Datasets:

phanerozoic
/

Coq-Stdpp

fact stringlengths 9 1.52k	type stringclasses 12 values	library stringclasses 3 values	imports listlengths 2 11	filename stringclasses 54 values	symbolic_name stringlengths 1 44	docstring stringclasses 420 values
#[projections(primitive=yes)] Record seal {A} (f : A) := { unseal : A; seal_eq : unseal = f }.	Record	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	seal	* Sealing off definitions
TCNoBackTrack (P : Prop) := TCNoBackTrack_intro { tc_no_backtrack : P }.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCNoBackTrack	The type class [TCNoBackTrack P] can be used to establish [P] without ever backtracking on the instance of [P] that has been found. Backtracking may normally happen when [P] contains evars that could be instanciated in different ways depending on which instance is picked, and type class search somewhere else depends on...
TCIf (P Q R : Prop) : Prop := \| TCIf_true : P → Q → TCIf P Q R \| TCIf_false : R → TCIf P Q R.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCIf	The type class [TCNoBackTrack P] can be used to establish [P] without ever backtracking on the instance of [P] that has been found. Backtracking may normally happen when [P] contains evars that could be instanciated in different ways depending on which instance is picked, and type class search somewhere else depends on...
Class TCIf.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
tc_opaque {A} (x : A) : A := x.	Definition	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	tc_opaque	The constant [tc_opaque] is used to make definitions opaque for just type class search. Note that [simpl] is set up to always unfold [tc_opaque].
Class or.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Class and.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
TCOr (P1 P2 : Prop) : Prop := \| TCOr_l : P1 → TCOr P1 P2 \| TCOr_r : P2 → TCOr P1 P2.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCOr	Below we define type class versions of the common logical operators. It is important to note that we duplicate the definitions, and do not declare the existing logical operators as type classes. That is, we do not say: Existing Class or. Existing Class and. If we could define the existing logical operators as cla...
Class TCOr.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCOr_l \| 9.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
Instance TCOr_r \| 10.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1 → P2 → TCAnd P1 P2.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCAnd	null
Class TCAnd.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCAnd_intro.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCTrue : Prop := TCTrue_intro : TCTrue.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCTrue	null
Class TCTrue.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCTrue_intro.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCFalse : Prop :=.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCFalse	The class [TCFalse] is not stricly necessary as one could also use [False]. However, users might expect that TCFalse exists and if it does not, it can cause hard to diagnose bugs due to automatic generalization.
Class TCFalse.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
TCForall {A} (P : A → Prop) : list A → Prop := \| TCForall_nil : TCForall P [] \| TCForall_cons x xs : P x → TCForall P xs → TCForall P (x :: xs).	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCForall	null
Class TCForall.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCForall_nil.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
Instance TCForall_cons.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCForall2 {A B} (P : A → B → Prop) : list A → list B → Prop := \| TCForall2_nil : TCForall2 P [] [] \| TCForall2_cons x y xs ys : P x y → TCForall2 P xs ys → TCForall2 P (x :: xs) (y :: ys).	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCForall2	The class [TCForall2 P l k] is commonly used to transform an input list [l] into an output list [k], or the converse. Therefore there are two modes, either [l] input and [k] output, or [k] input and [l] input.
Class TCForall2.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCForall2_nil.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
Instance TCForall2_cons.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCExists {A} (P : A → Prop) : list A → Prop := \| TCExists_cons_hd x l : P x → TCExists P (x :: l) \| TCExists_cons_tl x l: TCExists P l → TCExists P (x :: l).	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCExists	null
Class TCExists.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCExists_cons_hd \| 10.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
Instance TCExists_cons_tl \| 20.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCElemOf {A} (x : A) : list A → Prop := \| TCElemOf_here xs : TCElemOf x (x :: xs) \| TCElemOf_further y xs : TCElemOf x xs → TCElemOf x (y :: xs).	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCElemOf	null
Class TCElemOf.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCElemOf_here.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
Instance TCElemOf_further.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCEq {A} (x : A) : A → Prop := TCEq_refl : TCEq x x.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCEq	The intended use of [TCEq x y] is to use [x] as input and [y] as output, but this is not enforced. We use output mode [-] (instead of [!]) for [x] to ensure that type class search succeed on goals like [TCEq (if ? then e1 else e2) ?y], see https://gitlab.mpi-sws.org/iris/iris/merge_requests/391 for a use case. Mode [-]...
Class TCEq.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCEq_refl.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
TCEq_eq {A} (x1 x2 : A) : TCEq x1 x2 ↔ x1 = x2.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCEq_eq	null
TCSimpl {A} (x x' : A) := TCSimpl_TCEq : TCEq x x'.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCSimpl	The [TCSimpl x y] type class is similar to [TCEq] but performs [simpl] before proving the goal by reflexivity. Similar to [TCEq], the argument [x] is the input and [y] the output. When solving [TCEq x y], the argument [x] should be a concrete term and [y] an evar for the [simpl]ed result.
TCSimpl_eq {A} (x1 x2 : A) : TCSimpl x1 x2 ↔ x1 = x2.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCSimpl_eq	null
TCDiag {A} (C : A → Prop) : A → A → Prop := \| TCDiag_diag x : C x → TCDiag C x x.	Inductive	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TCDiag	null
Class TCDiag.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Class	null
Instance TCDiag_diag.	Existing	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Instance	null
tc_to_bool (P : Prop) {p : bool} `{TCIf P (TCEq p true) (TCEq p false)} : bool := p.	Definition	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	tc_to_bool	Given a proposition [P] that is a type class, [tc_to_bool P] will return [true] iff there is an instance of [P]. It is often useful in Ltac programming, where one can do [lazymatch tc_to_bool P with true => .. \| false => .. end].
Equiv A := equiv: relation A.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Equiv	We define an operational type class for setoid equality, i.e., the "canonical" equivalence for a type. The typeclass is tied to the \equiv symbol. This is based on (Spitters/van der Weegen, 2011).
equiv_rewrite_relation `{Equiv A} : RewriteRelation (@equiv A _) \| 150 := {}.	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	equiv_rewrite_relation	We instruct setoid rewriting to infer [equiv] as a relation on type [A] when needed. This allows setoid_rewrite to solve constraints of shape [Proper (eq ==> ?R) f] using [Proper (eq ==> (equiv (A:=A))) f] when an equivalence relation is available on type [A]. We put this instance at level 150 so it does not take prece...
LeibnizEquiv A `{Equiv A} := leibniz_equiv (x y : A) : x ≡ y → x = y.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	LeibnizEquiv	The type class [LeibnizEquiv] collects setoid equalities that coincide with Leibniz equality. We provide the tactic [fold_leibniz] to transform such setoid equalities into Leibniz equalities, and [unfold_leibniz] for the reverse. Various std++ tactics assume that this class is only instantiated if [≡] is an equivalenc...
leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (≡@{A})} (x y : A) : x ≡ y ↔ x = y.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	leibniz_equiv_iff	null
equivL {A} : Equiv A := (=).	Definition	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	equivL	null
equiv_default_relation `{Equiv A} : DefaultRelation (≡@{A}) \| 3 := {}.	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	equiv_default_relation	The following instance forces [setoid_replace] to use setoid equality (for types that have an [Equiv] instance) rather than the standard Leibniz equality.
Decision (P : Prop) := decide : {P} + {¬P}.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Decision	This type class by (Spitters/van der Weegen, 2011) collects decidable propositions.
RelDecision {A B} (R : A → B → Prop) := decide_rel x y :: Decision (R x y).	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	RelDecision	Although [RelDecision R] is just [∀ x y, Decision (R x y)], we make this an explicit class instead of a notation for two reasons: - It allows us to control [Hint Mode] more precisely. In particular, if it were defined as a notation, the above [Hint Mode] for [Decision] would not prevent diverging instance search w...
Inhabited (A : Type) : Type := populate { inhabitant : A }.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Inhabited	This type class collects types that are inhabited.
ProofIrrel (A : Type) : Prop := proof_irrel (x y : A) : x = y.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	ProofIrrel	This type class collects types that are proof irrelevant. That means, all elements of the type are equal. We use this notion only used for propositions, but by universe polymorphism we can generalize it.
Inj {A B} (R : relation A) (S : relation B) (f : A → B) : Prop := inj x y : S (f x) (f y) → R x y.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Inj	These operational type classes allow us to refer to common mathematical properties in a generic way. For example, for injectivity of [(k ++.)] it allows us to write [inj (k ++.)] instead of [app_inv_head k].
Inj2 {A B C} (R1 : relation A) (R2 : relation B) (S : relation C) (f : A → B → C) : Prop := inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2) → R1 x1 y1 ∧ R2 x2 y2.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Inj2	null
Cancel {A B} (S : relation B) (f : A → B) (g : B → A) : Prop := cancel x : S (f (g x)) x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Cancel	null
Surj {A B} (R : relation B) (f : A → B) := surj y : ∃ x, R (f x) y.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Surj	null
IdemP {A} (R : relation A) (f : A → A → A) : Prop := idemp x : R (f x x) x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	IdemP	null
Comm {A B} (R : relation A) (f : B → B → A) : Prop := comm x y : R (f x y) (f y x).	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Comm	null
LeftId {A} (R : relation A) (i : A) (f : A → A → A) : Prop := left_id x : R (f i x) x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	LeftId	null
RightId {A} (R : relation A) (i : A) (f : A → A → A) : Prop := right_id x : R (f x i) x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	RightId	null
Assoc {A} (R : relation A) (f : A → A → A) : Prop := assoc x y z : R (f x (f y z)) (f (f x y) z).	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Assoc	null
LeftAbsorb {A} (R : relation A) (i : A) (f : A → A → A) : Prop := left_absorb x : R (f i x) i.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	LeftAbsorb	null
RightAbsorb {A} (R : relation A) (i : A) (f : A → A → A) : Prop := right_absorb x : R (f x i) i.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	RightAbsorb	null
AntiSymm {A} (R S : relation A) : Prop := anti_symm x y : S x y → S y x → R x y.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	AntiSymm	null
Total {A} (R : relation A) := total x y : R x y ∨ R y x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Total	null
Trichotomy {A} (R : relation A) := trichotomy x y : R x y ∨ x = y ∨ R y x.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	Trichotomy	null
TrichotomyT {A} (R : relation A) := trichotomyT x y : {R x y} + {x = y} + {R y x}.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TrichotomyT	null
involutive {A} {R : relation A} (f : A → A) `{Involutive R f} x : R (f (f x)) x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	involutive	null
not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y → ¬R y x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	not_symmetry	null
symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y ↔ R y x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	symmetry_iff	null
not_inj `{Inj A B R R' f} x y : ¬R x y → ¬R' (f x) (f y).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	not_inj	null
not_inj2_1 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 : ¬R x1 x2 → ¬R'' (f x1 y1) (f x2 y2).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	not_inj2_1	null
not_inj2_2 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 : ¬R' y1 y2 → ¬R'' (f x1 y1) (f x2 y2).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	not_inj2_2	null
inj_iff {A B} {R : relation A} {S : relation B} (f : A → B) `{!Inj R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y) ↔ R x y.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	inj_iff	null
inj2_inj_1 `{Inj2 A B C R1 R2 R3 f} y : Inj R1 R3 (λ x, f x y).	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	inj2_inj_1	null
inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	inj2_inj_2	null
cancel_inj `{Cancel A B R1 f g, !Equivalence R1, !Proper (R2 ==> R1) f} : Inj R1 R2 g.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	cancel_inj	Smart constructors for [Inj] and [Surj] based on [Cancel]. These are not instances because they will blow-up the search space due to diamonds: in every node of the search tree for [Inj] or [Surj] of composed functions these instances could be applied. Note that [f] and [g] do not need to be given, these are infered usi...
cancel_surj `{Cancel A B R1 f g} : Surj R1 f.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	cancel_surj	null
idemp_L {A} f `{!@IdemP A (=) f} x : f x x = x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	idemp_L	The following lemmas are specific versions of the projections of the above type classes for Leibniz equality. These lemmas allow us to enforce Coq not to use the setoid rewriting mechanism.
comm_L {A B} f `{!@Comm A B (=) f} x y : f x y = f y x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	comm_L	null
left_id_L {A} i f `{!@LeftId A (=) i f} x : f i x = x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	left_id_L	null
right_id_L {A} i f `{!@RightId A (=) i f} x : f x i = x.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	right_id_L	null
assoc_L {A} f `{!@Assoc A (=) f} x y z : f x (f y z) = f (f x y) z.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	assoc_L	null
left_absorb_L {A} i f `{!@LeftAbsorb A (=) i f} x : f i x = i.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	left_absorb_L	null
right_absorb_L {A} i f `{!@RightAbsorb A (=) i f} x : f x i = i.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	right_absorb_L	null
strict {A} (R : relation A) : relation A := λ X Y, R X Y ∧ ¬R Y X.	Definition	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	strict	The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary relation [R] instead of [⊆] to support multiple orders on the same type.
PartialOrder {A} (R : relation A) : Prop := { partial_order_pre :: PreOrder R; partial_order_anti_symm :: AntiSymm (=) R }.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	PartialOrder	null
TotalOrder {A} (R : relation A) : Prop := { total_order_partial :: PartialOrder R; total_order_trichotomy :: Trichotomy (strict R) }.	Class	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	TotalOrder	null
prop_inhabited : Inhabited Prop := populate True.	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	prop_inhabited	* Logic
or_l P Q : ¬Q → P ∨ Q ↔ P.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	or_l	null
or_r P Q : ¬P → P ∨ Q ↔ Q.	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	or_r	null
and_wlog_l (P Q : Prop) : (Q → P) → Q → (P ∧ Q).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	and_wlog_l	null
and_wlog_r (P Q : Prop) : P → (P → Q) → (P ∧ Q).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	and_wlog_r	null
impl_trans (P Q R : Prop) : (P → Q) → (Q → R) → (P → R).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	impl_trans	null
forall_proper {A} (P Q : A → Prop) : (∀ x, P x ↔ Q x) → (∀ x, P x) ↔ (∀ x, Q x).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	forall_proper	null
exist_proper {A} (P Q : A → Prop) : (∀ x, P x ↔ Q x) → (∃ x, P x) ↔ (∃ x, Q x).	Lemma	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	exist_proper	null
eq_comm {A} : Comm (↔) (=@{A}).	Instance	stdpp	[ "Stdlib.Morphisms", "Stdlib.RelationClasses", "Stdlib.List", "Stdlib.Bool", "Stdlib.Setoid", "Stdlib.Peano", "Stdlib.Utf8", "Stdlib.Permutation", "Stdlib.Program.Basics", "Stdlib.Program.Syntax", "stdpp.options" ]	stdpp/base.v	eq_comm	null

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Coq-Stdpp

Structured dataset of formalizations from the std++ library (used by Iris).

Schema

Column	Type	Description
fact	string	Full declaration (name, signature, body)
type	string	Lemma, Definition, Instance, Class, etc.
library	string	Sub-library (stdpp, stdpp_bitvector, stdpp_unstable)
imports	list	Import statements
filename	string	Source file path
symbolic_name	string	Declaration identifier
docstring	string	Documentation comment (9% coverage)