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Require Import abp_base.
Require Import abp_defs.
Goal
forall (b : bit) (x y : proc),
D + (fun d : D => seq (ia D r1 d) (enc H (mer (seq (Sn_d d b) y) x))) =
enc H (Lmer (seq (Sn b) y) x).
intros.
elim ProcSn.
elim (SUM5 D (fun d : D => seq (ia D r1 d) (Sn_d d b)) y).
elimtype
((fun d : D => seq (ia D r1 d) (seq (S... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Module_util.
Require Export Sub_module.
Require Export Group_kernel.
Section Def.
Variable R : RING.
Variable Mod Mod2 : MODULE R.
Variable f : Hom Mod Mod2.
Definition Ker : submodule Mod.
Proof.
(* Goal: @submodule R Mod *)
apply (Build_submodule (R:=R)... |
From mathcomp
Require Import ssreflect ssrbool eqtype seq ssrfun.
From LemmaOverloading
Require Import heaps rels hprop stmod stsep stlogR.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Definition llist (T : Type) := ptr.
Section LList.
Variable T : Type.
Notation llist := (llist ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_betweennotequal.
Section Euclid.
Context `{Ax1:euclidean_neutral}.
Lemma lemma_raystrict :
forall A B C,
Out A B C ->
neq A C.
Proof.
(* Goal: forall (A B C : @Point Ax1) (_ : @Out Ax1 A B C), @neq Ax1 A C *)
intros.
(* Goal: @neq Ax1 A C *)
let Tf:=fres... |
Require Export GeoCoq.Tarski_dev.Ch13_6_Desargues_Hessenberg.
Section T14_sum.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
Lemma Pj_exists : forall A B C,
exists D, Pj A B C D.
Proof.
(* Goal: forall A B C : @Tpoint Tn, @ex (@Tpoint Tn) (fun D : @Tpoint Tn => @Pj Tn A B C D) *)
intros.
(* Go... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq div choice.
From mathcomp
Require Import fintype prime finset fingroup morphism automorphism.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GroupScope.
Section Cosets... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_group.
Require Export Abelian_group_cat.
Section Group.
Variable E : Setoid.
Variable genlaw : E -> E -> E.
Variable e : E.
Variable geninv : E -> E.
Hypothesis
fcomp :
forall x x' y y' : E,
Equal x x' -> Equal y y' -> Equal (genlaw x y) (gen... |
From mathcomp
Require Import ssreflect ssrnat seq.
From LemmaOverloading
Require Import prefix.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Definition invariant A s r i (e : A) := onth r i = Some e /\ prefix s r.
Example unit_test A (x1 x2 x3 x y : A):
(forall s e (f : XFind ... |
Require Import sur_les_relations.
Section YokouchiS.
Variable A : Set.
Variable R S : A -> A -> Prop.
Hypothesis C : explicit_confluence _ R.
Hypothesis N : explicit_noetherian _ R.
Hypothesis SC : explicit_strong_confluence _ S.
Definition Rstar_S_Rstar :=
explicit_comp_rel _ (explicit_star _ R)
(expli... |
Require Import securite.
Lemma POinvprel3 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Import securite.
Lemma POinvprel2 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_paralleldef2B.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelNC.
Require Export GeoCoq.Elements.OriginalProofs.lemma_planeseparation.
Section Euclid.
Context `{Ax:euclidean_neutral_... |
Require Import Coq.Lists.List.
Require Import Coq.Logic.Eqdep_dec.
Require Import Coq.omega.Omega.
Fixpoint arrow (xs : list Type) (res : Type) : Type :=
match xs with
| nil => res
| cons y ys => y -> arrow ys res
end.
Fixpoint tuple (xs : list Type) : Type :=
match xs with
| nil => unit
| con... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearorder.
Section Euclid.
Context `{Ax1:euclidean_neutral}.
Lemma lemma_NCorder :
forall A B C,
nCol A B C ->
nCol B A C /\ nCol B C A /\ nCol C A B /\ nCol A C B /\ nCol C B A.
Proof.
(* Goal: forall (A B C : @Point Ax1) (_ : @nCol Ax1 A B C), and... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_47.
Require Export GeoCoq.Elements.OriginalProofs.lemma_squaresequal.
Require Export GeoCoq.Elements.OriginalProofs.lemma_rectanglerotate.
Require Export GeoCoq.Elements.OriginalProofs.lemma_paste5.
Require Export GeoCoq.Elements.OriginalProofs.proposition_48A.
... |
From mathcomp
Require Import ssreflect ssrbool seq eqtype.
From LemmaOverloading
Require Import heaps.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Definition scan_axiom h s :=
def h -> uniq s /\ forall x, x \in s -> x \in dom h.
Lemma scanE x h (f : Scan h): def h -> x \in seq... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_congruenceflip.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_TGsymmetric :
forall A B C a b c,
TG A a B b C c ->
TG B b A a C c.
Proof.
(* Goal: forall ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_8_7.
Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_rectangleparallelogram :
forall A B C D,
RE A B C D ->
PG A B C D.
Proof.
(* Goal: forall (A B C D : @Point Ax... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_monoid.
Require Export Abelian_group_cat.
Section Monoid.
Variable E : Setoid.
Variable genlaw : E -> E -> E.
Variable e : E.
Hypothesis
fcomp :
forall x x' y y' : E,
Equal x x' -> Equal y y' -> Equal (genlaw x y) (genlaw x' y').
Hypothesis
... |
Require Import Bool Arith Div2.
Require Import BellantoniCook.Lib BellantoniCook.Bitstring BellantoniCook.BC.
Definition zero_e (n s:nat) : BC :=
comp n s zero nil nil.
Lemma zero_correct n s l1 l2:
bs2nat (sem (zero_e n s) l1 l2) = 0.
Proof.
(* Goal: @eq nat (bs2nat (sem (zero_e n s) l1 l2)) O *)
intros; simpl; ... |
Require Export GeoCoq.Tarski_dev.Ch02_cong.
Section T2_1.
Context `{Tn:Tarski_neutral_dimensionless}.
Lemma bet_col : forall A B C, Bet A B C -> Col A B C.
Proof.
(* Goal: forall (A B C : @Tpoint Tn) (_ : @Bet Tn A B C), @Col Tn A B C *)
intros;unfold Col;auto.
Qed.
Lemma between_trivial : forall A B : Tpoint, Bet ... |
Require Import Arith.
Require Import ZArith.
Require Import Wf_nat.
Require Import lemmas.
Require Import natZ.
Require Import dec.
Require Import list.
Require Import exp.
Require Import divides.
Require Import prime.
Require Import modulo.
Require Import modprime.
Require Import fermat.
Definition Order (b : Z) (... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq path div choice.
From mathcomp
Require Import fintype tuple finfun bigop prime ssralg poly finset.
From mathcomp
Require Import fingroup morphism perm automorphism quotient finalg action.
From mathcomp
Require Im... |
Definition identity (A:Set) := fun a:A=> a.
Definition compose (A B C:Set) (g:B->C) (f:A->B) := fun a:A=>g(f a).
Section Denumerability.
Definition same_cardinality (A:Set) (B:Set) := {f:A->B & { g:B->A | (forall b,(compose _ _ _ f g) b= (identity B) b)
... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq path fintype div.
From mathcomp
Require Import bigop prime finset fingroup morphism automorphism quotient.
From mathcomp
Require Import commutator gproduct gfunctor center gseries cyclic.
Set Implicit Arguments.... |
Require Import Bool.
Require Import Arith.
Require Import Compare_dec.
Require Import Peano_dec.
Require Import General.
Require Import MyList.
Require Import MyRelations.
Require Export Main.
Require Export SortV6.
Section CoqV6Beta.
Definition trm_v6 := term srt_v6.
Definition env_v6 := env srt_v6.
Definit... |
Require Import Bool.
Require Import Omega.
Section registers.
Inductive register : nat -> Set :=
| regO : register 0
| regS : forall m : nat, bool -> register m -> register (S m).
Definition register_zero :=
nat_rec register regO (fun m : nat => regS m false).
Definition register_max := nat_rec register regO ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_TCreflexive.
Section Euclid.
Context `{Ax:area}.
Lemma lemma_ETreflexive :
forall A B C,
Triangle A B C ->
ET A B C A B C.
Proof.
(* Goal: forall (A B C : @Point Ax0) (_ : @Triangle Ax0 A B C), @ET Ax0 Ax1 Ax2 Ax A B C A B C *)
intros.
(* Goal: @ET Ax0 A... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_together :
forall A B C D F G P Q a b c,
TG A a B b C c -> Cong D F A a -> Cong F G B b ... |
Require Export Numerals.
Require Export Compare_Nat.
Section compare_num.
Variable BASE : BT.
Let Digit := digit BASE.
Let valB := val BASE.
Let ValB := Val BASE.
Let Num := num BASE.
Let Val_bound := val_bound BASE.
Let Cons := cons Digit.
Let Nil := nil Digit.
Lemma Comp_dif :
forall (n : nat... |
Require Export GeoCoq.Tarski_dev.Ch13_5_Pappus_Pascal.
Section Desargues_Hessenberg.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
Lemma l13_15_1 : forall A B C A' B' C' O ,
~ Col A B C -> ~ Par O B A C ->
Par_strict A B A' B' -> Par_strict A C A' C'->
Col O A A' -> Col O B B' -> Col O C... |
From Coq Require Import ssreflect ssrbool ssrfun.
From mathcomp Require Import ssrnat eqtype seq path.
From fcsl Require Import ordtype finmap pcm.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module UM.
Section UM.
Variables (K : ordType) (V : Type) (p : pred K).
Inductive base ... |
From mathcomp
Require Import ssreflect ssrfun ssrnat div ssrbool seq.
From LemmaOverloading
Require Import prelude finmap ordtype.
From mathcomp
Require Import path eqtype.
Require Import Eqdep.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Notation eqn_addl := eqn_add2l.
Notation ... |
Require Export nat_trees.
Require Import Lt.
Inductive min (p : nat) (t : nat_tree) : Prop :=
min_intro : (forall q : nat, occ t q -> p < q) -> min p t.
Hint Resolve min_intro: searchtrees.
Inductive maj (p : nat) (t : nat_tree) : Prop :=
maj_intro : (forall q : nat, occ t q -> q < p) -> maj p t.
Hint Reso... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Paths.
Variables (n0 : nat) (T : Type).
Section Path.
Variables (x0_cycle : T) (e : rel T).
Fixpoint path x (p : se... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Subcat.
Require Export Set_cat.
Definition law_of_composition (E : SET) := Hom (cart E E:SET) E.
Definition associative (E : SET) (f : law_of_composition E) :=
forall x y z : E,
Equal (f (couple (f (couple x y)) z)) (f (couple x (f (couple y z)))).
R... |
Require Import utf_AMM11262.
Import NatSet GeneralProperties.
Section example_three_inhabitants.
Definition town_1:= {1}∪({2}∪({3}∪∅)).
Remark population₁ : |town_1| = 2×1 +1.
Proof.
(* Goal: @Logic.eq nat (cardinal town_1) (Nat.add (Nat.mul (S (S O)) (S O)) (S O)) *)
reflexivity.
Qed.
Definition familiarity₁ (m n... |
Require Import General.
Require Export Relations.
Unset Standard Proposition Elimination Names.
Section SortsOfECC.
Inductive calc : Set :=
| Pos : calc
| Neg : calc.
Inductive srt_ecc : Set :=
| Sprop : calc -> srt_ecc
| Stype : calc -> nat -> srt_ecc.
Inductive axiom_ecc : srt_ecc -> srt_e... |
Require Import List.
Import ListNotations.
Require Import StructTact.StructTactics.
Require Import StructTact.ListTactics.
Require Import StructTact.ListUtil.
Set Implicit Arguments.
Section dedup.
Variable A : Type.
Hypothesis A_eq_dec : forall x y : A, {x = y} + {x <> y}.
Fixpoint dedup (xs : list A) : list ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_congruencetransitive.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_congruenceflip :
forall A B C D,
Cong A B C D ->
Cong B A D C /\ Cong B A C D /\ Cong A B D C.
Proof.
(* Goal: forall (A B C D : @Point Ax) (_ : @Cong Ax A B C D), and (@C... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_group.
Require Export Module_facts.
Require Export Module_util.
Require Export Monoid_util.
Require Export Group_util.
Section Def.
Variable R : RING.
Variable M : MODULE R.
Section Sub_module.
Variable N : subgroup M.
Hypothesis
Nop : forall (a : R) ... |
Require Import List.
Import ListNotations.
Require Import StructTact.StructTactics.
Set Implicit Arguments.
Fixpoint Prefix {A} (l1 : list A) l2 : Prop :=
match l1, l2 with
| a :: l1', b :: l2' => a = b /\ Prefix l1' l2'
| [], _ => True
| _, _ => False
end.
Section prefix.
Variable A : Type.
Lem... |
Require Export GeoCoq.Tarski_dev.Ch08_orthogonality.
Require Export GeoCoq.Tarski_dev.Annexes.coplanar.
Ltac clean_reap_hyps :=
clean_duplicated_hyps;
repeat
match goal with
| H:(Midpoint ?A ?B ?C), H2 : Midpoint ?A ?C ?B |- _ => clear H2
| H:(Col ?A ?B ?C), H2 : Col ?A ?C ?B |- _ => clear H2
| H:(Col ?... |
Set Implicit Arguments.
Require Export inductive_wqo.
Require Export tree.
Require Export higman_aux.
Section higman.
Variable A : Set.
Variable leA : A -> A -> Prop.
Hypothesis eqA_dec : forall a a' : A, {a = a'} + {a <> a'}.
Hypothesis leA_dec : forall a a', {leA a a'} + {~ leA a a'}.
Hypothesis leA_trans : forall... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_crisscross.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_30helper :
forall A B E F G H,
Par A B E F -> BetS A G B -> BetS E H F -> ~ CR A F G H ->
CR A E G H.
Proof.
(* Goal: forall (A B E F G H : @Point Ax) (_ : @Par Ax A ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_diagonalsmeet.
Require Export GeoCoq.Elements.OriginalProofs.proposition_29B.
Require Export GeoCoq.Elements.OriginalProofs.proposition_26A.
Section Euclid.
Context `{Ax:euclidean_euclidean}.
Lemma proposition_34 :
forall A B C D,
PG A C D B ->
Cong A B ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglestransitive.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_RTcongruence :
forall A B C D E F P Q R,
RT A B C D E F -> CongA A B C P Q R ->
RT P Q R D E F.
Proof.
(* Goal: forall (A B C D E F P Q R : @Point Ax0) (_ :... |
Require Import List.
Import ListNotations.
Require Import StructTact.StructTactics.
Set Implicit Arguments.
Fixpoint before {A: Type} (x : A) y l : Prop :=
match l with
| [] => False
| a :: l' =>
a = x \/
(a <> y /\ before x y l')
end.
Section before.
Variable A : Type.
Lemma before_In :... |
Require Import abp_base.
Require Import abp_defs.
Require Import abp_lem1.
Require Import abp_lem2.
Require Import abp_lem25.
Require Import abp_lem3.
Require Import abp_lem1.
Parameter X' Y' : proc.
Parameter X1' X2' Y1' Y2' : D -> proc.
Axiom
Lin2' :
forall d : D,
alt (seq (ia one int i) (seq (ia D s4 d... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_TGsymmetric.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_TTorder :
forall A B C D E F G H,
TT A B C D E F G H ->
TT C D A B E F G H.
Proof.
(* Goal: forall (A B C D E F G H : @Point Ax0) (_ : @TT Ax0 A B C D E F G H), @TT A... |
Require Export Lib_Eq_Le_Lt.
Require Export Lib_Prop.
Lemma pred_diff_O : forall n : nat, n <> 0 -> n <> 1 -> pred n <> 0.
Proof.
(* Goal: forall (n : nat) (_ : not (@eq nat n O)) (_ : not (@eq nat n (S O))), not (@eq nat (Init.Nat.pred n) O) *)
simple induction n; auto with arith.
Qed.
Hint Resolve pred_diff_O.
Lem... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
From mathcomp
Require Import bigop finset prime binomial fingroup morphism perm automorphism.
From mathcomp
Require Import presentation quotient action commutator gproduct gfunctor.
From mathc... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_PGflip.
Require Export GeoCoq.Elements.OriginalProofs.proposition_34.
Section Euclid.
Context `{Ax:area}.
Lemma proposition_43 :
forall A B C D E F G H K,
PG A B C D -> BetS A H D -> BetS A E B -> BetS D F C -> BetS B G C -> BetS A K C -> PG E A H K -> PG G... |
Require Import Arith.
Require Import Terms.
Require Import Reduction.
Require Import Redexes.
Require Import Test.
Fixpoint lift_rec_r (L : redexes) : nat -> nat -> redexes :=
fun k n : nat =>
match L with
| Var i => Var (relocate i k n)
| Fun M => Fun (lift_rec_r M (S k) n)
| Ap b M N => Ap b (lift_rec_r M... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype tuple.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Def.
Variables (aT : finType) (rT : Type).
Inductive finfun_type : predArgType := Finfun of #|aT... |
From mathcomp
Require Import ssreflect ssrbool eqtype ssrfun seq.
Require Import Eqdep ClassicalFacts.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Axiom pext : forall p1 p2 : Prop, (p1 <-> p2) -> p1 = p2.
Axiom fext : forall A (B : A -> Type) (f1 f2 : forall x, B x),
... |
Require Import Classical.
Require Export GeoCoq.Elements.OriginalProofs.euclidean_defs.
Require Export GeoCoq.Elements.OriginalProofs.general_tactics.
Ltac remove_double_neg :=
repeat
match goal with
H: ~ ~ ?X |- _ => apply NNPP in H
end.
Section basic_lemmas.
Context `{Ax:euclidean_neutral}.
Lemma Col_or_nCol : ... |
Require Import securite.
Lemma POinvprel8 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Import securite.
Lemma POinv1rel1 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Import ZArith.
Require Import Wf_nat.
Require Import lemmas.
Require Import natZ.
Require Import dec.
Require Import exp.
Definition Divides (n m : nat) : Prop := exists q : nat, m = n * q.
Lemma div_refl : forall a : nat, Divides a a.
Proof.
(* Goal: forall a : nat, Divides a a *)
intros.
(* Goal: Divides ... |
Require Import StructTact.StructTactics.
Set Implicit Arguments.
Lemma or_false :
forall P : Prop, P -> (P \/ False).
Proof.
(* Goal: forall (P : Prop) (_ : P), or P False *)
firstorder.
Qed.
Lemma if_sum_bool_fun_comm :
forall A B C D (b : {A}+{B}) (c1 c2 : C) (f : C -> D),
f (if b then c1 else c2) = if b t... |
Require Export c_completeness.
Set Implicit Arguments.
Module Type hilbert_mod (B: base_mod) (S: sound_mod B) (C: complete_mod B S).
Import B S C.
Reserved Notation "Γ ⊢H A" (at level 80).
Inductive AxiomH : PropF -> Prop :=
| HOrI1 : forall A B , AxiomH (A → A∨B)
| HOrI2 : forall A B , AxiomH (B → A∨B)
| HAndI ... |
Require Import Ensembles.
Require Import Laws.
Require Import Group_definitions.
Require Import gr.
Parameter U : Type.
Parameter Gr : Group U.
Definition G : Ensemble U := G_ U Gr.
Definition star : U -> U -> U := star_ U Gr.
Definition inv : U -> U := inv_ U Gr.
Definition e : U := e_ U Gr.
Definition G0' : for... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_triangletoparallelogram.
Require Export GeoCoq.Elements.OriginalProofs.lemma_PGrotate.
Require Export GeoCoq.Elements.OriginalProofs.proposition_36.
Section Euclid.
Context `{Ax:area}.
Lemma proposition_38 :
forall A B C D E F P Q,
Par P Q B C -> Col P Q A ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglesflip.
Require Export GeoCoq.Elements.OriginalProofs.lemma_9_5.
Require Export GeoCoq.Elements.OriginalProofs.proposition_14.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_angleaddition :
forall A B C D E F P Q R a b c d e... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_samesideflip.
Require Export GeoCoq.Elements.OriginalProofs.proposition_39A.
Section Euclid.
Context `{Ax:area}.
Lemma proposition_39 :
forall A B C D,
Triangle A B C -> Triangle D B C -> OS A D B C -> ET A B C D B C -> neq A D ->
Par A D B C.
Proof.
(* ... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq path div fintype.
From mathcomp
Require Import tuple finfun.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Reserved Notation "\big [ op / idx ]_ i F"
(at level 36, F at lev... |
Require Import Arith.
Require Import Terms.
Require Import Reduction.
Require Import Redexes.
Require Import Test.
Require Import Marks.
Require Import Substitution.
Require Import Residuals.
Lemma mark_lift_rec :
forall (M : lambda) (n k : nat),
lift_rec_r (mark M) k n = mark (lift_rec M k n).
Proof.
(* Goal: fora... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sgroup_cat.
Section Lemmas.
Variable E : SGROUP.
Lemma SGROUP_assoc :
forall x y z : E,
Equal (sgroup_law _ (sgroup_law _ x y) z)
(sgroup_law _ x (sgroup_law _ y z)).
Proof.
(* Goal: forall x y z : Carrier (sgroup_set E), @Equal (sgroup_set E) (sgroup... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_ray.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_ray1 :
forall A B P,
Out A B P ->
(BetS A P B \/ eq B P \/ BetS A B P).
Proof.
(* Goal: forall (A B P : @Point Ax0) (_ : @Out Ax0 A B P), or (@BetS Ax0 A P B) (or (@eq Ax0 B ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglestransitive.
Require Export GeoCoq.Elements.OriginalProofs.lemma_supplements.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_supplements2 :
forall A B C D E F J K L P Q R,
RT A B C P Q R -> CongA A B C J K L -> RT J K ... |
From mathcomp
Require Import ssreflect ssrfun ssrbool ssrnat seq.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Prefix.
Variable A : Type.
Fixpoint onth (s : seq A) n : option A :=
if s is x :: s' then
if n is n'.+1 then onth s' n' else Some x
else None.
Definitio... |
Require Export Lib_Minus.
Lemma plus_opp : forall n m : nat, n + m - m = n.
Proof.
(* Goal: forall n m : nat, @eq nat (Init.Nat.sub (Init.Nat.add n m) m) n *)
intros n m; elim (plus_comm m n); apply minus_plus.
Qed.
Hint Resolve plus_opp.
Lemma S_plus : forall n : nat, S n = n + 1.
Proof.
(* Goal: forall n : nat, @e... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthantransitive.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_midpointunique :
forall A B C D,
Midpoint A B C -> Midpoint A D C ->
eq B D.
Proof.
(* Goal: forall (A B C D : @Point Ax0) (_ : @Midpoint Ax0 A B C) (_ : @Midp... |
Require Import Decidable DecidableTypeEx MSetFacts Setoid.
Module WDecide_fun (E : DecidableType)(Import M : WSetsOn E).
Module F := MSetFacts.WFactsOn E M.
Module FSetLogicalFacts.
Export Decidable.
Export Setoid.
Tactic Notation "fold" "any" "not" :=
repeat (
match goal with
... |
Require Export GeoCoq.Tarski_dev.Ch12_parallel.
Section L13_1.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Lemma per2_col_eq : forall A P P' B, A <> P -> A <> P' -> Per A P B -> Per A P' B -> Col P A P' -> P = P'.
Proof.
(* Goal: forall (A P P' B : @Tpoint Tn) (_ : not (@eq (@Tpoi... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice path.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module Finite.
Section RawMixin.
Variable T : eqType.
Definition axiom e := forall x : T, count_mem x e = 1.
Le... |
Require Export Linear_Structures.
Require Export Factorization.
Section Factorization_for_Verification.
Variable A : Set.
Variable BASE : BT.
Let b := base BASE.
Let Num := num BASE.
Let Digit := digit BASE.
Let Val_bound := val_bound BASE.
Variable R : forall n : nat, A -> inf n -> inf ... |
Require Export Bool.
Require Export Arith.
Require Export Compare_dec.
Require Export Peano_dec.
Require Export MyList.
Require Export MyRelations.
Set Implicit Arguments.
Unset Strict Implicit.
Definition max_nat (n m : nat) :=
match le_gt_dec n m with
| left _ => m
| right _ => n
end.
Lemma le... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Delimit Scope seq_scope with SEQ.
Open Scope seq_scope.
Notation seq := list.
Prenex Implicits cons.
Notation Cons T := (@cons T) ... |
Require Export GeoCoq.Tarski_dev.Ch03_bet.
Section T3.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Lemma l4_2 : forall A B C D A' B' C' D', IFSC A B C D A' B' C' D' -> Cong B D B' D'.
Proof.
(* Goal: forall (A B C D A' B' C' D' : @Tpoint Tn) (_ : @IFSC Tn A B C D A' B' C' D'), @Cong ... |
Require Export Qhomographic.
Require Export quadraticAcc_Qquadratic_sign.
Require Import general_Q Zaux.
Lemma Qquadratic_sg_denom_nonzero_always :
forall (k e f g h : Z) (p1 p2 : Qpositive),
k <> 0%Z ->
(0 < e)%Z ->
(0 < f)%Z ->
(0 < g)%Z ->
(0 < h)%Z ->
Qquadratic_sg_denom_nonzero (k * e) (k * f) (k * g) (k ... |
Require Import syntax.
Require Import List.
Require Import utils.
Require Import environments.
Require Import typecheck.
Inductive valid_env : OS_env -> Prop :=
| valid_nil : valid_env nil
| valid_cons :
forall (v : vari) (t : ty) (e : tm) (A : OS_env),
TC (OS_Dom_ty A) e t -> valid_env A -> valid_en... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_Euclid4.
Require Export GeoCoq.Elements.OriginalProofs.proposition_28C.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelflip.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_twoperpsparallel :
forall A B C D,
Per A B ... |
From mathcomp
Require Import ssreflect ssrnat seq.
From LemmaOverloading
Require Import prefix.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section XFind.
Variable A : Type.
Definition invariant s r i (e : A) := onth r i = Some e /\ prefix s r.
Structure xtagged := XTag {xunta... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_ray4.
Require Export GeoCoq.Elements.OriginalProofs.lemma_layoffunique.
Require Export GeoCoq.Elements.OriginalProofs.lemma_trichotomy2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_outerc... |
Require Import securite.
Lemma POinvprel4 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_06a.
Require Export GeoCoq.Elements.OriginalProofs.lemma_trichotomy1.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma proposition_06 :
forall A B C,
Triangle A B C -> CongA A B C A C B ->
Cong A B A C.
Proof.
(* Goal: forall (A... |
Require Import securite.
Lemma POinvprel5 :
forall (l l0 : list C) (k k0 k1 k2 : K) (c c0 c1 c2 : C)
(d d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19
d20 : D),
inv0
(ABSI (MBNaKab d7 d8 d9 k0) (MANbKabCaCb d4 d5 d6 k c c0)
(MABNaNbKeyK d d0 d1 d2 d3) l) ->
inv1
(ABSI (MBN... |
Require Import Bool Arith Div2 List.
Require Import BellantoniCook.Lib.
Notation bs := (list bool).
Definition unary (v : bs) := forallb id v.
Definition bs2bool (v:bs) : bool := hd false v.
Definition bool2bs (b:bool) : bs :=
if b then true::nil else nil.
Lemma bs_nat2bool_true : forall v,
bs2bool v = true ->... |
From mathcomp
Require Import ssreflect ssrbool seq ssrfun.
From LemmaOverloading
Require Import heaps rels hprop stmod stsep.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Lemma bnd_is_try (A B : Type) (s1 : spec A) (s2 : A -> spec B) i r :
verify (try_s s1 s2 (fun y => fr ... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype.
From mathcomp
Require Import bigop ssralg poly.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory.
Local Open Scope ring_scope.
Reserved Nota... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_26helper.
Require Export GeoCoq.Elements.OriginalProofs.lemma_trichotomy1.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma proposition_26B :
forall A B C D E F,
Triangle A B C -> Triangle D E F -> CongA A B C D E F -> CongA B C A E F D ... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_19.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence2.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_legsmallerhypotenuse :
forall A B C,
Per A B C ->
Lt A B A C /\ Lt B C A C.
Proof.
(* Goal: for... |
Require Export GeoCoq.Tarski_dev.Ch10_line_reflexivity.
Require Import GeoCoq.Meta_theory.Dimension_axioms.upper_dim_2.
Section T10_1.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Lemma cop__cong_on_bissect : forall A B M P X,
Coplanar A B X P -> Midpoint M A B -> Perp_at M A B P M -... |
Require Import Factorization_Synth.
Require Import Comparator_Relation.
Parameter BASE : BT.
Definition b := base BASE.
Definition Num := num BASE.
Definition Val_bound := val_bound BASE.
Lemma Comparator :
forall (n : nat) (o : order) (X Y : Num n),
{o' : order | R (exp b n) o (Val_bound n X) (Val_bound n Y) o'}... |
Require Import Arith.
Require Import ZArith.
Require Import Wf_nat.
Require Import lemmas.
Require Import natZ.
Require Import dec.
Require Import list.
Require Import exp.
Require Import divides.
Require Import prime.
Require Import modulo.
Require Import gcd.
Lemma prime_div_or_gcd1 :
forall (p : nat) (a : Z),
P... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq choice fintype.
From mathcomp
Require Import div path bigop prime finset.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Delimit Scope group_scope with g.
Delimit Scope Group_... |
Require Import Coq.Arith.Peano_dec.
Require Import Coq.Logic.Eqdep Coq.Logic.Eqdep_dec Coq.Program.Equality.
Theorem eq_rect_nat_double : forall T (a b c : nat) x ab bc,
eq_rect b T (eq_rect a T x b ab) c bc = eq_rect a T x c (eq_trans ab bc).
Proof.
(* Goal: forall (T : forall _ : nat, Type) (a b c : nat) (x : T a)... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_rightangleNC.
Require Export GeoCoq.Elements.OriginalProofs.lemma_ABCequalsCBA.
Require Export GeoCoq.Elements.OriginalProofs.lemma_supplements.
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglestransitive.
Section Euclid.
Context `{Ax:euclidean_neutral... |
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