statement stringlengths 1 510 | proof stringlengths 0 12.8k | type stringclasses 24
values | symbolic_name stringlengths 1 50 | library stringclasses 14
values | filename stringclasses 114
values | imports listlengths 1 47 | deps listlengths 0 64 | docstring stringclasses 241
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
continuity_pt_nbhs (f : R -> R) x :
continuity_pt f x <->
forall eps : {posnum R}, nbhs x (fun u => `|f u - f x| < eps%:num). | Proof.
split=> [fcont e|fcont _/RltP/posnumP[e]]; last first.
have [_/posnumP[d] xd_fxe] := fcont e.
exists d%:num; split; first by apply/RltP; have := [gt0 of d%:num].
by move=> y [_ /RltP yxd]; apply/RltP/xd_fxe; rewrite /= distrC.
have /RltP egt0 := [gt0 of e%:num].
have [_ [/RltP/posnumP[d] dx_fxe]] := fcont ... | Lemma | continuity_pt_nbhs | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"RltP",
"nbhs",
"split"
] | TODO: express using ball? | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
continuity_pt_cvg (f : R -> R) (x : R) :
continuity_pt f x <-> {for x, continuous f}. | Proof.
eapply iff_trans; first exact: continuity_pt_nbhs.
apply iff_sym.
have FF : Filter (f @ x)%classic.
by typeclasses eauto.
(*by apply fmap_filter; apply: @filter_filter' (locally_filter _).*)
case: (@fcvg_ballP _ _ (f @ x)%classic FF (f x)) => {FF}H1 H2.
(* TODO: in need for lemmas and/or refactoring of alrea... | Lemma | continuity_pt_cvg | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"Filter",
"ball",
"classic",
"continuity_pt_nbhs",
"continuous",
"fcvg_ballP",
"split",
"x0"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
continuity_ptE (f : R -> R) (x : R) :
continuity_pt f x <-> {for x, continuous f}. | Proof. exact: continuity_pt_cvg. Qed. | Lemma | continuity_ptE | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"continuity_pt_cvg",
"continuous"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
continuity_pt_cvg' f x :
continuity_pt f x <-> f @ x^' --> f x. | Proof. by rewrite continuity_ptE continuous_withinNx. Qed. | Lemma | continuity_pt_cvg' | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"continuity_ptE",
"continuous_withinNx"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
continuity_pt_dnbhs f x :
continuity_pt f x <->
forall eps, 0 < eps -> x^' (fun u => `|f x - f u| < eps). | Proof.
by rewrite continuity_pt_cvg' -filter_fromP cvg_ballP -filter_fromP.
Qed. | Lemma | continuity_pt_dnbhs | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"continuity_pt_cvg'",
"cvg_ballP",
"filter_fromP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
nbhs_pt_comp (P : R -> Prop) (f : R -> R) (x : R) :
nbhs (f x) P -> continuity_pt f x -> \near x, P (f x). | Proof. by move=> Lf /continuity_pt_cvg; apply. Qed. | Lemma | nbhs_pt_comp | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"continuity_pt_cvg",
"nbhs",
"near"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
RexpE (x : R) : Rtrigo_def.exp x = expR x. | Proof.
apply/esym; rewrite /exp /exist_exp; case: Alembert_C3 => y.
rewrite /Pser /infinite_sum /= => exp_ub.
rewrite /expR /exp_coeff /series/=; apply: (@cvg_lim R^o) => //.
rewrite -cvg_shiftS /=; apply/cvgrPdist_lt => /= e /RltP /exp_ub[N Nexp_ub].
near=> n.
have nN : (n >= N)%coq_nat by apply/ssrnat.leP; near: n; e... | Lemma | RexpE | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"INRE",
"RdistE",
"RinvE",
"RltP",
"RpowE",
"cvg_lim",
"cvg_shiftS",
"cvgrPdist_lt",
"end_near",
"exp",
"expR",
"exp_coeff",
"factE",
"mulrC",
"nbhs_infty_ge",
"near",
"series",
"sum_f_R0E"
] | proof by comparing the defining power series | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
RexpE | := RexpE.RexpE. | Definition | RexpE | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
RlnE (x : R) : Rpower.ln x = exp.ln x. | Proof.
rewrite /Rpower.ln /Rln.
have [xle0|xgt0] := leP x 0.
by case: Rlt_dec => //= /[dup] /RltP + ?; rewrite exp.ln0// ltNge xle0.
case: (Rlt_dec 0 x) => [/= ? | /RltP/[!xgt0]//].
by case: ln_exists => y ->; rewrite RexpE exp.expRK.
Qed. | Lemma | RlnE | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"RexpE",
"RltP",
"exp",
"expRK",
"ln",
"ln0"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
RealsE | := (RealsE, RexpE, RlnE). | Definition | RealsE | analysis_stdlib | analysis_stdlib/Rstruct_topology.v | [
"Stdlib",
"Rdefinitions",
"Raxioms",
"RIneq",
"Rbasic_fun",
"Zwf",
"Epsilon",
"FunctionalExtensionality",
"Ranalysis1",
"Rsqrt_def",
"Rtrigo1",
"Reals",
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"ssralg",
"ssrnum",
"archimedean",
"boolp",
"classical_sets",
"re... | [
"RexpE",
"RlnE"
] | extend RealsE from Rstruct.v | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
OuP (f : A -> R * R -> R) (g : R * R -> R) | :=
{ alp : R & { C : R |
0 < alp /\ 0 < C /\
forall X : A, forall dX : R * R,
sqrt (Rsqr (fst dX) + Rsqr (snd dX)) < alp -> P dX ->
Rabs (f X dX) <= C * Rabs (g dX)}}. | Definition | OuP | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
normedR2 : normedModType _ | := (R^o * R^o)%type. | Let | normedR2 | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"type"
] | first we replace sig with ex and the l^2 norm with the l^oo norm | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
OuPex (f : A -> R * R -> R^o) (g : R * R -> R^o) | :=
exists2 alp, 0 < alp & exists2 C, 0 < C &
forall X, forall dX : normedR2,
`|dX| < alp -> P dX -> `|f X dX| <= C * `|g dX|. | Definition | OuPex | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"normedR2"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
ler_norm2 (x : normedR2) :
`|x| <= sqrt (Rsqr (fst x) + Rsqr (snd x)) <= Num.sqrt 2 * `|x|. | Proof.
rewrite RsqrtE !Rsqr_pow2 !RpowE; apply/andP; split.
by rewrite ge_max; apply/andP; split;
rewrite -[`|_|]sqrtr_sqr ler_wsqrtr // (lerDl, lerDr) sqr_ge0.
wlog lex12 : x / (`|x.1| <= `|x.2|).
move=> ler_norm; case: (lerP `|x.1| `|x.2|) => [/ler_norm|] //.
rewrite lt_leAnge => /andP [lex21 _].
rewrite ... | Lemma | ler_norm2 | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"RplusE",
"RpowE",
"RsqrtE",
"addrC",
"normedR2",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
OuP_to_ex f g : OuP f g -> OuPex f g. | Proof.
move=> [_ [_ [/posnumP[a] [/posnumP[C] fOg]]]].
exists (a%:num / Num.sqrt 2) => //; exists C%:num => // x dx ltdxa Pdx.
apply: fOg; move: ltdxa; rewrite ltr_pdivlMr //; apply: le_lt_trans.
by rewrite mulrC; have /andP[] := ler_norm2 dx.
Qed. | Lemma | OuP_to_ex | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"OuP",
"OuPex",
"ler_norm2",
"mulrC"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
Ouex_to_P f g : OuPex f g -> OuP f g. | Proof.
move=> /exists2P /getPex; set Q := fun a => _ /\ _ => - [lt0getQ].
move=> /exists2P /getPex; set R := fun C => _ /\ _ => - [lt0getR fOg].
apply: existT (get Q) _; apply: exist (get R) _; split=> //; split => //.
move=> x dx ltdxgetQ; apply: fOg; apply: le_lt_trans ltdxgetQ.
by have /andP [] := ler_norm2 dx.
Qed. | Lemma | Ouex_to_P | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"OuP",
"OuPex",
"exists2P",
"get",
"getPex",
"ler_norm2",
"set",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
OuO (f : A -> R * R -> R^o) (g : R * R -> R^o) | :=
(fun x => f x.1 x.2) =O_ (filter_prod [set setT]%classic
(within P (nbhs (0%R:R^o, 0%R:R^o))(*[filter of 0 : R^o * R^o]*))) (fun x => g x.2). | Definition | OuO | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"classic",
"filter_prod",
"nbhs",
"set",
"setT",
"within"
] | then we replace the epsilon/delta definition with bigO | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
OuP_to_O f g : OuP f g -> OuO f g. | Proof.
move=> /OuP_to_ex [_/posnumP[a] [_/posnumP[C] fOg]].
apply/eqOP; near=> k; near=> x; apply: le_trans (fOg _ _ _ _) _; last 2 first.
- by near: x; exists (setT, P); [split=> //=; apply: withinT|move=> ? []].
- by rewrite ler_pM.
- near: x; exists (setT, ball (0 : R^o * R^o) a%:num).
by split=> //=; rewrite /w... | Lemma | OuP_to_O | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"OuO",
"OuP",
"OuP_to_ex",
"ball",
"ball_normE",
"end_near",
"eqOP",
"nbhsx_ballx",
"near",
"setT",
"split",
"within",
"withinT"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
OuO_to_P f g : OuO f g -> OuP f g. | Proof.
move=> fOg; apply/Ouex_to_P; move: fOg => /eqOP [k [kreal hk]].
have /hk [Q [->]] : k < maxr 1 (k + 1) by rewrite lt_max ltrDl orbC ltr01.
move=> [R [[_/posnumP[e1] Re1] [_/posnumP[e2] Re2]] sRQ] fOg.
exists (minr e1%:num e2%:num) => //.
exists (maxr 1 (k + 1)); first by rewrite lt_max ltr01.
move=> x dx dxe Pdx... | Lemma | OuO_to_P | analysis_stdlib.showcase | analysis_stdlib/showcase/uniform_bigO.v | [
"Stdlib",
"Reals",
"Corelib",
"ssreflect",
"ssrfun",
"ssrbool",
"mathcomp",
"ssrnat",
"eqtype",
"choice",
"fintype",
"bigop",
"order",
"ssralg",
"ssrnum",
"boolp",
"reals",
"Rstruct_topology",
"ereal",
"classical_sets",
"interval_inference",
"topology",
"normedtype",
"l... | [
"OuO",
"OuP",
"Ouex_to_P",
"eqOP",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
functional_extensionality_dep :
forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
(forall x : A, f x = g x) -> f = g. | Axiom | functional_extensionality_dep | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | ||
propositional_extensionality :
forall P Q : Prop, P <-> Q -> P = Q. | Axiom | propositional_extensionality | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | ||
constructive_indefinite_description :
forall (A : Type) (P : A -> Prop),
(exists x : A, P x) -> {x : A | P x}. | Axiom | constructive_indefinite_description | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | ||
cid | := constructive_indefinite_description. | Notation | cid | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"constructive_indefinite_description"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
cid2 (A : Type) (P Q : A -> Prop) :
(exists2 x : A, P x & Q x) -> {x : A | P x & Q x}. | Proof.
move=> PQA; suff: {x | P x /\ Q x} by move=> [a [*]]; exists a.
by apply: cid; case: PQA => x; exists x.
Qed. | Lemma | cid2 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"cid"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
existT_inj1 (T : Type) (P : T -> Type) (x y : T) (Px : P x) (Py : P y) :
existT P x Px = existT P y Py -> x = y. | Proof. by case. Qed. | Lemma | existT_inj1 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
existT_inj2 (T : eqType) (P : T -> Type) (x : T) (Px1 Px2 : P x) :
existT P x Px1 = existT P x Px2 -> Px1 = Px2. | Proof.
apply: internal_Eqdep_dec.inj_pair2_eq_dec => y z.
by have [|/eqP] := eqVneq y z; [left|right].
Qed. | Lemma | existT_inj2 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"inj_pair2_eq_dec"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
surjective_existT (T : Type) (P : T -> Type) (p : {x : T & P x}):
existT [eta P] (projT1 p) (projT2 p) = p. | Proof. by case: p. Qed. | Lemma | surjective_existT | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
mextensionality | := {
_ : forall (P Q : Prop), (P <-> Q) -> (P = Q);
_ : forall {T U : Type} (f g : T -> U),
(forall x, f x = g x) -> f = g;
}. | Record | mextensionality | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
extensionality : mextensionality. | Proof.
split.
- exact: propositional_extensionality.
- by move=> T U f g; apply: functional_extensionality_dep.
Qed. | Fact | extensionality | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"functional_extensionality_dep",
"mextensionality",
"propositional_extensionality",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
propext (P Q : Prop) : (P <-> Q) -> (P = Q). | Proof. by have [propext _] := extensionality; apply: propext. Qed. | Lemma | propext | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"extensionality"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eqProp | := apply: propext; split. | Ltac | eqProp | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propext",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funext {T U : Type} (f g : T -> U) : (f =1 g) -> f = g. | Proof. by case: extensionality=> _; apply. Qed. | Lemma | funext | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"extensionality"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
propeqE (P Q : Prop) : (P = Q) = (P <-> Q). | Proof. by apply: propext; split=> [->|/propext]. Qed. | Lemma | propeqE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propext",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
propeqP (P Q : Prop) : (P = Q) <-> (P <-> Q). | Proof. by rewrite propeqE. Qed. | Lemma | propeqP | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeqE {T U : Type} (f g : T -> U) : (f = g) = (f =1 g). | Proof. by rewrite propeqE; split=> [->//|/funext]. Qed. | Lemma | funeqE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funext",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeq2E {T U V : Type} (f g : T -> U -> V) : (f = g) = (f =2 g). | Proof.
by rewrite propeqE; split=> [->//|?]; rewrite funeqE=> x; rewrite funeqE.
Qed. | Lemma | funeq2E | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeqE",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeq3E {T U V W : Type} (f g : T -> U -> V -> W) :
(f = g) = (forall x y z, f x y z = g x y z). | Proof.
by rewrite propeqE; split=> [->//|?]; rewrite funeq2E=> x y; rewrite funeqE.
Qed. | Lemma | funeq3E | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq2E",
"funeqE",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeqP {T U : Type} (f g : T -> U) : (f = g) <-> (f =1 g). | Proof. by rewrite funeqE. Qed. | Lemma | funeqP | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeq2P {T U V : Type} (f g : T -> U -> V) : (f = g) <-> (f =2 g). | Proof. by rewrite funeq2E. Qed. | Lemma | funeq2P | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq2E"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
funeq3P {T U V W : Type} (f g : T -> U -> V -> W) :
(f = g) <-> (forall x y z, f x y z = g x y z). | Proof. by rewrite funeq3E. Qed. | Lemma | funeq3P | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq3E"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeqE {T} (P Q : T -> Prop) : (P = Q) = (forall x, P x <-> Q x). | Proof.
by rewrite propeqE; split=> [->//|?]; rewrite funeqE=> x; rewrite propeqE.
Qed. | Lemma | predeqE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeqE",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeq2E {T U} (P Q : T -> U -> Prop) :
(P = Q) = (forall x y, P x y <-> Q x y). | Proof.
by rewrite propeqE; split=> [->//|?]; rewrite funeq2E=> ??; rewrite propeqE.
Qed. | Lemma | predeq2E | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq2E",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeq3E {T U V} (P Q : T -> U -> V -> Prop) :
(P = Q) = (forall x y z, P x y z <-> Q x y z). | Proof.
by rewrite propeqE; split=> [->//|?]; rewrite funeq3E=> ???; rewrite propeqE.
Qed. | Lemma | predeq3E | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq3E",
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeqP {T} (A B : T -> Prop) : (A = B) <-> (forall x, A x <-> B x). | Proof. by rewrite predeqE. Qed. | Lemma | predeqP | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"predeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeq2P {T U} (P Q : T -> U -> Prop) :
(P = Q) <-> (forall x y, P x y <-> Q x y). | Proof. by rewrite predeq2E. Qed. | Lemma | predeq2P | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"predeq2E"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
predeq3P {T U V} (P Q : T -> U -> V -> Prop) :
(P = Q) <-> (forall x y z, P x y z <-> Q x y z). | Proof. by rewrite predeq3E. Qed. | Lemma | predeq3P | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"predeq3E"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
propT {P : Prop} : P -> P = True. | Proof. by move=> p; rewrite propeqE. Qed. | Lemma | propT | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
Prop_irrelevance (P : Prop) (x y : P) : x = y. | Proof. by move: x (x) y => /propT-> [] []. Qed. | Lemma | Prop_irrelevance | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propT"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
mclassic | := {
_ : forall (P : Prop), {P} + {~P};
_ : forall T, hasChoice T
}. | Record | mclassic | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
choice X Y (P : X -> Y -> Prop) :
(forall x, exists y, P x y) -> {f & forall x, P x (f x)}. | Proof. by move=> /(_ _)/constructive_indefinite_description -/all_tag. Qed. | Lemma | choice | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"constructive_indefinite_description"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
EM P : P \/ ~ P. | Proof.
pose U val := fun Q : bool => Q = val \/ P.
have Uex val : exists b, U val b by exists val; left.
pose f val := projT1 (cid (Uex val)).
pose Uf val : U val (f val) := projT2 (cid (Uex val)).
have : f true != f false \/ P.
have [] := (Uf true, Uf false); rewrite /U.
by move=> [->|?] [->|?] ; do ?[by right]; l... | Theorem | EM | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"cid",
"predeqE",
"split"
] | Diaconescu Theorem | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
pselect (P : Prop): {P} + {~P}. | Proof.
have : exists b, if b then P else ~ P.
by case: (EM P); [exists true|exists false].
by move=> /cid [[]]; [left|right].
Qed. | Lemma | pselect | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"EM",
"cid"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
pselectT T : (T -> False) + T. | Proof.
have [/cid[]//|NT] := pselect (exists t : T, True); first by right.
by left=> t; case: NT; exists t.
Qed. | Lemma | pselectT | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"cid",
"pselect"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
classic : mclassic. | Proof.
split=> [|T]; first exact: pselect.
exists (fun (P : pred T) (n : nat) =>
if pselect (exists x, P x) isn't left ex then None
else Some (projT1 (cid ex)))
=> [P n x|P [x Px]|P Q /funext -> //].
by case: pselect => // ex [<- ]; case: cid.
by exists 0; case: pselect => // -[]; exists x.
Qed. | Lemma | classic | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"cid",
"funext",
"mclassic",
"pselect",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
gen_choiceMixin (T : Type) : hasChoice T. | Proof. by case: classic. Qed. | Lemma | gen_choiceMixin | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"classic"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
lem (P : Prop): P \/ ~P. | Proof. by case: (pselect P); tauto. Qed. | Lemma | lem | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"pselect"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
trueE : true = True :> Prop. | Proof. by rewrite propeqE; split. Qed. | Lemma | trueE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
falseE : false = False :> Prop. | Proof. by rewrite propeqE; split. Qed. | Lemma | falseE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
propF (P : Prop) : ~ P -> P = False. | Proof. by move=> p; rewrite propeqE; tauto. Qed. | Lemma | propF | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_fun T rT (U V : T -> rT) :
(forall x : T, U x = V x) -> (fun x => U x) = (fun x => V x). | Proof. by move=> /funext->. Qed. | Lemma | eq_fun | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funext",
"rT"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq2_fun T1 T2 rT (U V : T1 -> T2 -> rT) :
(forall x y, U x y = V x y) -> (fun x y => U x y) = (fun x y => V x y). | Proof. by move=> UV; rewrite funeq2E => x y; rewrite UV. Qed. | Lemma | eq2_fun | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq2E",
"rT"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_fun2 | := eq2_fun (only parsing). | Notation | eq_fun2 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq2_fun"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq3_fun T1 T2 T3 rT (U V : T1 -> T2 -> T3 -> rT) :
(forall x y z, U x y z = V x y z) ->
(fun x y z => U x y z) = (fun x y z => V x y z). | Proof. by move=> UV; rewrite funeq3E => x y z; rewrite UV. Qed. | Lemma | eq3_fun | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"funeq3E",
"rT"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_fun3 | := eq3_fun (only parsing). | Notation | eq_fun3 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq3_fun"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_forall T (U V : T -> Prop) :
(forall x : T, U x = V x) -> (forall x, U x) = (forall x, V x). | Proof. by move=> e; rewrite propeqE; split=> ??; rewrite (e,=^~e). Qed. | Lemma | eq_forall | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq2_forall T S (U V : forall x : T, S x -> Prop) :
(forall x y, U x y = V x y) -> (forall x y, U x y) = (forall x y, V x y). | Proof. by move=> UV; apply/eq_forall => x; exact/eq_forall. Qed. | Lemma | eq2_forall | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq_forall"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_forall2 | := eq2_forall (only parsing). | Notation | eq_forall2 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq2_forall"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq3_forall T S R (U V : forall (x : T) (y : S x), R x y -> Prop) :
(forall x y z, U x y z = V x y z) ->
(forall x y z, U x y z) = (forall x y z, V x y z). | Proof. by move=> UV; apply/eq2_forall => x y; exact/eq_forall. Qed. | Lemma | eq3_forall | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq2_forall",
"eq_forall"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_forall3 | := eq3_forall (only parsing). | Notation | eq_forall3 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq3_forall"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_exists T (U V : T -> Prop) :
(forall x : T, U x = V x) -> (exists x, U x) = (exists x, V x). | Proof.
by move=> e; rewrite propeqE; split=> - [] x ?; exists x; rewrite (e,=^~e).
Qed. | Lemma | eq_exists | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq2_exists T S (U V : forall x : T, S x -> Prop) :
(forall x y, U x y = V x y) -> (exists x y, U x y) = (exists x y, V x y). | Proof. by move=> UV; apply/eq_exists => x; exact/eq_exists. Qed. | Lemma | eq2_exists | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq_exists"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq3_exists T S R (U V : forall (x : T) (y : S x), R x y -> Prop) :
(forall x y z, U x y z = V x y z) ->
(exists x y z, U x y z) = (exists x y z, V x y z). | Proof. by move=> UV; apply/eq2_exists => x y; exact/eq_exists. Qed. | Lemma | eq3_exists | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq2_exists",
"eq_exists"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_exists3 | := eq3_exists (only parsing). | Notation | eq_exists3 | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eq3_exists"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_exist T (P : T -> Prop) (s t : T) (p : P s) (q : P t) :
s = t -> exist P s p = exist P t q. | Proof. by move=> st; case: _ / st in q *; apply/congr1. Qed. | Lemma | eq_exist | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
forall_swap T S (U : forall (x : T) (y : S), Prop) :
(forall x y, U x y) = (forall y x, U x y). | Proof. by rewrite propeqE; split. Qed. | Lemma | forall_swap | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
exists_swap T S (U : forall (x : T) (y : S), Prop) :
(exists x y, U x y) = (exists y x, U x y). | Proof. by rewrite propeqE; split => -[x [y]]; exists y, x. Qed. | Lemma | exists_swap | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
reflect_eq (P : Prop) (b : bool) : reflect P b -> P = b. | Proof. by rewrite propeqE; exact: rwP. Qed. | Lemma | reflect_eq | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propeqE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asbool (P : Prop) | := if pselect P then true else false. | Definition | asbool | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"pselect"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
"`[< P >]" | := (asbool P) : bool_scope. | Notation | `[< P >] | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asbool"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolE (P : Prop) : `[<P>] = P :> Prop. | Proof. by rewrite propeqE /asbool; case: pselect; split. Qed. | Lemma | asboolE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asbool",
"propeqE",
"pselect",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolP (P : Prop) : reflect P `[<P>]. | Proof. by apply: (equivP idP); rewrite asboolE. Qed. | Lemma | asboolP | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolb (b : bool) : `[< b >] = b. | Proof. by apply/asboolP/idP. Qed. | Lemma | asboolb | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolPn (P : Prop) : reflect (~ P) (~~ `[<P>]). | Proof. by rewrite /asbool; case: pselect=> h; constructor. Qed. | Lemma | asboolPn | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asbool",
"pselect"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolW (P : Prop) : `[<P>] -> P. | Proof. by case: asboolP. Qed. | Lemma | asboolW | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
orW A B : A \/ B -> A + B. | Proof.
have [|NA] := asboolP A; first by left.
have [|NB] := asboolP B; first by right.
by move=> AB; exfalso; case: AB.
Qed. | Lemma | orW | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
or3W A B C : [\/ A, B | C] -> A + B + C. | Proof.
have [|NA] := asboolP A; first by left; left.
have [|NB] := asboolP B; first by left; right.
have [|NC] := asboolP C; first by right.
by move=> ABC; exfalso; case: ABC.
Qed. | Lemma | or3W | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
or4W A B C D : [\/ A, B, C | D] -> A + B + C + D. | Proof.
have [|NA] := asboolP A; first by left; left; left.
have [|NB] := asboolP B; first by left; left; right.
have [|NC] := asboolP C; first by left; right.
have [|ND] := asboolP D; first by right.
by move=> ABCD; exfalso; case: ABCD.
Qed. | Lemma | or4W | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
asboolT (P : Prop) : P -> `[<P>]. | Proof. by case: asboolP. Qed. | Lemma | asboolT | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | Shall this be a coercion ? | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
asboolF (P : Prop) : ~ P -> `[<P>] = false. | Proof. by apply/introF/asboolP. Qed. | Lemma | asboolF | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eq_opE (T : eqType) (x y : T) : (x == y : Prop) = (x = y). | Proof. by apply/propext; split=> /eqP. Qed. | Lemma | eq_opE | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"propext",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
is_true_inj : injective is_true. | Proof. by move=> [] []; rewrite ?(trueE, falseE) ?propeqE; tauto. Qed. | Lemma | is_true_inj | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"falseE",
"propeqE",
"trueE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
gen_eq (T : Type) (u v : T) | := `[<u = v>]. | Definition | gen_eq | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
gen_eqP (T : Type) : Equality.axiom (@gen_eq T). | Proof. by move=> x y; apply: (iffP (asboolP _)). Qed. | Lemma | gen_eqP | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"asboolP",
"gen_eq"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
gen_eqMixin (T : Type) : hasDecEq T | :=
hasDecEq.Build T (@gen_eqP T). | Definition | gen_eqMixin | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"Build",
"gen_eqP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
classicType | := T. | Definition | classicType | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
"'{classic' T }" | := (classicType T)
(format "'{classic' T }") : type_scope. | Notation | '{classic' T } | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"classicType"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
eclassicType : Type | := T. | Definition | eclassicType | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
"'{eclassic' T }" | := (eclassicType T)
(format "'{eclassic' T }") : type_scope. | Notation | '{eclassic' T } | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"eclassicType"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
canonical_of T U (sort : U -> T) | := forall (G : T -> Type),
(forall x', G (sort x')) -> forall x, G x. | Definition | canonical_of | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
canonical_ sort | := (@canonical_of _ _ sort). | Notation | canonical_ | classical | classical/boolp.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"mathcomp_extra",
"internal_Eqdep_dec",
"Order.TTheory",
"FunOrder.Exports"
] | [
"canonical_of"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
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