInt (n1 `quot` n2)
| RemOp => Some $ LitInt (n1 `rem` n2)
| AndOp => Some $ LitInt (Z.land n1 n2)
... | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | bin_op_eval_int | |
bin_op_eval_bool(op : bin_op) (b1 b2 : bool) : option base_lit :=
match op with
| PlusOp | MinusOp | MultOp | QuotOp | RemOp => None
| AndOp => Some (LitBool (b1 && b2))
| OrOp => Some (LitBool (b1 || b2))
| XorOp => Some (LitBool (xorb b1 b2))
| ShiftLOp | ShiftROp => None
| LeOp | LtOp => None
| EqOp ... | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | bin_op_eval_bool | |
bin_op_eval_loc(op : bin_op) (l1 : loc) (v2 : base_lit) : option base_lit :=
match op, v2 with
| OffsetOp, LitInt off => Some $ LitLoc (l1 +ₗ off)
| LeOp, LitLoc l2 => Some $ LitBool (bool_decide (l1 ≤ₗ l2))
| LtOp, LitLoc l2 => Some $ LitBool (bool_decide (l1 <ₗ l2))
| _, _ => None
end. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | bin_op_eval_loc | |
bin_op_eval(op : bin_op) (v1 v2 : val) : option val :=
if decide (op = EqOp) then | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | bin_op_eval | |
state_upd_heap(f : gmap loc (option val) → gmap loc (option val))
(σ : state) : state :=
{| heap := f σ.(heap); used_proph_id := σ.(used_proph_id) |}. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | state_upd_heap | |
Argumentsstate_upd_heap _ !_ /. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Arguments | |
state_upd_used_proph_id(f : gset proph_id → gset proph_id)
(σ : state) : state :=
{| heap := σ.(heap); used_proph_id := f σ.(used_proph_id) |}. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | state_upd_used_proph_id | |
Argumentsstate_upd_used_proph_id _ !_ /. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Arguments | |
heap_array(l : loc) (vs : list val) : gmap loc (option val) :=
match vs with
| [] => ∅
| v :: vs' => {[l := Some v]} ∪ heap_array (l +ₗ 1) vs'
end. | Fixpoint | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | heap_array | |
heap_array_singletonl v : heap_array l [v] = {[l := Some v]}.
Proof. by rewrite /heap_array right_id. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | heap_array_singleton | |
heap_array_lookupl vs ow k :
heap_array l vs !! k = Some ow ↔
∃ j w, (0 ≤ j)%Z ∧ k = l +ₗ j ∧ ow = Some w ∧ vs !! (Z.to_nat j) = Some w.
Proof.
revert k l; induction vs as [|v' vs IH]=> l' l /=.
{ rewrite lookup_empty. naive_solver lia. }
rewrite -insert_union_singleton_l lookup_insert_Some IH. split.
- int... | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | heap_array_lookup | |
heap_array_map_disjoint(h : gmap loc (option val)) (l : loc) (vs : list val) :
(∀ i, (0 ≤ i)%Z → (i < length vs)%Z → h !! (l +ₗ i) = None) →
heap_array l vs ##ₘ h.
Proof.
intros Hdisj. apply map_disjoint_spec=> l' v1 v2.
intros (j&w&?&->&?&Hj%lookup_lt_Some%inj_lt)%heap_array_lookup.
move: Hj. rewrite Z2Nat.i... | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | heap_array_map_disjoint | |
state_init_heap(l : loc) (n : Z) (v : val) (σ : state) : state :=
state_upd_heap (λ h, heap_array l (replicate (Z.to_nat n) v) ∪ h) σ. | Definition | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | state_init_heap | |
state_init_heap_singletonl v σ :
state_init_heap l 1 v σ = state_upd_heap <[l:=Some v]> σ.
Proof.
destruct σ as [h p]. rewrite /state_init_heap /=. f_equiv.
rewrite right_id insert_union_singleton_l. done.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | state_init_heap_singleton | |
base_step: expr → state → list observation → expr → state → list expr → Prop :=
| RecS f x e σ :
base_step (Rec f x e) σ [] (Val $ RecV f x e) σ []
| PairS v1 v2 σ :
base_step (Pair (Val v1) (Val v2)) σ [] (Val $ PairV v1 v2) σ []
| InjLS v σ :
base_step (InjL $ Val v) σ [] (Val $ InjLV v) σ []
|... | Inductive | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | base_step | |
Instancefill_item_inj Ki : Inj (=) (=) (fill_item Ki).
Proof. induction Ki; intros ???; simplify_eq/=; auto with f_equal. Qed. | Global | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | Instance | |
fill_item_valKi e :
is_Some (to_val (fill_item Ki e)) → is_Some (to_val e).
Proof. intros [v ?]. induction Ki; simplify_option_eq; eauto. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | fill_item_val | |
val_base_stucke1 σ1 κ e2 σ2 efs : base_step e1 σ1 κ e2 σ2 efs → to_val e1 = None.
Proof. destruct 1; naive_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | val_base_stuck | |
base_ctx_step_valKi e σ1 κ e2 σ2 efs :
base_step (fill_item Ki e) σ1 κ e2 σ2 efs → is_Some (to_val e).
Proof. revert κ e2. induction Ki; inversion_clear 1; simplify_option_eq; eauto. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | base_ctx_step_val | |
fill_item_no_val_injKi1 Ki2 e1 e2 :
to_val e1 = None → to_val e2 = None →
fill_item Ki1 e1 = fill_item Ki2 e2 → Ki1 = Ki2.
Proof.
revert Ki1. induction Ki2; intros Ki1; induction Ki1; naive_solver eauto with f_equal.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | fill_item_no_val_inj | |
alloc_freshv n σ :
let l := Loc.fresh (dom σ.(heap)) in
(0 < n)%Z →
base_step (AllocN ((Val $ LitV $ LitInt $ n)) (Val v)) σ []
(Val $ LitV $ LitLoc l) (state_init_heap l n v σ) [].
Proof.
intros.
apply AllocNS; first done.
intros. apply not_elem_of_dom.
by apply Loc.fresh_fresh.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | alloc_fresh | |
new_proph_id_freshσ :
let p := fresh σ.(used_proph_id) in
base_step NewProph σ [] (Val $ LitV $ LitProphecy p) (state_upd_used_proph_id ({[ p ]} ∪.) σ) [].
Proof. constructor. apply is_fresh. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | new_proph_id_fresh | |
heap_lang_mixin: EctxiLanguageMixin of_val to_val fill_item base_step.
Proof.
split; apply _ || eauto using to_of_val, of_to_val, val_base_stuck,
fill_item_val, fill_item_no_val_inj, base_ctx_step_val.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | heap_lang_mixin | |
to_val_fill_someK e v : to_val (fill K e) = Some v → K = [] ∧ e = Val v.
Proof.
intro H. destruct K as [|Ki K]; first by apply of_to_val in H. exfalso.
assert (to_val e ≠ None) as He.
{ intro A. by rewrite fill_not_val in H. }
assert (∃ w, e = Val w) as [w ->].
{ destruct e; try done; eauto. }
assert (to_va... | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | to_val_fill_some | |
prim_step_to_val_is_base_stepe σ1 κs w σ2 efs :
prim_step e σ1 κs (Val w) σ2 efs → base_step e σ1 κs (Val w) σ2 efs.
Proof.
intro H. destruct H as [K e1 e2 H1 H2].
assert (to_val (fill K e2) = Some w) as H3; first by rewrite -H2.
apply to_val_fill_some in H3 as [-> ->]. subst e. done.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | prim_step_to_val_is_base_step | |
base_step_to_vale1 σ1 κ e2 σ2 efs σ1' κ' e2' σ2' efs' :
base_step e1 σ1 κ e2 σ2 efs →
base_step e1 σ1' κ' e2' σ2' efs' → is_Some (to_val e2) → is_Some (to_val e2').
Proof. destruct 1; inversion 1; naive_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | base_step_to_val | |
irreducible_resolvee v1 v2 σ :
irreducible e σ → irreducible (Resolve e (Val v1) (Val v2)) σ.
Proof.
intros H κs ? σ' efs [Ks e1' e2' Hfill -> step]. simpl in *.
induction Ks as [|K Ks _] using rev_ind; simpl in Hfill.
- subst e1'. inversion step. eapply H. by apply base_prim_step.
- rewrite fill_app /= in Hf... | Lemma | iris_heap_lang | [
"From stdpp Require Export binders strings",
"From stdpp Require Import gmap",
"From iris.algebra Require Export ofe",
"From iris.program_logic Require Export language ectx_language ectxi_language",
"From iris.heap_lang Require Export locations",
"From iris.prelude Require Import options"
] | iris_heap_lang/lang.v | irreducible_resolve | |
loc:= Loc { loc_car : Z }.
Add Printing Constructor loc. | Record | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | loc | |
OpenScope Z_scope. | Local | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Open | |
eq_specl1 l2 : l1 = l2 ↔ loc_car l1 = loc_car l2.
Proof. destruct l1, l2; naive_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | eq_spec | |
Instanceeq_dec : EqDecision loc.
Proof. solve_decision. Defined. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instanceinhabited : Inhabited loc := populate {|loc_car := 0 |}. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instancecountable : Countable loc.
Proof. by apply (inj_countable' loc_car Loc); intros []. Defined. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
ProgramInstance infinite : Infinite loc :=
inj_infinite (λ p, {| loc_car := p |}) (λ l, Some (loc_car l)) _.
Next Obligation. done. Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Program | |
add(l : loc) (off : Z) : loc :=
{| loc_car := loc_car l + off|}. | Definition | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | add | |
le(l1 l2 : loc) : Prop := loc_car l1 ≤ loc_car l2. | Definition | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | le | |
lt(l1 l2 : loc) : Prop := loc_car l1 < loc_car l2. | Definition | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | lt | |
add_assocl i j : l +ₗ i +ₗ j = l +ₗ (i + j).
Proof. rewrite eq_spec /=. lia. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | add_assoc | |
add_0l : l +ₗ 0 = l.
Proof. rewrite eq_spec /=; lia. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | add_0 | |
Instanceadd_inj l : Inj eq eq (add l).
Proof. intros x1 x2. rewrite eq_spec /=. lia. Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instancele_dec l1 l2 : Decision (l1 ≤ₗ l2).
Proof. rewrite /le. apply _. Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instancelt_dec l1 l2 : Decision (l1 <ₗ l2).
Proof. rewrite /lt. apply _. Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instancele_po : PartialOrder le.
Proof.
rewrite /le. split; [split|].
- by intros ?.
- intros [x] [y] [z]; lia.
- intros [x] [y] ??; f_equal/=; lia.
Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
Instancele_total : Total le.
Proof. rewrite /Total /le. lia. Qed. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Instance | |
le_ngtl1 l2 : l1 ≤ₗ l2 ↔ ¬l2 <ₗ l1.
Proof. apply Z.le_ngt. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | le_ngt | |
le_lteql1 l2 : l1 ≤ₗ l2 ↔ l1 <ₗ l2 ∨ l1 = l2.
Proof. rewrite eq_spec. apply Z.le_lteq. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | le_lteq | |
add_le_monol1 l2 i1 i2 :
l1 ≤ₗ l2 → i1 ≤ i2 → l1 +ₗ i1 ≤ₗ l2 +ₗ i2.
Proof. apply Z.add_le_mono. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | add_le_mono | |
fresh(ls : gset loc) : loc :=
{| loc_car := set_fold (λ k r, (1 + loc_car k) `max` r) 1 ls |}. | Definition | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | fresh | |
fresh_freshls i : 0 ≤ i → fresh ls +ₗ i ∉ ls.
Proof.
intros Hi. cut (∀ l, l ∈ ls → loc_car l < loc_car (fresh ls) + i).
{ intros help Hf%help. simpl in *. lia. }
apply (set_fold_ind_L (λ r ls, ∀ l, l ∈ ls → (loc_car l < r + i)));
set_solver by eauto with lia.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | fresh_fresh | |
Opaquefresh. | Global | iris_heap_lang | [
"From stdpp Require Import countable numbers gmap",
"From iris.prelude Require Export prelude",
"From iris.prelude Require Import options"
] | iris_heap_lang/locations.v | Opaque | |
Definitionset_binder_insert (x : binder) (X : stringset) : stringset :=
match x with
| BAnon => X
| BNamed f => {[f]} ∪ X
end. | Local | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | Definition | |
is_closed_expr(X : stringset) (e : expr) : bool :=
match e with
| Val v => is_closed_val v
| Var x => bool_decide (x ∈ X)
| Rec f x e => is_closed_expr (set_binder_insert f (set_binder_insert x X)) e
| UnOp _ e | Fst e | Snd e | InjL e | InjR e | Fork e | Free e | Load e =>
is_closed_expr X e
| App e1 ... | Fixpoint | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | is_closed_expr | |
subst_map(vs : gmap string val) (e : expr) : expr :=
match e with
| Val _ => e
| Var y => if vs !! y is Some v then Val v else Var y
| Rec f y e => Rec f y (subst_map (binder_delete y (binder_delete f vs)) e)
| App e1 e2 => App (subst_map vs e1) (subst_map vs e2)
| UnOp op e => UnOp op (subst_map vs e)
| ... | Fixpoint | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_map | |
Instanceset_unfold_elem_of_insert_binder x y X Q :
SetUnfoldElemOf y X Q →
SetUnfoldElemOf y (set_binder_insert x X) (Q ∨ BNamed y = x).
Proof. destruct 1; constructor; destruct x; set_solver. Qed. | Local | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | Instance | |
is_closed_weakenX Y e : is_closed_expr X e → X ⊆ Y → is_closed_expr Y e.
Proof. revert X Y; induction e; naive_solver (eauto; set_solver). Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | is_closed_weaken | |
is_closed_weaken_emptyX e : is_closed_expr ∅ e → is_closed_expr X e.
Proof. intros. by apply is_closed_weaken with ∅, empty_subseteq. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | is_closed_weaken_empty | |
is_closed_substX e y v :
is_closed_val v →
is_closed_expr ({[y]} ∪ X) e →
is_closed_expr X (subst y v e).
Proof.
intros Hv. revert X.
induction e=> X /= ?; destruct_and?; split_and?; simplify_option_eq;
try match goal with
| H : ¬(_ ∧ _) |- _ => apply not_and_l in H as [?%dec_stable|?%dec_stable]
... | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | is_closed_subst | |
is_closed_subst'X e x v :
is_closed_val v →
is_closed_expr (set_binder_insert x X) e →
is_closed_expr X (subst' x v e).
Proof. destruct x; eauto using is_closed_subst. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | is_closed_subst' | |
subst_is_closedX e x es : is_closed_expr X e → x ∉ X → subst x es e = e.
Proof.
revert X. induction e=> X /=;
rewrite ?bool_decide_spec ?andb_True=> ??;
repeat case_decide; simplify_eq/=; f_equal; intuition eauto with set_solver.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_is_closed | |
subst_is_closed_emptye x v : is_closed_expr ∅ e → subst x v e = e.
Proof. intros. apply subst_is_closed with (∅:stringset); set_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_is_closed_empty | |
subst_subste x v v' :
subst x v (subst x v' e) = subst x v' e.
Proof.
intros. induction e; simpl; try (f_equal; by auto);
simplify_option_eq; auto using subst_is_closed_empty with f_equal.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_subst | |
subst_subst'e x v v' :
subst' x v (subst' x v' e) = subst' x v' e.
Proof. destruct x; simpl; auto using subst_subst. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_subst' | |
subst_subst_nee x y v v' :
x ≠ y → subst x v (subst y v' e) = subst y v' (subst x v e).
Proof.
intros. induction e; simpl; try (f_equal; by auto);
simplify_option_eq; auto using eq_sym, subst_is_closed_empty with f_equal.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_subst_ne | |
subst_subst_ne'e x y v v' :
x ≠ y → subst' x v (subst' y v' e) = subst' y v' (subst' x v e).
Proof. destruct x, y; simpl; auto using subst_subst_ne with congruence. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_subst_ne' | |
subst_rec'f y e x v :
x = f ∨ x = y ∨ x = BAnon →
subst' x v (Rec f y e) = Rec f y e.
Proof. intros. destruct x; simplify_option_eq; naive_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_rec' | |
subst_rec_ne'f y e x v :
(x ≠ f ∨ f = BAnon) → (x ≠ y ∨ y = BAnon) →
subst' x v (Rec f y e) = Rec f y (subst' x v e).
Proof. intros. destruct x; simplify_option_eq; naive_solver. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_rec_ne' | |
bin_op_eval_closedop v1 v2 v' :
is_closed_val v1 → is_closed_val v2 → bin_op_eval op v1 v2 = Some v' →
is_closed_val v'.
Proof.
rewrite /bin_op_eval /bin_op_eval_bool /bin_op_eval_int /bin_op_eval_loc;
repeat case_match; by naive_solver.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | bin_op_eval_closed | |
heap_closed_allocσ l n w :
(0 < n)%Z →
is_closed_val w →
map_Forall (λ _ v, from_option is_closed_val true v) (heap σ) →
(∀ i : Z, (0 ≤ i)%Z → (i < n)%Z → heap σ !! (l +ₗ i) = None) →
map_Forall (λ _ v, from_option is_closed_val true v)
(heap_array l (replicate (Z.to_nat n) w) ∪ heap σ).
Proof.
... | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | heap_closed_alloc | |
base_step_is_closede1 σ1 obs e2 σ2 es :
is_closed_expr ∅ e1 →
map_Forall (λ _ v, from_option is_closed_val true v) σ1.(heap) →
base_step e1 σ1 obs e2 σ2 es →
is_closed_expr ∅ e2 ∧ Forall (is_closed_expr ∅) es ∧
map_Forall (λ _ v, from_option is_closed_val true v) σ2.(heap).
Proof.
intros Cl1 Clσ1 STEP.
in... | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | base_step_is_closed | |
subst_map_emptye : subst_map ∅ e = e.
Proof.
assert (∀ x, binder_delete x (∅:gmap string val) = ∅) as Hdel.
{ intros [|x]; by rewrite /= ?delete_empty. }
induction e; simplify_map_eq; rewrite ?Hdel; auto with f_equal.
Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_map_empty | |
subst_map_insertx v vs e :
subst_map (<[x:=v]>vs) e = subst x v (subst_map (delete x vs) e).
Proof.
revert vs. induction e=> vs; simplify_map_eq; auto with f_equal.
- match goal with
| |- context [ <[?x:=_]> _ !! ?y ] =>
destruct (decide (x = y)); simplify_map_eq=> //
end. by case (vs !! _); simpli... | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_map_insert | |
subst_map_singletonx v e :
subst_map {[x:=v]} e = subst x v e.
Proof. by rewrite subst_map_insert delete_empty subst_map_empty. Qed. | Lemma | iris_heap_lang | [
"From stdpp Require Import gmap stringmap",
"From iris.heap_lang Require Export lang",
"From iris.prelude Require Import options"
] | iris_heap_lang/metatheory.v | subst_map_singleton |
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