statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
is_tautoT l f g :
match deduce f g with None => false | Some u => unsat u end ->
~~ NFeval l f || ~~ NFeval l g. | Proof.
case: nformula_plus_nformula (@eval_nformula_plus_nformula l f g) => [f'|//].
move=> /(_ f') + if'.
rewrite -negb_and; apply: contraPN => /andP[-> ->] /(_ erefl erefl erefl).
by rewrite (check_inconsistentT _ _ if').
Qed. | Lemma | is_tautoT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"NFeval",
"apply",
"check_inconsistentT",
"deduce",
"eval_nformula_plus_nformula",
"nformula_plus_nformula",
"unsat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_nth T n s (x0 : T) : ListDef.nth n s x0 = nth x0 s n. | Proof. by elim: s n => [| e s IH] /= [| n]. Qed. | Lemma | nth_nth | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_eval_Psatz l lf (w : Psatz C) : all (NFeval l) lf ->
forall f : NFormula C, eval_Psatz lf w = Some f -> NFeval l f. | Proof.
elim: w lf.
- move=> p1 IHp1 p2 IHp2 lf elf f /=.
case: eval_Psatz (IHp1 lf elf) => [f' /(_ _ erefl) ef'|//].
by apply: IHp2; rewrite /= ef'.
- move=> n lf /all_nthP elf f /=[<-].
case: (ltnP n (size lf)) => nslf; first by rewrite nth_nth elf.
by rewrite nth_nth nth_default/= ?rmorph0.
- by move=> e lf e... | Lemma | eval_eval_Psatz | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"NFeval",
"Peval_square",
"R_of_C_ge0",
"all",
"all_nthP",
"apply",
"c0",
"eval_Psatz",
"eval_nformula_plus_nformula",
"eval_nformula_times_nformula",
"eval_pexpr_times_nformula",
"f1",
"f2",
"last",
"ltNge",
"ltP",
"lt_neqAle",
"ltnP",
"nth_default",
"nth_nth",
"oppr_ge0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
check_normalised_formulasT l (lf : seq (NFormula C)) (w : Psatz C) :
check_normalised_formulas lf w -> has (fun f => ~~ NFeval l f) lf. | Proof.
rewrite has_predC; apply: contraL => /(@eval_eval_Psatz _ _ w).
rewrite /check_normalised_formulas; case: eval_Psatz => [f /(_ _ erefl)|//].
by apply: contraL => /check_inconsistentT->.
Qed. | Lemma | check_normalised_formulasT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"NFeval",
"apply",
"check_inconsistentT",
"check_normalised_formulas",
"eval_Psatz",
"eval_eval_Psatz",
"has",
"has_predC",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tauto_checker | := (tauto_checker
(fun c : seq (NFormula C * unit) => check_normalised_formulas (map fst c))). | Notation | tauto_checker | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"check_normalised_formulas",
"map",
"seq",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
check_normalised_formulasT | := (check_normalised_formulasT
R_of_C_ge0). | Notation | check_normalised_formulasT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"R_of_C_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tauto_checkerT l (f : cnf (NFormula C) unit) (w : seq (Psatz C)) :
tauto_checker f w -> eval_cnf (fun f => ~~ NFeval R_of_C l f) f. | Proof.
elim: f w => [//| f lf IH] [//| w lw] /=/andP[/(check_normalised_formulasT l)].
by rewrite has_map /eval_clause => -> /IH.
Qed. | Lemma | tauto_checkerT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"NFeval",
"check_normalised_formulasT",
"cnf",
"eval_clause",
"eval_cnf",
"has_map",
"seq",
"tauto_checker",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_Rxx u : unit_R u u. | Proof. by case: u; apply: tt_R. Qed. | Lemma | unit_Rxx | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"apply"
] | a bunch of helper lemmas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
option_Rxx A (RA : A -> A -> Type) (_ : forall x, RA x x) s :
option_R RA s s. | Proof. by case: s => [x |]; constructor. Qed. | Lemma | option_Rxx | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
list_Rxx A (RA : A -> A -> Type) (_ : forall x, RA x x) s : list_R RA s s. | Proof. by elim: s => [| x s IH]; constructor. Qed. | Lemma | list_Rxx | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eKind_Rxx k (rk : kind_R k k) (t : eKind k) : eKind_R rk t t. | Proof.
case: k rk t => rk t.
- by refine
(match rk in kind_R k1 k2
return
eKind_R rk
(match k1 with isProp => t | isBool => true end)
(match k2 with isProp => t | isBool => true end)
with
| isProp_R => fun _ _ => True
| isBool_R => true_R
end).
- b... | Lemma | eKind_Rxx | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"True",
"bool_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
erefl1 {A B} {f : A -> B} : forall a1 a2 (ra : a1 = a2), f a1 = f a2. | Proof. by move=> ? ? ->. Qed. | Lemma | erefl1 | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
erefl2 {A B C} {f : A -> B -> C} :
forall a1 a2 (ra : a1 = a2) b1 b2 (rb : b1 = b2), f a1 b1 = f a2 b2. | Proof. by move=> ? ? -> ? ? ->. Qed. | Lemma | erefl2 | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
erefl2b {A B} {f : A -> B -> bool} :
forall a1 a2 (ra : a1 = a2) b1 b2 (rb : b1 = b2), bool_R (f a1 b1) (f a2 b2). | Proof. by move=> ? ? -> ? ? ->; apply: bool_Rxx. Qed. | Lemma | erefl2b | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"a1",
"a2",
"apply",
"bool_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
erefl2n {A B} {f : A -> N -> B} :
forall a1 a2 (ra : a1 = a2) b1 b2 (rb : N_R b1 b2), f a1 b1 = f a2 b2. | Proof. by move=> _ ? -> _ ? /N_R_eq->. Qed. | Lemma | erefl2n | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"N_R_eq",
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Formula_R_map A B (RAB : A -> B -> Type) (f : A -> B) :
(forall a, RAB a (f a)) ->
forall g : Formula A, Formula_R RAB g (Fmap f g). | Proof.
move=> rf [lhs o rhs]; apply: Build_Formula_R.
- exact: PExpr_R_map.
- by case: o; constructor.
- exact: PExpr_R_map.
Qed. | Lemma | Formula_R_map | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"PExpr_R_map",
"apply",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BFormula_R_map A B (RAB : A -> B -> Type) (f : A -> B) :
(forall a, RAB a (f a)) ->
forall k (rk : kind_R k k) (g : BFormula (Formula A) k),
BFormula_R (Formula_R RAB) rk g (GFmap (Fmap f) g). | Proof.
move=> rf k rk g; elim: g rk.
- by move=> {}k rk; apply: TT_R.
- by move=> {}k rk; apply: FF_R.
- by move=> {}k t rk; apply/X_R/eKind_Rxx.
- by move=> {}k g u rk; apply: A_R; [apply: Formula_R_map | apply: unit_Rxx].
- by move=> {}k f1 IH1 f2 IH2 rk; apply: AND_R; [apply: IH1 | apply: IH2].
- by move=> {}k f1 IH... | Lemma | BFormula_R_map | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"Formula_R_map",
"apply",
"eKind_Rxx",
"f1",
"f2",
"option_Rxx",
"unit_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pol_R_map A B (RAB : A -> B -> Type) (f : A -> B) :
(forall a, RAB a (f a)) -> forall w : Pol A, Pol_R RAB w (Pmap f w). | Proof.
by move=> rf; elim=> [c | j P IH | P IHP i Q IHQ];
constructor=> //; apply: positive_Rxx.
Qed. | Lemma | Pol_R_map | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"apply",
"positive_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Psatz_R_map A B (RAB : A -> B -> Type) (f : A -> B) :
(forall a, RAB a (f a)) -> forall w : Psatz A, Psatz_R RAB w (Psatz_map f w). | Proof.
move=> rf; elim=> [p1 IH1 p2 IH2|n|p|re e IH|f1 IH1 f2 IH2|f1 IH1 f2 IH2|c|].
- by apply: PsatzLet_R; [apply: IH1 | apply: IH2].
- exact/PsatzIn_R/nat_Rxx.
- exact/PsatzSquare_R/Pol_R_map.
- by apply: PsatzMulC_R; [apply: Pol_R_map | apply: IH].
- by apply: PsatzMulE_R; [apply: IH1 | apply: IH2].
- by apply: Psa... | Lemma | Psatz_R_map | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Pol_R_map",
"apply",
"f1",
"f2",
"nat_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_of_AC_eq0 : {mono R_of_AC : x / x == 0}. | Hypothesis | R_of_AC_eq0 | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
R_of_AC_ge0 : {mono R_of_AC : x / 0 <= x}. | Hypothesis | R_of_AC_ge0 | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACD : forall c ac (_ : CAC c ac) c' ac' (_ : CAC c' ac'),
CAC (cadd c c') (ac + ac'). | Hypothesis | CACD | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACM : forall c ac (_ : CAC c ac) c' ac' (_ : CAC c' ac'),
CAC (cmul c c') (ac * ac'). | Hypothesis | CACM | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACB : forall c ac (_ : CAC c ac) c' ac' (_ : CAC c' ac'),
CAC (csub c c') (ac - ac'). | Hypothesis | CACB | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACN : forall c ac, CAC c ac -> CAC (copp c) (- ac). | Hypothesis | CACN | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACeq : forall c ac (_ : CAC c ac) c' ac' (_ : CAC c' ac'),
ceqb c c' = (ac == ac'). | Hypothesis | CACeq | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CACle : forall c ac (_ : CAC c ac) c' ac' (_ : CAC c' ac'),
cleb c c' = (ac <= ac'). | Hypothesis | CACle | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
R_of_C_R_of_AC : forall c ac, CAC c ac -> R_of_C c = R_of_AC ac. | Hypothesis | R_of_C_R_of_AC | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CAC_AC_of_C : forall c, CAC c (AC_of_C c). | Hypothesis | CAC_AC_of_C | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
CTautoChecker_map_AC_of_C f w :
@CTautoChecker C cO cI cadd cmul csub copp ceqb cleb f w
= @CTautoChecker AC 0 1 +%R *%R (fun x y : AC => x - y) -%R eq_op <=%R
(GFmap (Fmap AC_of_C) f) [seq Psatz_map AC_of_C ps | ps <- w]. | Proof.
have rf : BFormula_R (Formula_R CAC) isProp_R f (GFmap (Fmap AC_of_C) f).
exact: BFormula_R_map.
have rw : list_R (Psatz_R CAC) w [seq Psatz_map AC_of_C ps | ps <- w].
by apply: list_R_map => w'; apply: Psatz_R_map.
by apply: bool_R_eq; apply: CTautoChecker_R rw => //; apply: eq_bool_R2.
Qed. | Lemma | CTautoChecker_map_AC_of_C | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"AC",
"BFormula_R_map",
"Psatz_R_map",
"apply",
"bool_R_eq",
"eq_bool_R2",
"list_R_map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RBFeval_map_AC_of_C_bool l (f : BFormula (Formula C) isBool) :
RBFeval R_of_AC l (GFmap (Fmap AC_of_C) f) = RBFeval R_of_C l f. | Proof.
have rl : list_R eq l l by apply: list_Rxx.
have rf : BFormula_R (Formula_R CAC) isBool_R f (GFmap (Fmap AC_of_C) f).
exact: BFormula_R_map.
by rewrite /RBFeval (bool_R_eq (BFKFeval_R erefl erefl erefl2 erefl2 erefl1
erefl2n erefl2b erefl2b erefl2b erefl2b R_of_C_R_of_AC rl rf)).
Qed. | Lemma | RBFeval_map_AC_of_C_bool | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"BFormula_R_map",
"Formula",
"RBFeval",
"R_of_C_R_of_AC",
"apply",
"bool_R_eq",
"erefl1",
"erefl2",
"erefl2b",
"erefl2n",
"list_Rxx"
] | Unfortunately, we can only use derive.param2 for bool, not Prop | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
RBFeval_map_AC_of_C l k (f : BFormula (Formula C) k) :
hold true
(eIFF eqb k (RBFeval R_of_AC l (GFmap (Fmap AC_of_C) f))
(RBFeval R_of_C l f)). | Proof.
elim: f.
- by case.
- by case.
- by case=> f => [//|/=]; rewrite -/(f == f) eqxx.
- case=> f u; last by apply/eqP; rewrite -RBFeval_map_AC_of_C_bool.
have rN n n' : N_R n n' -> n = n' by move=> /N_R_eq->.
have renv_nth : forall i i' (ii' : positive_R i i')
(s s' : seq R) (ss' : list_R eq s s'), env_nth... | Lemma | RBFeval_map_AC_of_C | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"Formula_R_map",
"N_R_eq",
"RBFeval",
"RBFeval_map_AC_of_C_bool",
"RFevalP",
"R_of_C_R_of_AC",
"apply",
"bool_R_eq",
"env_nth",
"eqb",
"eqxx",
"erefl1",
"erefl2",
"erefl2b",
"hold",
"last",
"list_R_eq",
"list_Rxx",
"n'",
"positive_R_eq",
"seq",
"split"
] | So we have to do the Prop case by hand (but we can still use Feval_R there
and it isn't much work in the end). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
CTautoCheckerT l f w :
@CTautoChecker C cO cI cadd cmul csub copp ceqb cleb f w ->
RBFeval R_of_C l f. | Proof.
rewrite CTautoChecker_map_AC_of_C => checkfw.
apply: (RBFeval_map_AC_of_C l f).1.
rewrite -/(hold true (RBFeval R_of_AC l (GFmap (Fmap AC_of_C) f))).
apply: eval_cnf_of_GFormula => //; first exact: is_tautoT.
exact: tauto_checkerT checkfw.
Qed. | Lemma | CTautoCheckerT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"CTautoChecker_map_AC_of_C",
"RBFeval",
"RBFeval_map_AC_of_C",
"apply",
"eval_cnf_of_GFormula",
"hold",
"is_tautoT",
"tauto_checkerT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_of_Q (R : unitRingType) (q : Q) : R | :=
let: Qmake n d := q in
if d is xH then (int_of_Z n)%:~R
else if n is Zpos xH then (Pos.to_nat d)%:R ^-1
else (int_of_Z n)%:~R / (Pos.to_nat d)%:R. | Definition | R_of_Q | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"int_of_Z"
] | Refinement from rat to Q, for actual computation in the reflexive tactic. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
R_of_Q_ratr (R : numFieldType) q r : Qrat q r -> R_of_Q R q = ratr r. | Proof.
suff -> : R_of_Q R q = (int_of_Z (Qnum q))%:~R / (Pos.to_nat (Qden q))%:R.
by move/eqP => <-; rewrite fmorph_div/= ratr_int ratr_nat.
by case: q => n [d | d |]/=; [| |by rewrite Pos_to_nat1 divr1];
case: n => [//| [//|//|/=] |//]; rewrite Pos_to_nat1 div1r.
Qed. | Lemma | R_of_Q_ratr | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Pos_to_nat1",
"Qrat",
"R_of_Q",
"div1r",
"divr1",
"fmorph_div",
"int_of_Z",
"ratr",
"ratr_int",
"ratr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
QTautoCheckerT (R : realFieldType) (l : seq R) f w :
QTautoChecker f w -> RBFeval (R_of_Q R) l f. | Proof.
exact: (CTautoCheckerT (ler0q R) Qrat0 Qrat1 QratD QratM QratB QratN
Qrat_eq Qrat_le (R_of_Q_ratr R) Qrat_rat_of_Q).
Qed. | Lemma | QTautoCheckerT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"CTautoCheckerT",
"Qrat0",
"Qrat1",
"QratB",
"QratD",
"QratM",
"QratN",
"Qrat_eq",
"Qrat_le",
"Qrat_rat_of_Q",
"RBFeval",
"R_of_Q",
"R_of_Q_ratr",
"ler0q",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZTautoChecker | := @CTautoChecker Z
Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.eqb Z.leb. | Definition | ZTautoChecker | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"add",
"eqb",
"mul",
"opp",
"sub"
] | Refinement from int to Z, for actual computation in the reflexive tactic. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ZTautoCheckerT (R : realDomainType) (l : seq R) f w :
ZTautoChecker f w -> RBFeval (R_of_Z R) l f. | Proof.
exact: (CTautoCheckerT (ler0z R) (CAC:=Zint) Zint0 _ ZintD ZintM ZintB ZintN
Zint_eq Zint_le (R_of_Z_intr R) Zint_int_of_Z).
Qed. | Lemma | ZTautoCheckerT | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"CTautoCheckerT",
"RBFeval",
"R_of_Z",
"R_of_Z_intr",
"ZTautoChecker",
"Zint",
"Zint0",
"ZintB",
"ZintD",
"ZintM",
"ZintN",
"Zint_eq",
"Zint_int_of_Z",
"Zint_le",
"ler0z",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RFormula R | := { Rlhs : RExpr R; Rop : Op2; Rrhs : RExpr R }. | Record | RFormula | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"RExpr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_expr | := (Rnorm R_of_N (R_of_N false N0)
(R_of_N false (Npos xH)) add mul opp_intr exp inv true false). | Notation | Rnorm_expr | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"R_of_N",
"Rnorm",
"add",
"exp",
"inv",
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Reval_op2 k : Op2 -> R -> R -> eKind k | :=
if k is isBool then eval_op2 isBool beq bneq le lt
else eval_op2 isProp eq (fun x y => ~ x = y) le lt. | Definition | Reval_op2 | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"le",
"lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Reval_formula k (ff : RFormula R) : eKind k | :=
let: Build_RFormula lhs o rhs := ff in Reval_op2 k o (Reval lhs) (Reval rhs). | Definition | Reval_formula | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"RFormula",
"Reval",
"Reval_op2",
"ff",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_formula k (ff : RFormula R) | :=
let: Build_RFormula lhs o rhs := ff in
Reval_op2 k o (Rnorm_expr id lhs) (Rnorm_expr id rhs). | Definition | Rnorm_formula | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"RFormula",
"Reval_op2",
"Rnorm_expr",
"ff",
"id",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_formula_correct k (ff : RFormula R) :
Reval_formula k ff = Rnorm_formula k ff. | Proof.
case: ff => lhs o rhs /=.
have e : true = (if R is SemiRing _ then false else true) && true.
by case: R R_not_semiring.
rewrite !(@Rnorm_correct R true _ e) !R_of_NE addE opp_intrE mulE expE invE.
by congr Reval_op2; apply: Rnorm_eq_F_of_N.
Qed. | Lemma | Rnorm_formula_correct | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"RFormula",
"Reval_formula",
"Reval_op2",
"Rnorm_correct",
"Rnorm_eq_F_of_N",
"Rnorm_formula",
"SemiRing",
"addE",
"apply",
"expE",
"ff",
"mulE",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_bf_correct k (ff : BFormula (RFormula R) k) :
BFeval eqb Reval_formula ff = BFeval eqb Rnorm_formula ff. | Proof.
elim: ff => // {k}.
- by move=> k ff ?; exact: Rnorm_formula_correct.
- by move=> k ff1 IH1 ff2 IH2; congr eAND.
- by move=> k ff1 IH1 ff2 IH2; congr eOR.
- by move=> k ff IH; congr eNOT.
- by move=> k ff1 IH1 o ff2 IH2; congr eIMPL.
- by move=> k ff1 IH1 ff2 IH2; congr eIFF.
- by move=> ff1 IH1 ff2 IH2; congr e... | Lemma | Rnorm_bf_correct | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"RFormula",
"Reval_formula",
"Rnorm_formula",
"Rnorm_formula_correct",
"eqb",
"ff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FTautoChecker_sound (F : realFieldType)
(ff : BFormula (RFormula F) isProp)
(f : BFormula (Formula Q) isProp) (env : seq F) (w : seq (Psatz Q)) :
(forall F_of_Q add mul opp exp beq bneq le lt,
let norm_ff :=
let F_of_N b n :=
if b then F_of_Q (Qinv (Qmake (Z.of_N n) 1))
... | Proof.
pose F_of_N b n : Field F :=
if b then R_of_Q F (Qinv (Qmake (Z.of_N n) 1))
else R_of_Q F (Qmake (Z.of_N n) 1).
have F_of_NE : F_of_N =2 fun b n => @invi (Field F) b (N.to_nat n)%:R.
by move=> [] [|[]]; rewrite //= /inv_id/= ?invr0 ?invr1.
rewrite (Rnorm_bf_correct _ F_of_NE erefl erefl erefl erefl erefl)/... | Lemma | FTautoChecker_sound | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"F_of_N",
"Formula",
"KFeval",
"QTautoCheckerT",
"RFormula",
"R_of_Q",
"Reval_formula",
"Rnorm_bf_correct",
"Rnorm_formula",
"add",
"env",
"eqb",
"exp",
"ff",
"int",
"intr",
"inv",
"inv_id",
"invi",
"invr0",
"invr1",
"le",
"lt",
"mul",
"opp",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RTautoChecker_sound (R : realDomainType)
(ff : BFormula (RFormula R) isProp)
(f : BFormula (Formula Z) isProp) (env : seq R) (w : seq (Psatz Z)) :
(forall R_of_Z add mul opp exp beq bneq le lt,
let norm_ff :=
let R_of_N _ n := R_of_Z (Z.of_N n) in
let opp_intr := Some (opp, intr : i... | Proof.
pose R_of_N (b : bool) n : Ring R := R_of_Z R (Z.of_N n).
have R_of_NE : R_of_N =2 fun b n => @invi (Ring R) b (N.to_nat n)%:R.
by case=> [] [].
rewrite (Rnorm_bf_correct _ R_of_NE erefl erefl erefl erefl erefl)//.
by move/(_ (R_of_Z R)) => -> /(ZTautoCheckerT env).
Qed. | Lemma | RTautoChecker_sound | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"KFeval",
"RFormula",
"R_of_N",
"R_of_Z",
"Reval_formula",
"Ring",
"Rnorm_bf_correct",
"Rnorm_formula",
"ZTautoChecker",
"ZTautoCheckerT",
"add",
"env",
"eqb",
"exp",
"ff",
"int",
"intr",
"inv",
"invi",
"le",
"lt",
"mul",
"opp",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PExpr_Q2Z (e : PExpr Q) : option (PExpr Z) | := match e with
| PEO => Some PEO
| PEI => Some PEI
| PEc (Qmake z 1) => Some (PEc z) | PEc _ => None
| PEX n => Some (PEX _ n)
| PEadd e1 e2 => map_option2 PEadd (PExpr_Q2Z e1) (PExpr_Q2Z e2)
| PEsub e1 e2 => map_option2 PEsub (PExpr_Q2Z e1) (PExpr_Q2Z e2)
| PEmul e1 e2 => map_option2 PEmul (PExpr_Q2Z e1... | Fixpoint | PExpr_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | Translating formulas and witnesses from Q to Z for the realDomainType case | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Formula_Q2Z (ff : Formula Q) : option (Formula Z) | :=
map_option2
(fun l r => Build_Formula l (Fop ff) r)
(PExpr_Q2Z (Flhs ff)) (PExpr_Q2Z (Frhs ff)). | Definition | Formula_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"PExpr_Q2Z",
"ff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BFormula_Q2Z [k] (ff : BFormula (Formula Q) k) :
option (BFormula (Formula Z) k) | := match ff with
| TT k => Some (TT k)
| FF k => Some (FF k)
| X k P => Some (X k P)
| A k a aa => map_option (A k ^~ aa) (Formula_Q2Z a)
| AND _ f1 f2 =>
map_option2 (fun f => AND f) (BFormula_Q2Z f1) (BFormula_Q2Z f2)
| OR _ f1 f2 =>
map_option2 (fun f => OR f) (BFormula_Q2Z f1) (BFormula_Q2Z ... | Fixpoint | BFormula_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Formula",
"Formula_Q2Z",
"f1",
"f2",
"ff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pol_Q2Z (p : Pol Q) : Pol Z * positive | := match p with
| Pc (Qmake n d) => (Pc n, d)
| Pinj j p => let (p, n) := Pol_Q2Z p in (Pinj j p, n)
| PX p1 i p2 =>
let (p1, n1) := Pol_Q2Z p1 in
let (p2, n2) := Pol_Q2Z p2 in
let mulc c p := PmulC Z0 (Zpos 1) Z.mul Z.eqb p (Zpos c) in
(PX (mulc n2 p1) i (mulc n1 p2), Pos.mul n1 n2)
end... | Fixpoint | Pol_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"PmulC",
"eqb",
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Psatz_Q2Z (l : seq positive) (p : Psatz Q) : Psatz Z * positive | :=
match p with
| PsatzC (Qmake n d) => (PsatzC n, d)
| PsatzLet p1 p2 =>
let (p1, n1) := Psatz_Q2Z l p1 in
let (p2, n2) := Psatz_Q2Z (n1 :: l) p2 in
(PsatzLet p1 p2, n2)
| PsatzIn n => (PsatzIn _ n, nth 1%positive l n)
| PsatzSquare p => let (p, n) := Pol_Q2Z p in (PsatzSquare p, Pos.mul n ... | Fixpoint | Psatz_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Pol_Q2Z",
"mul",
"nth",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_Psatz_Q2Z : seq (Psatz Q) -> seq (Psatz Z) | :=
map (fun p => fst (Psatz_Q2Z [::] p)). | Definition | seq_Psatz_Q2Z | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"Psatz_Q2Z",
"map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mclra_witness n f | := let w := fresh "__wit" in mp_wlra_Q w f. | Ltac | mclra_witness | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | Main tactics, called from the elpi parser (c.f., lra.elpi) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mcnra_witness n f | := let w := fresh "__wit" in mp_wnra_Q w f. | Ltac | mcnra_witness | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mcpsatz_witness n f | :=
let w := fresh "__wit" in mp_wsos_Q w f || mp_wpsatz_Q n w f. | Ltac | mcpsatz_witness | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mctacF F hyps ff f varmap wit | :=
let nff := fresh "__ff" in
let nf := fresh "__f" in
let nvarmap := fresh "__varmap" in
pose (nff := ff);
pose (nf := f);
pose (nvarmap := varmap);
refine (hyps (@FTautoChecker_sound F nff nf nvarmap wit
(fun _ _ _ _ _ _ _ _ _ => erefl) _));
[ vm_compute; reflexivity ]. | Ltac | mctacF | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"FTautoChecker_sound",
"ff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mctacR R hyps ff f varmap wit | :=
let nff := fresh "__ff" in
let nf := fresh "__f" in
let nvarmap := fresh "__varmap" in
lazymatch eval vm_compute in (BFormula_Q2Z f) with
| Some ?f =>
pose (nff := ff);
pose (nf := f);
pose (nvarmap := varmap);
refine (hyps (@RTautoChecker_sound R nff f nvarmap (seq_Psatz_Q2Z wit)
... | Ltac | mctacR | algebra | algebra/arithmetic_tactic.v | [
"elpi",
"derive.std",
"param2",
"Corelib",
"BinNums",
"micromega_plugin",
"PosDef",
"NatDef",
"IntDef",
"RatDef",
"formula",
"witness",
"tactics",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
... | [
"BFormula_Q2Z",
"RTautoChecker_sound",
"eval",
"ff",
"seq_Psatz_Q2Z"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat (p : positive) (n : nat) | := Pos.to_nat p == n. | Definition | pos_nat | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_Pos_to_nat p : pos_nat p (Pos.to_nat p). | Proof. exact/eqP. Qed. | Lemma | pos_nat_Pos_to_nat | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_spec (p : positive) (n : nat) : positive -> nat -> bool -> Set | :=
| Pos_nat_spec_false : pos_nat_spec p n p n false
| Pos_nat_spec_xH : p = xH -> n = 1%N -> pos_nat_spec p n xH 1 true
| Pos_nat_spec_xO :
forall p' n', p = xO p' -> n = n'.*2 -> pos_nat p' n' ->
pos_nat_spec p n (xO p') n'.*2 true
| Pos_nat_spec_xI :
forall p' n', p = xI p' -> n = n'.*2.+... | Variant | pos_nat_spec | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"n'",
"nat",
"pos_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_opDdoubler p n : Pos.iter_op addn p n.*2 = (Pos.iter_op addn p n).*2. | Proof. by elim: p n => [p ip | p ip |//] n /=; rewrite !addnn ip ?doubleD. Qed. | Lemma | iter_opDdoubler | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"addn",
"addnn",
"doubleD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_opD2 p : Pos.iter_op addn p 2 = (Pos.iter_op addn p 1).*2. | Proof. by rewrite -[2]/1.*2 iter_opDdoubler. Qed. | Lemma | iter_opD2 | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"addn",
"iter_opDdoubler"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natP p n : pos_nat_spec p n p n (pos_nat p n). | Proof.
case: (boolP (pos_nat p n)) => /eqP pn; last exact: Pos_nat_spec_false.
elim: p n pn => [p IH | p IH |] n <-; rewrite /Pos.to_nat /= ?iter_opD2.
- exact: Pos_nat_spec_xI.
- exact: Pos_nat_spec_xO.
- exact: Pos_nat_spec_xH.
Qed. | Lemma | pos_natP | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"iter_opD2",
"last",
"pos_nat",
"pos_nat_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_ind (P : positive -> nat -> Prop) :
(P xH 1%N) ->
(forall p n, pos_nat p n -> P p n -> P (xO p) n.*2) ->
(forall p n, pos_nat p n -> P p n -> P (xI p) n.*2.+1) ->
forall p n, pos_nat p n -> P p n. | Proof.
move=> P1 Pd PdS; elim=> [p Ppn | p Ppn |] n; case: pos_natP => //.
- by move=> _ {}n [<-] _ pn _; apply: PdS (Ppn n pn).
- by move=> _ {}n [<-] _ pn _; apply: Pd (Ppn n pn).
Qed. | Lemma | pos_nat_ind | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"P1",
"apply",
"nat",
"pos_nat",
"pos_natP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat_gt0 p : (Pos.to_nat p > 0)%N. | Proof. by elim: p => // ? ?; rewrite /Pos.to_nat/= iter_opD2 double_gt0. Qed. | Lemma | Pos_to_nat_gt0 | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"double_gt0",
"iter_opD2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat0F p : (Pos.to_nat p == 0) = false. | Proof. by apply/negbTE; rewrite -lt0n Pos_to_nat_gt0. Qed. | Lemma | Pos_to_nat0F | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat_gt0",
"apply",
"lt0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_exS p n : pos_nat p n -> exists2 n', n = n'.+1 & pos_nat p n'.+1. | Proof.
by case: n => [|n pn]; [move: (Pos_to_nat_gt0 p) => /[swap]/eqP-> | exists n].
Qed. | Lemma | pos_nat_exS | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat_gt0",
"n'",
"pos_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat1 : pos_nat xH 1. | Proof. by []. Qed. | Lemma | pos_nat1 | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_double p n : pos_nat p~0 n.*2 = pos_nat p n. | Proof.
apply/idP/idP => [|/eqP<-]; last by rewrite /pos_nat [eqbLHS]iter_opD2.
by case: pos_natP => // _ _ [<-] /double_inj<-.
Qed. | Lemma | pos_nat_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"apply",
"double_inj",
"eqbLHS",
"iter_opD2",
"last",
"pos_nat",
"pos_natP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_doubleS p n : pos_nat p~1 n.*2.+1 = pos_nat p n. | Proof.
rewrite /pos_nat -[Pos.to_nat p~1]/(Pos.to_nat p~0).+1 eqSS.
exact: pos_nat_double.
Qed. | Lemma | pos_nat_doubleS | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"eqSS",
"pos_nat",
"pos_nat_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natE | := (pos_nat_double, pos_nat_doubleS, pos_nat1). | Definition | pos_natE | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat1",
"pos_nat_double",
"pos_nat_doubleS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natS p n : pos_nat p n -> pos_nat (Pos.succ p) (S n). | Proof. by elim/pos_nat_ind => // ? ? ? ?; rewrite -?doubleS pos_natE. Qed. | Lemma | pos_natS | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"doubleS",
"pos_nat",
"pos_natE",
"pos_nat_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natD p n p' n' :
pos_nat p n -> pos_nat p' n' -> pos_nat (Pos.add p p') (n + n'). | Proof.
move=> pn p'n'; suff: pos_nat (Pos.add p p') (n + n')
&& pos_nat (Pos.add_carry p p') (n + n').+1 by move=> /andP[].
elim/pos_nat_ind: pn p' n' p'n' => [p' n' {p n} ||];
[|move=> {}p {}n pn IH p' n'..].
- by case: pos_natP => //?? _ _ e; rewrite add1n -doubleS !pos_natE ?e pos_natS.
- case: pos_natP => [//... | Lemma | pos_natD | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"add",
"add1n",
"addSn",
"addn1",
"addnS",
"doubleD",
"doubleS",
"n'",
"pos_nat",
"pos_natE",
"pos_natP",
"pos_natS",
"pos_nat_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_pred_double p n :
pos_nat p n -> pos_nat (Pos.pred_double p) n.*2.-1. | Proof.
by elim/pos_nat_ind => [//||] {}p {}n; [move=> /pos_nat_exS[?-> pn]|];
rewrite /Pos.pred_double/= -/double -?doubleS !pos_natE.
(* when dropping support for Coq 8.20, replace above proof by
by elim/pos_nat_ind => [//||] {}p {}n; [move=> /pos_nat_exS[?-> pn]/=|];
rewrite !pos_natE. *)
Qed. | Lemma | pos_nat_pred_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"double",
"doubleS",
"pos_nat",
"pos_natE",
"pos_nat_exS",
"pos_nat_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natM p n p' n' :
pos_nat p n -> pos_nat p' n' -> pos_nat (Pos.mul p p') (n * n'). | Proof.
move=> pn; elim/pos_nat_ind: pn p' n' => [p' n' p'n'||];
[|move=> {}p {}n pn IH p' n' /[dup] p'n' /IH pp'nn' /=..].
- by rewrite mul1n.
- by rewrite -doubleMl pos_natE.
- by rewrite mulSn pos_natD// -doubleMl pos_natE.
Qed. | Lemma | pos_natM | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"doubleMl",
"mul",
"mul1n",
"mulSn",
"n'",
"pos_nat",
"pos_natD",
"pos_natE",
"pos_nat_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_nat m (n : nat) | :=
match m with
| Pos.IsNul => n == 0
| Pos.IsPos p => pos_nat p n
| Pos.IsNeg => false
end. | Definition | mask_nat | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"nat",
"pos_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_nat_double_pred p n : pos_nat p n ->
mask_nat (Pos.double_pred_mask p) n.-1.*2. | Proof.
by case: pos_natP => [//|//|{}p {}n _ _ pn _|{}p {}n _ _ pn _] /=;
rewrite !pos_natE ?pos_nat_pred_double.
Qed. | Lemma | mask_nat_double_pred | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"mask_nat",
"pos_nat",
"pos_natE",
"pos_natP",
"pos_nat_pred_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_nat_double m n : mask_nat m n ->
mask_nat (Pos.double_mask m) n.*2. | Proof. by case: m => [/eqP->//| p |//] /=; rewrite pos_natE. Qed. | Lemma | mask_nat_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"mask_nat",
"pos_natE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_nat_succ_double m n : mask_nat m n ->
mask_nat (Pos.succ_double_mask m) n.*2.+1. | Proof. by case: m => [/eqP->//| p |//] /=; rewrite pos_natE. Qed. | Lemma | mask_nat_succ_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"mask_nat",
"pos_natE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_natB p n (pn : pos_nat p n) p' n' (p'n' : pos_nat p' n') :
(n' <= n)%N -> mask_nat (Pos.sub_mask p p') (n - n'). | Proof.
suff: (n' <= n)%N -> (mask_nat (Pos.sub_mask p p') (n - n')%N
/\ ((n' < n)%N -> mask_nat (Pos.sub_mask_carry p p') (n - n').-1)).
by move=> /[apply] -[].
elim/pos_nat_ind: pn p' n' p'n' => [p' n' {p n} ||];
[|move=> {}p {}n pn IH p' n'..].
- case: n' => [|n'] /[swap]; first by rewrite /pos_nat Pos_to_n... | Lemma | mask_natB | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat0F",
"apply",
"double0",
"doubleB",
"doubleS",
"double_eq0",
"double_pred",
"eqSS",
"eq_sym",
"leq_Sdouble",
"leq_double",
"leqn0",
"ltnS",
"ltnW",
"ltn_double",
"ltnn",
"mask_nat",
"mask_nat_double",
"mask_nat_double_pred",
"mask_nat_succ_double",
"n'",
"neq_dou... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_sub_mask_Neg i j : (Pos.to_nat i < Pos.to_nat j)%N ->
Pos.sub_mask i j = Pos.IsNeg. | Proof.
suff: ((Pos.to_nat i < Pos.to_nat j)%N -> Pos.sub_mask i j = Pos.IsNeg)
/\ ((Pos.to_nat i <= Pos.to_nat j)%N -> Pos.sub_mask_carry i j = Pos.IsNeg).
by move=> [].
elim/pos_nat_ind: (pos_nat_Pos_to_nat i) j => [{i}|{}i n pin IH|{}i n pin IH] j.
- by case: pos_natP (pos_nat_Pos_to_nat j).
- case: pos_natP (p... | Lemma | Pos_sub_mask_Neg | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat0F",
"double0",
"doubleS",
"leq_Sdouble",
"leq_double",
"leqn0",
"ltn0",
"ltnS",
"ltn_double",
"pos_nat",
"pos_natP",
"pos_nat_Pos_to_nat",
"pos_nat_ind",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_natB p n (pn : pos_nat p n) p' n' (p'n' : pos_nat p' n') :
(n' < n)%N -> pos_nat (Pos.sub p p') (n - n'). | Proof.
move=> /[dup] /ltnW n'n; rewrite /Pos.sub.
case: Pos.sub_mask (mask_natB pn p'n' n'n) => [/=|//|//].
by rewrite -subn_gt0 => /eqP->.
Qed. | Lemma | pos_natB | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"ltnW",
"mask_natB",
"n'",
"pos_nat",
"sub",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_eq p n (pn : pos_nat p n) p' n' (p'n' : pos_nat p' n') :
Pos.eqb p p' = (n == n'). | Proof.
elim/pos_nat_ind: pn p' n' p'n' => [p' n' {p n} ||];
[|move=> {}p {}n pn IH p' n'..].
- case: pos_natP => [//|//||] {}p' {}n' _ _ p'n' _ /=.
+ by rewrite -double0 neq_doubleS_double.
+ apply/esym/negbTE; rewrite eqSS eq_sym double_eq0.
by rewrite -(eqP p'n') Pos_to_nat0F.
- case: pos_natP => [//|_ _ ... | Lemma | pos_nat_eq | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat0F",
"apply",
"double0",
"double_eq0",
"double_inj",
"eqSS",
"eq_sym",
"eqb",
"inj_eq",
"n'",
"neq_doubleS_double",
"pos_nat",
"pos_natP",
"pos_nat_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_compare p n (pn : pos_nat p n) p' n' (p'n' : pos_nat p' n') :
Pos.compare p p' = if n == n' then Eq else if (n < n')%N then Lt else Gt. | Proof.
rewrite /Pos.compare; elim/pos_nat_ind: pn Eq p' n' p'n' => [c p' n' {p n} ||];
[|move=> {}p {}n pn IH c p' n'..].
- case: pos_natP => [//|//||] /= {}p' {}n' _ _ p'n' _.
+ rewrite -double0 neq_doubleS_double -doubleS leq_double ifT//.
by rewrite -(eqP p'n') Pos_to_nat_gt0.
+ rewrite eqSS eq_sym doubl... | Lemma | pos_nat_compare | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat0F",
"Pos_to_nat_gt0",
"apply",
"compare",
"double0",
"doubleS",
"double_eq0",
"double_gt0",
"double_inj",
"eqSS",
"eq_sym",
"inj_eq",
"leq_double",
"leq_eqVlt",
"leqn0",
"lt0n_neq0",
"ltnS",
"ltn_double",
"ltnn",
"n'",
"neq_doubleS_double",
"pos_nat",
"pos_nat... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_le p n (pn : pos_nat p n) p' n' (p'n' : pos_nat p' n') :
Pos.leb p p' = (n <= n')%N. | Proof.
rewrite /Pos.leb (pos_nat_compare pn p'n') [RHS]leq_eqVlt.
by case: eqP => [//| _ /=]; case: ltnP.
Qed. | Lemma | pos_nat_le | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"leq_eqVlt",
"ltnP",
"n'",
"pos_nat",
"pos_nat_compare"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natI : injective Pos.to_nat. | Proof.
elim=> [i IH | i IH |]; rewrite /Pos.to_nat; case=> [j | j |]//=.
- by rewrite !iter_opD2 => -[/double_inj/IH->].
- by move/eqP; rewrite !iter_opD2 neq_doubleS_double.
- move: (Pos_to_nat_gt0 i); rewrite iter_opD2 => /[swap] -[/eqP].
by rewrite double_eq0 => /eqP<-; rewrite ltnn.
- by move/esym/eqP; rewrite !i... | Lemma | Pos_to_natI | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat_gt0",
"double0",
"double_eq0",
"double_inj",
"iter_opD2",
"ltnn",
"neq_doubleS_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_compare_spec (p p' : positive) :
positive -> positive -> comparison -> Set | :=
| Pos_nat_compare_spec_Eq : pos_nat_compare_spec p p' p p Eq
| Pos_nat_compare_spec_Lt :
(Pos.to_nat p < Pos.to_nat p')%N -> pos_nat_compare_spec p p' p p' Lt
| Pos_nat_compare_spec_Gt :
(Pos.to_nat p' < Pos.to_nat p)%N -> pos_nat_compare_spec p p' p p' Gt. | Variant | pos_nat_compare_spec | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_nat_compareP p p' : pos_nat_compare_spec p p' p p' (Pos.compare p p'). | Proof.
have := pos_nat_compare (pos_nat_Pos_to_nat p) (pos_nat_Pos_to_nat p').
(case: Pos.compare; [|do 2?[case: ifP => //]..]) => [|+ _ _|/[swap]+ + _].
- by case: eqP => [/Pos_to_natI e _|]; [rewrite -{2}e; constructor|case: ifP].
- by constructor.
- by move=> e /negbT; rewrite -leqNgt leq_eqVlt eq_sym e/=; construct... | Lemma | pos_nat_compareP | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_natI",
"compare",
"eq_sym",
"leqNgt",
"leq_eqVlt",
"pos_nat_Pos_to_nat",
"pos_nat_compare",
"pos_nat_compare_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat1 : Pos.to_nat xH = 1%N. | Proof. by []. Qed. | Lemma | Pos_to_nat1 | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat_double i : Pos.to_nat (xO i) = (Pos.to_nat i).*2. | Proof. exact/eqP/(eqbRL (pos_nat_double i (Pos.to_nat i))). Qed. | Lemma | Pos_to_nat_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat_doubleS i : Pos.to_nat (xI i) = (Pos.to_nat i).*2.+1. | Proof. exact/eqP/(eqbRL (pos_nat_doubleS i (Pos.to_nat i))). Qed. | Lemma | Pos_to_nat_doubleS | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat_doubleS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natS i : Pos.to_nat (Pos.succ i) = (Pos.to_nat i).+1. | Proof. exact: eqP (pos_natS (pos_nat_Pos_to_nat i)). Qed. | Lemma | Pos_to_natS | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_natS",
"pos_nat_Pos_to_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natD i j :
Pos.to_nat (Pos.add i j) = (Pos.to_nat i + Pos.to_nat j)%N. | Proof. exact: eqP (pos_natD (pos_nat_Pos_to_nat i) (pos_nat_Pos_to_nat j)). Qed. | Lemma | Pos_to_natD | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"add",
"pos_natD",
"pos_nat_Pos_to_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_nat_pred_double i :
Pos.to_nat (Pos.pred_double i) = (Pos.to_nat i).*2.-1. | Proof. exact: eqP (pos_nat_pred_double (pos_nat_Pos_to_nat i)). Qed. | Lemma | Pos_to_nat_pred_double | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_nat_Pos_to_nat",
"pos_nat_pred_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natM i j :
Pos.to_nat (Pos.mul i j) = (Pos.to_nat i * Pos.to_nat j)%N. | Proof. exact: eqP (pos_natM (pos_nat_Pos_to_nat i) (pos_nat_Pos_to_nat j)). Qed. | Lemma | Pos_to_natM | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"mul",
"pos_natM",
"pos_nat_Pos_to_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natB i j : (Pos.to_nat j < Pos.to_nat i)%N ->
Pos.to_nat (Pos.sub i j) = (Pos.to_nat i - Pos.to_nat j)%N. | Proof.
by move/(pos_natB (pos_nat_Pos_to_nat i) (pos_nat_Pos_to_nat j))/eqP.
Qed. | Lemma | Pos_to_natB | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"pos_natB",
"pos_nat_Pos_to_nat",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pos_to_natE | := (Pos_to_nat1, Pos_to_nat_double, Pos_to_nat_doubleS,
Pos_to_natS, Pos_to_natD, Pos_to_nat_pred_double, Pos_to_natM, Pos_to_natB). | Definition | Pos_to_natE | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Pos_to_nat1",
"Pos_to_natB",
"Pos_to_natD",
"Pos_to_natM",
"Pos_to_natS",
"Pos_to_nat_double",
"Pos_to_nat_doubleS",
"Pos_to_nat_pred_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nnat (i : N) (n : nat) | := N.to_nat i == n. | Definition | Nnat | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nnat_N_to_nat i : Nnat i (N.to_nat i). | Proof. exact/eqP. Qed. | Lemma | Nnat_N_to_nat | algebra | algebra/binnums.v | [
"micromega_plugin",
"PosDef",
"NatDef",
"Corelib",
"IntDef",
"RatDef",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"GRing.Theory",
"Num.Theory"
] | [
"Nnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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