fact stringlengths 16 1.35k | type stringclasses 10
values | library stringclasses 1
value | imports listlengths 1 4 | filename stringclasses 5
values | symbolic_name stringlengths 1 23 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
inv_unique : forall x y y', x*y == 1 -> y'*x == 1 -> y==y'. Proof. intros x y y' Hxy Hy'x. aac_instances <- Hy'x [dot]. aac_rewrite <- Hy'x at 1 [dot]. aac_rewrite Hxy. aac_reflexivity. Qed. | Lemma | theories | [
"From Stdlib Require NArith PeanoNat.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Caveats.v | inv_unique | |
eq_subr {X} {R} `{@Reflexive X R} : subrelation eq R. Proof. intros x y ->. reflexivity. Qed. | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | eq_subr | |
Obligation . unfold iff; split; intros; tauto. Qed. Solve All Obligations with firstorder. #[export] Instance aac_Prop_impl_iff_lift : AAC_lift Basics.impl iff := Build_AAC_lift _ _. | Next | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | Obligation | |
inter : relation T := fun x y => R x y /\ S x y. | Definition | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | inter | |
compo : relation T := fun x y => exists z : T, R x z /\ S z y. | Definition | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | compo | |
negr : relation T := fun x y => ~ R x y. (** union and converse are already defined in the standard library *) | Definition | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | negr | |
bot : relation T := fun _ _ => False. | Definition | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | bot | |
top : relation T := fun _ _ => True. | Definition | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | top | |
Obligation . firstorder. Qed. | Next | theories | [
"From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.",
"From Stdlib Require QArith Qminmax Relations.",
"From AAC_tactics Require Export AAC."
] | theories/Instances.v | Obligation | |
Z_abs_triangle : forall x y, Z.abs (x + y) <= Z.abs x + Z.abs y. Proof Z.abs_triangle. | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith List Permutation Lia.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Tutorial.v | Z_abs_triangle | |
Z_add_opp_diag_r : forall x, x + -x = 0. Proof Z.add_opp_diag_r. (** The following morphisms are required to perform the required rewrites: *) #[local] Instance Z_opp_ge_le_compat : Proper (Z.ge ==> Z.le) Z.opp. Proof. intros x y. lia. Qed. #[local] Instance Z_add_le_compat : Proper (Z.le ==> Z.le ==> Z.le) Z.add. Proo... | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith List Permutation Lia.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Tutorial.v | Z_add_opp_diag_r | |
Hbin1 : forall x y, (x+y)^2 = x^2 + y^2 + 2⋅x*y. Proof. intros; ring. Qed. | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith List Permutation Lia.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Tutorial.v | Hbin1 | |
Hbin2 : forall x y, x^2 + y^2 = (x+y)^2 + -(2⋅x*y). Proof. intros; ring. Qed. | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith List Permutation Lia.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Tutorial.v | Hbin2 | |
Hopp : forall x, x + -x = 0. Proof. apply Zplus_opp_r. Qed. Variables a b c : Z. Hypothesis H : c^2 + 2⋅(a+1)*b = (a+1+b)^2. Goal a^2 + b^2 + 2⋅a + 1 = c^2. aacu_rewrite <- Hbin1. rewrite Hbin2. aac_rewrite <- H. aac_rewrite Hopp. aac_reflexivity. Qed. (** Note: after the [aac_rewrite <- H], one could use [ring] to clo... | Lemma | theories | [
"From Stdlib Require PeanoNat ZArith List Permutation Lia.",
"From AAC_tactics Require Import AAC.",
"From AAC_tactics Require Instances."
] | theories/Tutorial.v | Hopp | |
idx := positive. | Notation | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | idx | |
eq_idx_bool i j := match i,j with | xH, xH => true | xO i, xO j => eq_idx_bool i j | xI i, xI j => eq_idx_bool i j | _, _ => false end. (** Specification predicate for boolean binary functions *) | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | eq_idx_bool | |
decide_spec {A} {B} (R : A -> B -> Prop) (x : A) (y : B) : bool -> Prop := | decide_true : R x y -> decide_spec R x y true | decide_false : ~(R x y) -> decide_spec R x y false. | Inductive | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | decide_spec | |
eq_idx_spec : forall i j, decide_spec (@eq _) i j (eq_idx_bool i j). Proof. induction i; destruct j; simpl; try (constructor; congruence). case (IHi j); constructor; congruence. case (IHi j); constructor; congruence. Qed. (** Weak specification predicate for comparison functions: only the [Eq] case is specified *) | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | eq_idx_spec | |
compare_weak_spec A: A -> A -> comparison -> Prop := | pcws_eq: forall i, compare_weak_spec i i Eq | pcws_lt: forall i j, compare_weak_spec i j Lt | pcws_gt: forall i j, compare_weak_spec i j Gt. | Inductive | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | compare_weak_spec | |
pos_compare_weak_spec : forall i j, compare_weak_spec i j (Pos.compare i j). Proof. intros. case Pos.compare_spec; try constructor. intros <-; constructor. Qed. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | pos_compare_weak_spec | |
pos_compare_reflect_eq : forall i j, Pos.compare i j = Eq -> i=j. Proof. intros ??. case pos_compare_weak_spec; intros; congruence. Qed. (** ** Dependent types utilities *) | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | pos_compare_reflect_eq | |
cast T H u := (eq_rect _ T u _ H). | Notation | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | cast | |
cast_eq : (forall u v: U, {u=v}+{u<>v}) -> forall A (H: A=A) (u: T A), cast T H u = u. Proof. intros. rewrite <- Eqdep_dec.eq_rect_eq_dec; trivial. Qed. Variable f: forall A B, T A -> T B -> comparison. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | cast_eq | |
reflect_eqdep := forall A u B v (H: A=B), @f A B u v = Eq -> cast T H u = v. (** These lemmas have to remain transparent to get structural recursion in the lemma [tcompare_weak_spec] below *) | Definition | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | reflect_eqdep | |
reflect_eqdep_eq : reflect_eqdep -> forall A u v, @f A A u v = Eq -> u = v. Proof. intros H A u v He. apply (H _ _ _ _ eq_refl He). Defined. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | reflect_eqdep_eq | |
reflect_eqdep_weak_spec : reflect_eqdep -> forall A u v, compare_weak_spec u v (@f A A u v). Proof. intros. case_eq (f u v); try constructor. intro H'. apply reflect_eqdep_eq in H'. subst. constructor. assumption. Defined. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | reflect_eqdep_weak_spec | |
nelist (A : Type) : Type := | nil : A -> nelist A | cons : A -> nelist A -> nelist A. Register nil as aac_tactics.nelist.nil. Register cons as aac_tactics.nelist.cons. | Inductive | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | nelist | |
nelist_map (A B: Type) (f: A -> B) l := match l with | nil x => nil ( f x) | cons x l => cons ( f x) (nelist_map f l) end. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | nelist_map | |
appne A l l' : nelist A := match l with nil x => cons x l' | cons t q => cons t (appne A q l') end. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | appne | |
mset A := nelist (A*positive). (** Lexicographic composition of comparisons (this is a notation to keep it lazy) *) | Definition | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | mset | |
lex e f := (match e with Eq => f | _ => e end). | Notation | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | lex | |
list_compare h k := match h,k with | nil x, nil y => compare x y | nil x, _ => Lt | _, nil x => Gt | u::h, v::k => lex (compare u v) (list_compare h k) end. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | list_compare | |
mset_compare A B compare: mset A -> mset B -> comparison := list_compare (fun un vm => let '(u,n) := un in let '(v,m) := vm in lex (compare u v) (Pos.compare n m)). | Definition | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | mset_compare | |
list_compare_weak_spec : forall h k, compare_weak_spec h k (list_compare compare h k). Proof. induction h as [|u h IHh]; destruct k as [|v k]; simpl; try constructor. case (Hcompare a a0 ); try constructor. case (Hcompare u v ); try constructor. case (IHh k); intros; constructor. Defined. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | list_compare_weak_spec | |
mset_compare_weak_spec : forall h k, compare_weak_spec h k (mset_compare compare h k). Proof. apply list_compare_weak_spec. intros [u n] [v m]. case (Hcompare u v); try constructor. case (pos_compare_weak_spec n m); try constructor. Defined. | Lemma | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | mset_compare_weak_spec | |
insert n1 h1 := let fix insert_aux l2 := match l2 with | nil (h2,n2) => match compare h1 h2 with | Eq => nil (h1,Pplus n1 n2) | Lt => (h1,n1):: nil (h2,n2) | Gt => (h2,n2):: nil (h1,n1) end | (h2,n2)::t2 => match compare h1 h2 with | Eq => (h1,Pplus n1 n2):: t2 | Lt => (h1,n1)::l2 | Gt => (h2,n2)::insert_aux t2 end end... | Definition | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | insert | |
merge_msets l1 := match l1 with | nil (h1,n1) => fun l2 => insert n1 h1 l2 | (h1,n1)::t1 => let fix merge_aux l2 := match l2 with | nil (h2,n2) => match compare h1 h2 with | Eq => (h1,Pplus n1 n2) :: t1 | Lt => (h1,n1):: merge_msets t1 l2 | Gt => (h2,n2):: l1 end | (h2,n2)::t2 => match compare h1 h2 with | Eq => (h1,Pp... | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | merge_msets | |
reduce_mset : mset A -> mset A := nelist_map (fun x => (fst x,xH)). (** Interpretation of a list with a constant and a binary operation *) Variable B: Type. Variable map: A -> B. Variable b2: B -> B -> B. | Definition | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | reduce_mset | |
fold_map l := match l with | nil x => map x | u::l => b2 (map u) (fold_map l) end. (** Mapping and merging *) Variable merge: A -> nelist B -> nelist B. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | fold_map | |
merge_map (l: nelist A): nelist B := match l with | nil x => nil (map x) | u::l => merge u (merge_map l) end. Variable ret : A -> B. Variable bind : A -> B -> B. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | merge_map | |
fold_map' (l : nelist A) : B := match l with | nil x => ret x | u::l => bind u (fold_map' l) end. | Fixpoint | theories | [
"From Stdlib Require Import Arith NArith List RelationClasses."
] | theories/Utils.v | fold_map' |
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