text stringlengths 107 2.17M |
|---|
From mathcomp
Require Import ssreflect ssrbool ssrfun.
From LemmaOverloading
Require Import heaps rels stmod stsep stlog.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Structure tagged_heap := Tag {untag :> heap}.
Definition right_tag := Tag.
Definition left_tag := right_tag.
Cano... |
Require Import syntax.
Require Import utils.
Inductive FV (z : vari) : tm -> Prop :=
| FV_abs :
forall e : tm,
FV z e -> forall v : vari, z <> v -> forall t : ty, FV z (abs v t e)
| FV_fix :
forall e : tm,
FV z e -> forall v : vari, z <> v -> forall t : ty, FV z (Fix v t e)
| FV_appl1 : ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_sameside2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_10_12.
Require Export GeoCoq.Elements.OriginalProofs.proposition_07.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_erectedperpendicularunique :
forall A B C E,
Per A... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_extension.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence2.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_TTflip2 :
forall A B C D E F G H,
TT A B C D E F G H ->
TT A B C D H G F E.
Proof.
(* Goal: for... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_31.
Require Export GeoCoq.Elements.OriginalProofs.lemma_crossbar2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_supplementinequality.
Require Export GeoCoq.Elements.OriginalProofs.lemma_angletrichotomy2.
Require Export GeoCoq.Elements.OriginalProofs.lemma... |
Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Z_group_facts.
Section Zup1.
Variable R : RING.
Hint Resolve Z_to_group_nat_eq_pos: algebra.
Hint Resolve Z_to_group_nat_unit: algebra.
Hint Resolve Zl1: algebra.
Hint Resolve Zl2: algebra.
Lemma nat_to_group_mult :
forall... |
Require Import Ensf.
Require Import Words.
Require Import more_words.
Require Import Rat.
Require Import need.
Require Import fonctions.
Require Import Relations.
Require Import gram.
Require Import gram2.
Require Import gram3.
Section gram4.
Variable X V1 R1 : Ensf.
Variable S1 : Elt.
Variable V2 R2 : Ensf.
Variab... |
Require Import Ensf.
Require Import Words.
Require Import more_words.
Require Import Rat.
Require Import need.
Require Import fonctions.
Require Import Relations.
Require Import gram.
Require Import gram2.
Require Import gram3.
Require Import gram4.
Section gram5.
Variable X : Ensf.
Variable V1 R1 : Ensf.
Variable ... |
Require Import Coq.Arith.Div2.
Require Import Coq.micromega.Lia.
Require Import Coq.NArith.NArith.
Require Import Coq.ZArith.ZArith.
Require Import bbv.N_Z_nat_conversions.
Require Export bbv.Nomega.
Set Implicit Arguments.
Fixpoint mod2 (n : nat) : bool :=
match n with
| 0 => false
| 1 => true
| S (S n... |
Require Import Ensf.
Require Import Max.
Require Import Words.
Require Import fonctions.
Require Import need.
Require Import Relations.
Section pushdown_automata.
Variable X P : Ensf.
Variable wd : Word.
Variable wa : Word.
Variable d : Ensf.
Definition eps := natural (sup X).
Lemma not_dans_X_eps : ~ dans eps X.
... |
Require Import ZArith.
Require Import ZArithRing.
Require Import Zcomplements.
Unset Standard Proposition Elimination Names.
Inductive divide (a b : Z) : Prop :=
divide_intro : forall q : Z, b = (q * a)%Z -> divide a b.
Notation "( x | y )" := (divide x y) (at level 0) : Z_scope.
Local Open Scope Z_scope.
Lem... |
Lemma pair_fst_snd : forall (A B : Set) (c : A * B), (fst c, snd c) = c.
Proof.
(* Goal: forall (A B : Set) (c : prod A B), @eq (prod A B) (@pair A B (@fst A B c) (@snd A B c)) c *)
intros.
(* Goal: @eq (prod A B) (@pair A B (@fst A B c) (@snd A B c)) c *)
pattern c in |- *; elim c; auto.
Qed.
Inductive prod_3 (A B C... |
Require Export GeoCoq.Tarski_dev.Ch13_3_angles.
Ltac anga_instance_o a A B P C :=
assert(tempo_anga:= anga_const_o a A B P);
match goal with
|H: Q_CongA_Acute a |- _ => assert(tempo_H:= H); apply tempo_anga in tempo_H; ex_elim tempo_H C
end;
clear tempo_anga.
Section Cosin... |
From Coq Require Import ssreflect ssrbool ssrfun.
From mathcomp Require Import ssrnat eqtype seq.
From fcsl Require Import pred ordtype pcm unionmap.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Helpers.
Variable A : Type.
Fixpoint onth (s : seq A) n : option A :=
if ... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Cfield_facts.
Section Def.
Variable R : CFIELD.
Record Ctype : Type := {real :> R; imag : R}.
Definition Cadd (z z' : Ctype) : Ctype :=
Build_Ctype (sgroup_law R (real z) (real z'))
(sgroup_law R (imag z) (imag z')).
Definition Cmult (z z' : Ctype... |
Require Export Arith.
Require Export Compare.
Lemma not_S_eq : forall n m : nat, S n <> S m -> n <> m.
Proof.
(* Goal: forall (n m : nat) (_ : not (@eq nat (S n) (S m))), not (@eq nat n m) *)
red in |- *; intros.
(* Goal: False *)
apply H; auto with arith.
Qed.
Hint Resolve not_S_eq.
Lemma neq_O_le : forall n : nat,... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Fpart.
Require Export Inter.
Require Export Arith.
Section fparts2_def.
Variable E : Setoid.
Definition disjoint (A B : part_set E) := Equal (inter A B) (empty E).
Lemma disjoint_comp :
forall A A' B B' : part_set E,
Equal A A' -> Equal B B' -> disjoint... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_27.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearparallel.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelflip.
Section Euclid.
Context `{Ax:euclidean_neutral_r... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearorder.
Require Export GeoCoq.Elements.OriginalProofs.lemma_inequalitysymmetric.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_parallelflip :
forall A B C D,
Par A B C D ->
Par B A C D /\ Par A B D C /\ Par B A D C.
Proof.
(* Goal:... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_congruenceflip.
Require Export GeoCoq.Elements.OriginalProofs.lemma_extensionunique.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_rightreverse :
forall A B C D,
Per A B C -> BetS A B D -> Cong A B B D ->
Cong A C D C.
Proof.
(* Goal: fora... |
Require Import Arith List.
Require Import BellantoniCook.Lib BellantoniCook.Bitstring BellantoniCook.BC BellantoniCook.BCI.
Fixpoint conv n s (e : BCI) : BC :=
match e with
| zeroI => comp n s zero nil nil
| projIn i => proj n s i
| projIs i => proj n s (n + i)
| succI b => comp n s (succ b) nil [pro... |
Require Import Bool Arith Div2 List Permutation.
Require Export Omega.
Global Obligation Tactic := idtac.
Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) .. ).
Lemma length_nil : forall A (l : list A),
length l = 0 -> l = nil.
Proof.
(* Goal: forall (A : Type) (l : list A) (_ : @eq nat (@length A l) O), @eq ... |
Set Implicit Arguments.
Require Export List.
Section Wrap.
Variable A : Set.
Variable leA : A -> A -> Prop.
Variable leA_dec : forall a a', {leA a a'} + {~ leA a a'}.
Inductive greater : A -> list A -> Prop :=
| Gr0 : forall a a' w, leA a' a -> greater a (a'::w)
| Gr1 : forall a a' w, greater a w -> greater a (a':... |
Require Export GeoCoq.Tarski_dev.Ch05_bet_le.
Ltac eCol := eauto with col.
Section T6_1.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Lemma bet_out : forall A B C, B <> A -> Bet A B C -> Out A B C.
Proof.
(* Goal: forall (A B C : @Tpoint Tn) (_ : not (@eq (@Tpoint Tn) B A)) (_ : @Bet... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_planeseparation.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_samesidetransitive :
forall A B P Q R,
OS P Q A B -> OS Q R A B ->
OS P R A B.
Proof.
(* Goal: forall (A B P Q R : @Point Ax) (_ : @OS Ax P Q A B) (_ : @OS Ax Q ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_8_2.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_rectanglereverse :
forall A B C D,
RE A B C D ->
RE D C B A.
Proof.
(* Goal: forall (A B C D : @Point Ax0) (_ : @RE Ax0 A B C D), @RE Ax0 D C B A *)
intros.
(* Goal: @RE Ax0 ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinear2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinear1.
Section Euclid.
Context `{Ax1:euclidean_neutral}.
Lemma lemma_collinearorder :
forall A B C,
Col A B C ->
Col B A C /\ Col B C A /\ Col C A B /\ Col A C B /\ Col C B A.
Proof.
(*... |
Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Zring.
Section Int_power.
Variable G : GROUP.
Set Strict Implicit.
Unset Implicit Arguments.
Definition group_square (x : G) : G := sgroup_law G x x.
Set Implicit Arguments.
Unset Strict Implicit.
Fixpoint group_power_pos (... |
Require Export GeoCoq.Tarski_dev.Ch14_prod.
Section Order.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
Lemma l14_36_a : forall O E E' A B C,
Sum O E E' A B C -> Out O A B -> Bet O A C.
Proof.
(* Goal: forall (O E E' A B C : @Tpoint Tn) (_ : @Sum Tn O E E' A B C) (_ : @Out Tn O A B), @Bet Tn ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_samenotopposite.
Require Export GeoCoq.Elements.OriginalProofs.lemma_crisscross.
Require Export GeoCoq.Elements.OriginalProofs.proposition_33.
Section Euclid.
Context `{Ax:euclidean_euclidean}.
Lemma proposition_33B :
forall A B C D,
Par A B C D -> Cong A B... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_samesidereflexive.
Require Export GeoCoq.Elements.OriginalProofs.lemma_sameside2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_samesidesymmetric.
Require Export GeoCoq.Elements.OriginalProofs.proposition_12.
Require Export GeoCoq.Elements.OriginalProofs.lemma_8... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_raystrict.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinear4.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_crossbar :
forall A B C E U V,
Tria... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_10.
Require Export GeoCoq.Elements.OriginalProofs.proposition_15.
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglesreflexive.
Require Export GeoCoq.Elements.OriginalProofs.lemma_angleorderrespectscongruence.
Require Export GeoCoq.Elements.OriginalP... |
Require Export Lib_Pred.
Lemma minus_SS_n : forall n : nat, S (S n) - n = 2.
Proof.
(* Goal: forall n : nat, @eq nat (Init.Nat.sub (S (S n)) n) (S (S O)) *)
intros n.
(* Goal: @eq nat (Init.Nat.sub (S (S n)) n) (S (S O)) *)
elim minus_Sn_m.
(* Goal: le n (S n) *)
(* Goal: @eq nat (S (Init.Nat.sub (S n) n)) (S (S O)) ... |
Require Import Le.
Require Import Lt.
Require Import Plus.
Require Import Gt.
Require Import Minus.
Require Import Mult.
Require Import TS.
Require Import sigma_lift.
Require Import comparith.
Definition e_P1 (b : wsort) (U : TS b) : nat :=
(fix F (w : wsort) (t : TS w) {struct t} : nat :=
match t with
... |
From mathcomp
Require Import ssreflect ssrbool ssrnat ssrfun eqtype seq.
From LemmaOverloading
Require Import prelude prefix perms heaps.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Structure ctx := Context {heap_ctx : seq heap; ptr_ctx : seq ptr}.
Definition empc := Context [::... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_NChelper.
Require Export GeoCoq.Elements.OriginalProofs.proposition_16.
Require Export GeoCoq.Elements.OriginalProofs.lemma_crossbar.
Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.
Require Export GeoCoq.Elements.OriginalProofs.lemma_NCorder.
Section ... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sets.
Comments
"We define here the set of parts of a set, inclusion, union of a part,".
Comments
"and we prove that there is no surjection from a set in its part set".
Section Subtype.
Comments "In Coq type theory, there is no primitive notion of subty... |
Require Export GeoCoq.Tarski_dev.Ch10_line_reflexivity_2.
Ltac permut :=
match goal with
|H : (Col ?X ?Y ?Z) |- Col ?X ?Y ?Z => assumption
|H : (Col ?X ?Y ?Z) |- Col ?Y ?Z ?X => apply col_permutation_1; assumption
|H : (Col ?X ?Y ?Z) |- Col ?Z ?X ?Y => apply col_permutation_2; assumption
|H : (Col ?X ?Y ?Z... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq div fintype prime.
From mathcomp
Require Import bigop finset fingroup morphism automorphism quotient action.
From mathcomp
Require Import cyclic gproduct gfunctor commutator pgroup center nilpotent.
Set Implicit... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun ssrnat eqtype seq choice div fintype.
From mathcomp
Require Import path bigop finset prime ssralg poly polydiv mxpoly.
From mathcomp
Require Import generic_quotient countalg closed_field ssrnum ssrint rat intdiv.
From mathcomp
Req... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearorder.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_parallelNC :
forall A B C D,
Par A B C D ->
nCol A B C /\ nCol A C D /\ nCol B C D /\ nCol A B D.
Proof.
(* Goal: forall (A B C D : @Point Ax) (_ : @Par Ax A B C D), and (@nCol ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthannotequal.
Require Export GeoCoq.Elements.OriginalProofs.lemma_together.
Require Export GeoCoq.Elements.OriginalProofs.lemma_ray5.
Require Export GeoCoq.Elements.OriginalProofs.lemma_subtractequals.
Require Export GeoCoq.Elements.OriginalProofs.lemma_ondiameter... |
Require Export GeoCoq.Elements.OriginalProofs.euclidean_tactics.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_congruencesymmetric :
forall A B C D,
Cong B C A D ->
Cong A D B C.
Proof.
(* Goal: forall (A B C D : @Point Ax) (_ : @Cong Ax B C A D), @Cong Ax A D B C *)
intros.
(* Goal: @Cong ... |
Require Export GeoCoq.Elements.OriginalProofs.proposition_20.
Require Export GeoCoq.Elements.OriginalProofs.lemma_TGsymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_TGflip.
Require Export GeoCoq.Elements.OriginalProofs.proposition_22.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma... |
Require Import Arith List.
Require Import BellantoniCook.Lib BellantoniCook.MultiPoly BellantoniCook.Cobham BellantoniCook.CobhamLib.
Lemma Zero_correct n l:
length (Sem (Zero_e n) l) = 0.
Proof.
(* Goal: @eq nat (@length bool (Sem (Zero_e n) l)) O *)
trivial.
Qed.
Lemma One_correct n l:
length (Sem (One_e n) l... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalitysymmetric.
Section Euclid.
Context `{Ax:euclidean_neutral}.
Lemma lemma_inequalitysymmetric :
forall A B,
neq A B ->
neq B A.
Proof.
(* Goal: forall (A B : @Point Ax) (_ : @neq Ax A B), @neq Ax B A *)
intros.
(* Goal: @neq Ax B A *)
assert (~ eq ... |
Require Import Eqdep_dec.
Require Export Field.
Require Export Q_order.
Lemma Q_Ring_Theory :
ring_theory Zero Qone Qplus Qmult Qminus Qopp (eq(A:=Q)).
Proof.
(* Goal: @ring_theory Q Zero Qone Qplus Qmult Qminus Qopp (@eq Q) *)
split; intros n m p || intros n m || intros n; solve [ first [ apply Qplus_sym | apply... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Monoid_cat.
Require Export Sgroup_facts.
Require Export Monoid_facts.
Require Export Monoid_util.
Require Export Abelian_group_facts.
Section Free_abelian_monoid_def.
Variable V : SET.
Inductive FaM : Type :=
| Var : V -> FaM
| Law : FaM -> FaM -> FaM
... |
Require Import Omega.
Require Import Zcomplements.
Require Import Zpower.
Require Import Zlogarithm.
Require Import Diadic.
Require Import IEEE754_def.
Section basic_verifs.
Lemma max_abstract_wf :
forall (b : bool) (t : float_type), abstract_wf t (max_abstract t b).
Proof.
(* Goal: forall (b : bool) (t : float_typ... |
Require Import List.
Import ListNotations.
Require Import StructTact.StructTactics.
Require Import StructTact.ListTactics.
Fixpoint before_all {A : Type} (x : A) y l : Prop :=
match l with
| [] => True
| a :: l' =>
~ In x l' \/
(y <> a /\ before_all x y l')
end.
Section before_all.
Variabl... |
Require Export Relation_Definitions.
Require Export Relation_Operators.
Require Export Operators_Properties.
Require Export Inclusion.
Require Export Transitive_Closure.
Require Export Union.
Require Export Inverse_Image.
Require Export Lexicographic_Product.
Hint Resolve rt_step rt_refl rst_step rst_refl t_step: co... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_8_2.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_squareflip :
forall A B C D,
SQ A B C D ->
SQ B A D C.
Proof.
(* Goal: forall (A B C D : @Point Ax) (_ : @SQ Ax A B C D), @SQ Ax B A D C *)
intros.
(* Goal: @SQ Ax B A D C *... |
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 polydiv finset.
From mathcomp
Require Import fingroup morphism perm automorphism quotient finalg action zmodp.
From mathc... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Monoid_cat.
Section Def.
Variable E : SET.
Definition endo_comp : law_of_composition (Hom E E).
Proof.
(* Goal: Carrier (law_of_composition (@Hom SET E E)) *)
unfold law_of_composition in |- *.
(* Goal: Carrier (@Hom SET (cart (@Hom SET E E) (@Hom SET E E... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Group_util.
Require Export Abelian_group_facts.
Section Free_abelian_group_def.
Variable V : SET.
Inductive FaG : Type :=
| Var : V -> FaG
| Law : FaG -> FaG -> FaG
| Unit : FaG
| Inv : FaG -> FaG.
Inductive eqFaG : FaG -> FaG -> Prop :=
| eqFaG... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Union.
Require Export Singleton.
Require Export Diff.
Require Export Classical_Prop.
Section fparts_in_def.
Variable E : Setoid.
Definition add_part (A : part_set E) (x : E) := union A (single x).
Lemma add_part_comp :
forall (A A' : part_set E) (x x' : ... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import eqtype choice ssreflect ssrbool ssrnat ssrfun seq.
From mathcomp
Require Import ssralg generic_quotient.
Import GRing.Theory.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local... |
Require Import Arith.
Require Import Terms.
Require Import Reduction.
Require Import Redexes.
Require Import Test.
Require Import Substitution.
Inductive residuals : redexes -> redexes -> redexes -> Prop :=
| Res_Var : forall n : nat, residuals (Var n) (Var n) (Var n)
| Res_Fun :
forall U V W : redexes,
... |
Require Import basis.
Require Import part1.
Require Import part2.
Require Import part3.
Require Import affinity.
Require Import orthogonality.
Theorem pb9_1 :
forall (a : Point) (l : Line), ex (fun b : Point => Incident b l).
Proof.
(* Goal: forall (_ : Point) (l : Line), @ex Point (fun b : Point => Incident b l) *)... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_3_5b.
Section Euclid.
Context `{Ax1:euclidean_neutral_ruler_compass}.
Lemma lemma_3_6b :
forall A B C D,
BetS A B C -> BetS A C D ->
BetS A B D.
Proof.
(* Goal: forall (A B C D : @Point Ax) (_ : @BetS Ax A B C) (_ : @BetS Ax A C D), @BetS Ax A B D *)
int... |
From mathcomp
Require Import ssreflect ssrbool.
From LemmaOverloading
Require Import prelude xfind heaps cancel.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Structure tagged_prop := Tag {puntag :> Prop}.
Definition default_tag := Tag.
Definition dyneq_tag := default_tag.
Lemma ... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Parts2.
Section Restrictions1.
Variable E F : Setoid.
Variable f : MAP E F.
Definition restrict : forall A : part_set E, MAP A F.
Proof.
(* Goal: forall A : Carrier (part_set E), Carrier (MAP (@set_of_subtype_image E (@part E A)) F) *)
intros A; try assum... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_monoid.
Require Export Group_facts.
Section Def.
Variable G : GROUP.
Section Sub_group.
Variable H : submonoid G.
Hypothesis Hinv : forall x : G, in_part x H -> in_part (group_inverse _ x) H.
Definition subgroup_inv : MAP H H.
Proof.
(* Goal: Carrier (... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Union.
Section Inter1.
Variable E : Setoid.
Definition inter : part_set E -> part_set E -> part_set E.
Proof.
(* Goal: forall (_ : Carrier (part_set E)) (_ : Carrier (part_set E)), Carrier (part_set E) *)
intros A B.
(* Goal: Carrier (part_set E) *)
apply... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_crossbar.
Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglesreflexive.
Require Export GeoCoq.Elements.OriginalProofs.lemma_sameside2.
Require Export GeoCoq.Elements.OriginalProofs.proposition_07.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_comp... |
Require Import ZArith.
Require Import lemmas.
Require Import natZ.
Require Import exp.
Require Import divides.
Definition Mod (a b : Z) (n : nat) :=
exists q : Z, a = (b + Z_of_nat n * q)%Z.
Lemma modpq_modp : forall (a b : Z) (p q : nat), Mod a b (p * q) -> Mod a b p.
Proof.
(* Goal: forall (a b : Z) (p q : nat)... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Classical_Prop.
Require Export Parts.
Comments "We define here complement of a part, image of a part by a map.".
Section Complement1.
Variable E : Setoid.
Lemma not_in_comp_l :
forall (E : Setoid) (A : part_set E) (x y : E),
~ in_part x A -> Equal y x -... |
Require Export Coq.Strings.String.
Require Import Coq.Strings.Ascii.
Require Import Coq.NArith.NArith.
Local Open Scope char_scope.
Local Open Scope N_scope.
Definition hexDigitToN (c : ascii) : option N :=
match c with
| "0" => Some 0
| "1" => Some 1
| "2" => Some 2
| "3" => Some 3
| "4" => S... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq div fintype finset.
From mathcomp
Require Import prime fingroup morphism automorphism quotient action gproduct.
From mathcomp
Require Import gfunctor commutator center pgroup finmodule nilpotent sylow.
From mathc... |
Require Export bbv.HexNotation.
Require Export bbv.ReservedNotations.
Require Import Coq.ZArith.BinInt.
Notation "'Ox' a" := (Z.of_N (hex a)).
Goal Ox"41" = 65%Z.
|
From mathcomp
Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.
From mathcomp
Require Import choice path finset finfun fintype bigop.
Require Import finmap.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Lemma sumn_map I (f : I -> nat) s :
sumn [seq f i | i <- s] = \sum... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
From mathcomp
Require Import finfun bigop prime binomial ssralg finset fingroup finalg.
From mathcomp
Require Import perm zmodp matrix.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Pr... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelNC.
Require Export GeoCoq.Elements.OriginalProofs.lemma_crossimpliesopposite.
Require Export GeoCoq.Elements.OriginalProofs.proposition_34.
Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.
Section Euclid.
Context `{Ax:euclidean_euclidean}.
Le... |
From Coq Require Import ssreflect ssrbool ssrfun.
From mathcomp Require Import ssrnat eqtype seq path.
From fcsl Require Import prelude ordtype pcm finmap unionmap.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module Type NMSig.
Parameter tp : Type -> Type.
Section Params.
Variab... |
Require Import Coq.Arith.Arith Coq.micromega.Lia Coq.NArith.NArith.
Require Import Coq.ZArith.ZArith.
Local Open Scope N_scope.
Hint Rewrite Nplus_0_r nat_of_Nsucc nat_of_Nplus nat_of_Nminus
N_of_nat_of_N nat_of_N_of_nat
nat_of_P_o_P_of_succ_nat_eq_succ nat_of_P_succ_morphism : N.
Theorem nat_of_N_eq : forall n... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
From mathcomp
Require Import finfun bigop prime binomial ssralg finset fingroup finalg.
From mathcomp
Require Import perm zmodp countalg.
Set Implicit Arguments.
Unset Strict Implicit.
Unset ... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq path choice fintype.
From mathcomp
Require Import div tuple finfun bigop ssralg finalg zmodp matrix vector poly.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope... |
Require Import NF.
Require Import List.
Require Import syntax.
Require Import typecheck.
Require Import environments.
Require Import freevars.
Require Import utils.
Goal forall e : tm, F e -> forall H : ty_env, ~ TC H e nat_ty.
simple induction 1; intros.
red in |- *; intro.
specialize inv_TC_abs with (1 := ... |
Require Import sur_les_relations.
Require Import TS.
Require Import sigma_lift.
Definition e_invSL (b : wsort) (M N : TS b) :=
match M, N with
| lift M1, id => M1 = id
| lift M1, lift N1 => e_relSL _ M1 N1
| lambda M1, lambda N1 => e_relSL _ M1 N1
| app M1 M2, app N1 N2 =>
e_relSL _ M1 N1 /\ M2 = N2 \... |
Require Import Ensf_types.
Require Import Ensf_dans.
Require Import Ensf_union.
Require Import Ensf_inclus.
Fixpoint map (f : Elt -> Elt) (e : Ensf) {struct e} : Ensf :=
match e with
| empty => empty
| add y e => add (f y) (map f e)
end.
Lemma dans_map :
forall (f : Elt -> Elt) (a : Ensf) (x : Elt),
dans x... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_diagonalsbisect.
Section Euclid.
Context `{Ax:euclidean_euclidean}.
Lemma lemma_trapezoiddiagonals :
forall A B C D E,
PG A B C D -> BetS A E D ->
exists X, BetS B X D /\ BetS C X E.
Proof.
(* Goal: forall (A B C D E : @Point Ax0) (_ : @PG Ax0 A B C D) (... |
Require Import TS.
Require Import sur_les_relations.
Require Import sigma_lift.
Require Import determinePC_SL.
Goal
forall a b : terms,
exists u : terms,
e_relSLstar _ (app (env a id) (env b id)) u /\ e_relSLstar _ (app a b) u.
intros; exists (app a b); split; red in |- *.
apply star_trans1 with (app a (env b id));... |
Require Export Qsyntax.
Require Export Field_Theory_Q.
Require Export Q_ordered_field_properties.
Lemma Qpositive_in_Q_Archimedean_inf:forall qp:Qpositive, {z:Z | (Qpos qp)<=z /\ (z-(Qpos qp))<= Qone}.
Theorem Q_Archimedean_inf:forall q:Q, {z:Z | q<=z /\ (z-q)<= Qone}.
Definition up_Q q:= proj1_sig (Q_Archimedean_i... |
Require Export GeoCoq.Tarski_dev.Ch13_2_length.
Section Angles_1.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Lemma ang_exists : forall A B C, A <> B -> C <> B -> exists a, Q_CongA a /\ a A B C.
Proof.
(* Goal: forall (A B C : @Tpoint Tn) (_ : not (@eq (@Tpoint Tn) A B)) (_ : not (... |
Require Import abp_base.
Section BOOL.
Variable b : bool.
Parameter andb orb : bool -> bool -> bool.
Parameter notb : bool -> bool.
Axiom andb1 : b = andb true b.
Axiom andb2 : false = andb false b.
Axiom orb1 : true = orb true b.
Axiom orb2 : b = orb false b.
Axiom notb1 : false = notb true.
Axiom notb2 : true = not... |
Require Export GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_trans_NID.
Section T13.
Context `{TE:Tarski_euclidean}.
Lemma cop_npar__inter_exists : forall A1 B1 A2 B2,
Coplanar A1 B1 A2 B2 -> ~ Par A1 B1 A2 B2 -> exists X, Col X A1 B1 /\ Col X... |
Require Import securite.
Lemma POinv1rel6 :
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 mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype.
Require Import BinNat.
Require BinPos Ndec.
Require Export Ring.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Delimit Scope coq_nat_scope with coq_nat.
Notation "m + n" := (plus m n)... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_sgroup.
Require Export Monoid_facts.
Section Def.
Variable G : MONOID.
Section Sub_monoid.
Variable H : subsgroup G.
Hypothesis Hunit : in_part (monoid_unit G) H.
Definition submonoid_monoid : monoid.
Proof.
(* Goal: monoid *)
apply (Build_monoid (mono... |
Section Relations.
Variable A : Set.
Variable R : A -> A -> Prop.
Definition Rstar (x y : A) :=
forall P : A -> A -> Prop,
(forall u : A, P u u) ->
(forall u v w : A, R u v -> P v w -> P u w) -> P x y.
Theorem Rstar_reflexive : forall x : A, Rstar x x.
Proof
fun (x : A) (P : A -> A -> Prop) (h1 ... |
Require Import securite.
Lemma POinv1rel7 :
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... |
Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Z_group.
Section Lemmas.
Variable G : GROUP.
Lemma Z_to_group_nat_eq_pos :
forall (n : Z) (g : G), Equal (Z_to_group_nat_fun g n) (Z_to_group_fun g n).
Proof.
(* Goal: forall (n : Z) (g : Carrier (sgroup_set (monoid_sgroup... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_oppositesidesymmetric.
Require Export GeoCoq.Elements.OriginalProofs.proposition_27.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma proposition_28A :
forall A B C D E G H,
BetS A G B -> BetS C H D -> BetS E G H -> CongA E G B G H D -> ... |
Require Import Arith.
Require Import Test.
Require Import Terms.
Inductive red1 : lambda -> lambda -> Prop :=
| beta : forall M N : lambda, red1 (App (Abs M) N) (subst N M)
| abs_red : forall M N : lambda, red1 M N -> red1 (Abs M) (Abs N)
| app_red_l :
forall M1 N1 : lambda,
red1 M1 N1 -> forall M2 ... |
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat div seq choice tuple.
From mathcomp
Require Import bigop ssralg poly polydiv generic_quotient.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory.
Local Open Scope... |
Require Import basis.
Theorem Uniqueness_of_constructed_lines :
forall (x : Segment) (l : Line),
Incident (origin x) l -> Incident (extremity x) l -> EqLn l (ln x).
Proof.
(* Goal: forall (x : Segment) (l : Line) (_ : Incident (origin x) l) (_ : Incident (extremity x) l), EqLn l (ln x) *)
intros x l.
(* Goal: foral... |
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Endo_set.
Section Def.
Variable M : MONOID.
Variable S : SET.
Definition operation := Hom M (Endo_SET S).
Variable op : operation.
Lemma operation_assoc :
forall (x y : M) (s : S), Equal (op (sgroup_law _ x y) s) (op x (op y s)).
Proof.
(* Goal: forall ... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_extension.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_layoff :
forall A B C D,
neq A B -> neq C D ->
exists X, Out A B X /\ Cong A X C D.
Proof.
(* Goal: forall (A B C D : @Point Ax0) (_ : @neq Ax0 A B) (_ : @neq Ax0 C D),... |
Require Export GeoCoq.Elements.OriginalProofs.lemma_inequalitysymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_3_7b.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_partnotequalwhole :
forall A B C,
BetS A B C ->
~ Cong A B A C.
Proof.
(* Goal: forall (A B C : @Poi... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.