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 |
|---|---|---|---|---|---|---|---|---|---|---|
natrP x : reflect (exists n, x = n%:R) (nat_num x). | Proof.
rewrite natrE.
by apply: (iffP eqP) => [<-|[n ->]]; [exists (truncn x) | rewrite natrK].
Qed. | Fact | natrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"nat_num",
"natrE",
"natrK",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound x | := sval (sigW (archi_bound_subproof x)). | Definition | bound | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"sigW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
boundP x : 0 <= x -> x < (bound x)%:R. | Proof. by move/ger0_norm=> {1}<-; rewrite /bound; case: (sigW _). Qed. | Fact | boundP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"bound",
"ger0_norm",
"sigW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_sig x : {m | 0 <= x -> m%:R <= x < m.+1%:R}. | Proof.
have [Rx | _] := boolP (0 <= x); last by exists 0%N.
have/ex_minnP[n lt_x_n1 min_n]: exists n, x < n.+1%:R.
by exists (bound x); rewrite (lt_trans (boundP Rx)) ?ltr_nat.
exists n => _; rewrite {}lt_x_n1 andbT; case: n min_n => //= n min_n.
rewrite real_leNgt ?rpred_nat ?ger0_real //; apply/negP => /min_n.
by r... | Fact | truncn_sig | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"bound",
"boundP",
"ex_minnP",
"ger0_real",
"last",
"lt_trans",
"ltnn",
"ltr_nat",
"real_leNgt",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn x | := if 0 <= x then sval (truncn_sig x) else 0%N. | Definition | truncn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn_sig"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_subproof x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N. | Proof.
rewrite /truncn; case: truncn_sig => // n hn.
by case: ifP => x_ge0; rewrite ?(ifT _ _ x_ge0) ?(ifF _ _ x_ge0) // hn.
Qed. | Fact | truncn_subproof | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn",
"truncn_sig"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat | := nat_num. | Notation | nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num"
] | Not to pollute the local namespace, we define Num.nat and Num.int here. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
int | := int_num. | Notation | int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ArchiDomain T | := (ArchiRealDomain T) (only parsing). | Notation | ArchiDomain | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type | := ArchiRealDomain.type (only parsing). | Notation | type | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copy T C | := (ArchiRealDomain.copy T C) (only parsing). | Notation | copy | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
on T | := (ArchiRealDomain.on T) (only parsing). | Notation | on | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ArchiField T | := (ArchiRealField T) (only parsing). | Notation | ArchiField | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type | := ArchiRealField.type (only parsing). | Notation | type | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copy T C | := (ArchiRealField.copy T C) (only parsing). | Notation | copy | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
on T | := (ArchiRealField.on T) (only parsing). | Notation | on | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorD | := real_floorDzr (only parsing). | Notation | floorD | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_floorDzr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilD | := real_ceilDzr (only parsing). | Notation | ceilD | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_ceilDzr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceilD | := real_ceilDzr (only parsing). | Notation | real_ceilD | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_ceilDzr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
archiDomainType | := archiRealDomainType (only parsing). | Notation | archiDomainType | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
archiFieldType | := archiRealFieldType (only parsing). | Notation | archiFieldType | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
env_nth | := (env_nth 0). | Notation | env_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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Peval | := (Peval
+%R *%R N.to_nat (@GRing.exp R) R_of_C (@env_jump R) env_nth). | Notation | Peval | 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",
... | [
"env_jump",
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Psquare | := (Psquare 0 1 +%R *%R eq_op). | Notation | Psquare | 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 | |
Peval_square l P : Peval l (Psquare P) = Peval l P ^+ 2. | Proof.
elim: P l => [c | j Q IH | P IHP i Q IHQ] l; rewrite /= ?rmorphM ?expr2 ?IH//.
rewrite Peval_mkPX PevalD Peval_mkPX PevalM Peval_mkPinj PevalMC IHP.
rewrite IHQ /= rmorph0 addr0 [in LHS]mulrDl -mulrA mulrACA.
set p := Peval l P * env_nth 1 l ^+ Pos.to_nat i; set q := Peval _ Q.
rewrite rmorphD rmorph1 [X in _ + ... | Lemma | Peval_square | 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",
... | [
"Peval",
"PevalD",
"PevalM",
"PevalMC",
"Peval_mkPX",
"Peval_mkPinj",
"Psquare",
"addr0",
"env_nth",
"expr2",
"mulrA",
"mulrAC",
"mulrACA",
"mulrDl",
"mulr_natr",
"rmorph0",
"rmorph1",
"rmorphD",
"rmorphM",
"sqrrD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_tautoT : forall t t', is_tauto t t' -> eval t || eval t'. | Hypothesis | 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",
... | [
"eval",
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
clause | := (clause Term Annot). | Notation | clause | 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 | |
cnf | := (cnf Term Annot). | Notation | cnf | 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 | |
cnf_tt | := (cnf_tt Term Annot). | Notation | cnf_tt | 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 | |
cnf_ff | := (cnf_ff Term Annot). | Notation | cnf_ff | 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 | |
add_term | := (add_term is_tauto). | Notation | add_term | 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",
... | [
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
or_clause | := (or_clause is_tauto). | Notation | or_clause | 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",
... | [
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
or_clause_cnf | := (or_clause_cnf is_tauto). | Notation | or_clause_cnf | 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",
... | [
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
or_cnf_aux | := (or_cnf_aux is_tauto). | Notation | or_cnf_aux | 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",
... | [
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
or_cnf | := (or_cnf is_tauto). | Notation | or_cnf | 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",
... | [
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_clause : clause -> bool | := has (fun t => eval t.1). | Definition | eval_clause | 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",
... | [
"clause",
"eval",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf : cnf -> bool | := all eval_clause. | Definition | eval_cnf | 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",
... | [
"all",
"cnf",
"eval_clause"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_tt : eval_cnf cnf_tt. | Proof. by []. Qed. | Lemma | eval_cnf_tt | 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",
... | [
"cnf_tt",
"eval_cnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_ff : eval_cnf cnf_ff = false. | Proof. by []. Qed. | Lemma | eval_cnf_ff | 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",
... | [
"cnf_ff",
"eval_cnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_cnf_ttT f : is_cnf_tt f -> f = cnf_tt. | Proof. by case: f. Qed. | Lemma | is_cnf_ttT | 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",
... | [
"cnf_tt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_cnf_ffT f : is_cnf_ff f -> f = cnf_ff. | Proof. by case: f => [//| [|//] []]. Qed. | Lemma | is_cnf_ffT | 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",
... | [
"cnf_ff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_termP t cl :
match add_term t cl with
| None => eval t.1 || eval_clause cl
| Some cl' => eval_clause cl' = eval t.1 || eval_clause cl
end. | Proof.
elim: cl t => [| t' cl IH] t /=.
by case: ifP => [/is_tautoT/[!orbb]->//|_].
case: ifP => [|_]; first by rewrite orbA => /is_tautoT->.
by case: add_term (IH t) => [?|] {}IH; rewrite /= orbCA IH ?orbT.
Qed. | Lemma | add_termP | 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_term",
"eval",
"eval_clause",
"is_tautoT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
or_clauseP cl1 cl2 :
match or_clause cl1 cl2 with
| None => eval_clause cl1 || eval_clause cl2
| Some cl' => eval_clause cl' = eval_clause cl1 || eval_clause cl2
end. | Proof.
elim: cl1 cl2 => [//| t cl IH] cl2 /=.
case: add_term (add_termP t cl2) => [cl'|]; last by rewrite orbAC => ->.
rewrite [eval t.1 || eval_clause cl]orbC -orbA.
by case: or_clause (IH cl') => [_->->|/[swap]->].
Qed. | Lemma | or_clauseP | 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_term",
"add_termP",
"eval",
"eval_clause",
"last",
"or_clause"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_or_clause_cnf cl f :
eval_cnf (or_clause_cnf cl f) = eval_clause cl || eval_cnf f. | Proof.
case: cl => [//| t cl].
rewrite /or_clause_cnf; move: (t :: cl) => {t}cl; set orcl := fun acc cl => _.
have orclP acc cl' : eval_cnf (orcl acc cl')
= (eval_clause cl || eval_clause cl') && eval_cnf acc.
rewrite /orcl; case: or_clause (or_clauseP cl cl') => [cl''/=->//|].
by move=> /orP[]->//=; rewrite or... | Lemma | eval_or_clause_cnf | 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",
... | [
"eval_clause",
"eval_cnf",
"or_clause",
"or_clauseP",
"or_clause_cnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_rev_append f1 f2 :
eval_cnf (ListDef.rev_append f1 f2) = eval_cnf f1 && eval_cnf f2. | Proof. by elim: f1 f2 => [//|cl f1 IH] f2 /=; rewrite IH andbCA andbA. Qed. | Lemma | eval_rev_append | 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",
... | [
"eval_cnf",
"f1",
"f2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_or_cnf_aux f1 f2 :
eval_cnf (or_cnf_aux f1 f2) = eval_cnf f1 || eval_cnf f2. | Proof.
elim: f1 => [//|cl f1 IH] /=.
by rewrite eval_rev_append IH eval_or_clause_cnf -orb_andl andbC.
Qed. | Lemma | eval_or_cnf_aux | 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",
... | [
"eval_cnf",
"eval_or_clause_cnf",
"eval_rev_append",
"f1",
"f2",
"or_cnf_aux"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_or_cnf f1 f2 : eval_cnf (or_cnf f1 f2) = eval_cnf f1 || eval_cnf f2. | Proof.
rewrite /or_cnf.
case: ifP => [/orP[]/is_cnf_ttT->|_]; rewrite ?eval_cnf_tt ?orbT//.
by case: ifP => [/is_cnf_ffT->|_]; rewrite ?eval_cnf_ff ?orbF ?eval_or_cnf_aux.
Qed. | Lemma | eval_or_cnf | 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",
... | [
"eval_cnf",
"eval_cnf_ff",
"eval_cnf_tt",
"eval_or_cnf_aux",
"f1",
"f2",
"is_cnf_ffT",
"is_cnf_ttT",
"or_cnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_and_cnf f1 f2 :
eval_cnf (and_cnf f1 f2) = eval_cnf f1 && eval_cnf f2. | Proof.
rewrite /and_cnf.
case: ifP => [/orP[]/is_cnf_ffT->|_]; rewrite ?eval_cnf_ff ?andbF//.
by case: ifP => [/is_cnf_ttT->|_]; rewrite ?eval_cnf_tt ?andbT ?eval_rev_append.
Qed. | Lemma | eval_and_cnf | 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",
... | [
"and_cnf",
"eval_cnf",
"eval_cnf_ff",
"eval_cnf_tt",
"eval_rev_append",
"f1",
"f2",
"is_cnf_ffT",
"is_cnf_ttT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_of_C_ge0 : {mono R_of_C : x / 0 <= x}. | Hypothesis | R_of_C_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 | ||
Peval | := (Peval
+%R *%R N.to_nat (@GRing.exp R) R_of_C (@env_jump R) (env_nth 0)). | Notation | Peval | 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",
... | [
"env_jump",
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval | := (PEeval 0 1 +%R *%R (fun x y : R => x - y) -%R
N.to_nat (@GRing.exp R) R_of_C (env_nth 0)). | Notation | PEeval | 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",
... | [
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Feval | := (Feval 0 1 +%R *%R (fun x y : R => x - y) -%R
N.to_nat (@GRing.exp R) isBool eq_op (cneqb eq_op) <=%R <%R R_of_C
(env_nth 0)). | Notation | Feval | 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",
... | [
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Pol_of_PExpr | := (Pol_of_PExpr
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op). | Notation | Pol_of_PExpr | 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 | |
normalise | := (normalise
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op). | Notation | normalise | 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 | |
check_inconsistent | := (check_inconsistent 0%R eq_op <=%R). | Notation | check_inconsistent | 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 | |
normalise_aux | := (normalise_aux -%R). | Notation | normalise_aux | 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 | |
negate_aux | := (negate_aux -%R). | Notation | negate_aux | 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 | |
cnf_of_list | := (cnf_of_list 0 eq_op <=%R). | Notation | cnf_of_list | 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 | |
cnf_normalise | := (cnf_normalise
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op <=%R). | Notation | cnf_normalise | 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 | |
cnf_negate | := (cnf_negate
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op <=%R). | Notation | cnf_negate | 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 | |
eval_op1 (o : Op1) (x : R) : bool | :=
match o with
| Equal => x == 0
| NonEqual => x != 0
| Strict => 0 < x
| NonStrict => 0 <= x
end. | Definition | eval_op1 | 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 | |
NFeval (env : seq R) (f : NFormula C) : bool | :=
let: (p, op) := f in eval_op1 op (Peval env p). | Definition | NFeval | 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",
... | [
"Peval",
"env",
"eval_op1",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NFeval_normalise l f : NFeval l (normalise f) = Feval l f. | Proof.
by case: f => lhs [|||||] rhs /=;
rewrite PevalB !Peval_Pol_of_PExpr ?subr_eq0 ?subr_ge0 ?subr_gt0.
Qed. | Lemma | NFeval_normalise | 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",
... | [
"Feval",
"NFeval",
"PevalB",
"Peval_Pol_of_PExpr",
"normalise",
"rhs",
"subr_eq0",
"subr_ge0",
"subr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
check_inconsistentT l f : check_inconsistent f -> NFeval l f = false. | Proof.
case: f => -[] // c [] /= + /ltac:(apply: negbTE).
- by apply: contraNN; rewrite !eq_le -oppr_ge0 -rmorphN !R_of_C_ge0 oppr_ge0.
- by move=> /eqP->; rewrite rmorph0 negbK.
- by rewrite -leNgt -!oppr_ge0 -rmorphN R_of_C_ge0.
- by rewrite /cltb andbC -lt_neqAle ltNge R_of_C_ge0.
Qed. | Lemma | check_inconsistentT | 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",
"R_of_C_ge0",
"apply",
"check_inconsistent",
"eq_le",
"leNgt",
"ltNge",
"lt_neqAle",
"oppr_ge0",
"rmorph0",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_normalise_aux T (tg : T) l f :
eval_cnf (fun f => ~~ NFeval l f) [seq [:: (nf, tg)] | nf <- normalise_aux f]
= NFeval l f. | Proof.
case: f => e [] /=; rewrite andbT !orbF// PevalN.
- by rewrite oppr_gt0 -!leNgt eq_le.
- by rewrite oppr_ge0 -ltNge.
- by rewrite oppr_gt0 -leNgt.
Qed. | Lemma | eval_normalise_aux | 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",
"PevalN",
"eq_le",
"eval_cnf",
"leNgt",
"ltNge",
"normalise_aux",
"oppr_ge0",
"oppr_gt0",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_negate_aux T (tg : T) l f :
eval_cnf (fun f => ~~ NFeval l f) [seq [:: (nf, tg)] | nf <- negate_aux f]
= ~~ NFeval l f. | Proof.
by case: f => e[]; rewrite /= andbT !orbF// PevalN oppr_gt0 -!leNgt negbK eq_le.
Qed. | Lemma | eval_negate_aux | 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",
"PevalN",
"eq_le",
"eval_cnf",
"leNgt",
"negate_aux",
"oppr_gt0",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_of_list T (tg : T) l lf :
eval_cnf (fun f => ~~ NFeval l f) (cnf_of_list lf tg)
= eval_cnf (fun f => ~~ NFeval l f) [seq [:: (f, tg)] | f <- lf]. | Proof.
elim: lf => [//|f lf /= <-]; rewrite orbF.
case: (boolP (check_inconsistent f)) => [/check_inconsistentT->//|_/=].
by rewrite orbF.
Qed. | Lemma | eval_cnf_of_list | 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_inconsistent",
"check_inconsistentT",
"cnf_of_list",
"eval_cnf",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_normalise T (tg : T) l f :
eval_cnf (fun f => ~~ NFeval l f) (cnf_normalise f tg) = Feval l f. | Proof.
rewrite /cnf_normalise/= -NFeval_normalise; set nf := normalise f.
case: (boolP (check_inconsistent nf)) => [/check_inconsistentT->//|_].
by rewrite eval_cnf_of_list eval_normalise_aux.
Qed. | Lemma | eval_cnf_normalise | 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",
... | [
"Feval",
"NFeval",
"NFeval_normalise",
"check_inconsistent",
"check_inconsistentT",
"cnf_normalise",
"eval_cnf",
"eval_cnf_of_list",
"eval_normalise_aux",
"normalise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_negate T (tg : T) l f :
eval_cnf (fun f => ~~ NFeval l f) (cnf_negate f tg) = ~~ Feval l f. | Proof.
rewrite /cnf_negate/= -NFeval_normalise; set nf := normalise f.
case: (boolP (check_inconsistent nf)) => [/check_inconsistentT->//|_].
by rewrite eval_cnf_of_list eval_negate_aux.
Qed. | Lemma | eval_cnf_negate | 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",
... | [
"Feval",
"NFeval",
"NFeval_normalise",
"check_inconsistent",
"check_inconsistentT",
"cnf_negate",
"eval_cnf",
"eval_cnf_of_list",
"eval_negate_aux",
"normalise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
KFeval k : seq R -> Formula C -> eKind k | :=
let ev k eq neq le lt := Feval rO rI radd rmul (fun x y => radd x (ropp y))
ropp id rpow k eq neq le lt R_of_C (env_nth rO) in
if k is isBool then ev isBool req rneq rle rlt else
ev isProp eq (fun x y => x <> y) (fun x y => rle x y) (fun x y => rlt x y). | Definition | KFeval | 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",
... | [
"Feval",
"Formula",
"env_nth",
"id",
"le",
"lt",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BFKFeval (l : seq R) : forall k, BFormula (Formula C) k -> eKind k | :=
BFeval eqb (fun k f => KFeval k l f). | Definition | BFKFeval | 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",
"eqb",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RFeval (R : numDomainType) C (R_of_C : C -> R) | := KFeval 0 1
+%R *%R -%R (fun x n => x ^+ N.to_nat n) eq_op (cneqb eq_op) <=%R <%R R_of_C. | Definition | RFeval | 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",
... | [
"KFeval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RBFeval (R : numDomainType) C (R_of_C : C -> R) | := BFKFeval 0 1
+%R *%R -%R (fun x n => x ^+ N.to_nat n) eq_op (cneqb eq_op) <=%R <%R R_of_C. | Definition | RBFeval | 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",
... | [
"BFKFeval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RFevalP (R : numDomainType) C (R_of_C : C -> R) l f :
reflect (RFeval R_of_C isProp l f) (RFeval R_of_C isBool l f). | Proof.
case: f => lhs [] rhs; try exact/idP; first exact/(iffP eqP) => [/eqP|/=->].
by apply/(iffP idP) => [/eqP//|]; apply: contra_notN => /eqP.
Qed. | Lemma | RFevalP | 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",
... | [
"RFeval",
"apply",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_tauto | := (is_tauto 0 +%R eq_op <=%R). | Notation | is_tauto | 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 | |
cnf_tt | := (cnf_tt (NFormula C) unit). | Notation | cnf_tt | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cnf_ff | := (cnf_ff (NFormula C) unit). | Notation | cnf_ff | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
and_cnf | := (@and_cnf (NFormula C) unit). | Notation | and_cnf | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalise | := (cnf_normalise
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op <=%R). | Notation | normalise | 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",
... | [
"cnf_normalise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
negate | := (cnf_negate
0 1 +%R *%R (fun x y : C => x - y) -%R eq_op <=%R). | Notation | negate | 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",
... | [
"cnf_negate"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cnf_of_GFormula | := (@cnf_of_GFormula
_ _ _ cnf_tt cnf_ff or_cnf and_cnf normalise negate). | Notation | cnf_of_GFormula | 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",
... | [
"and_cnf",
"cnf_ff",
"cnf_tt",
"negate",
"normalise",
"or_cnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_tautoT : forall l (f g : NFormula C),
is_tauto f g -> ~~ NFeval R_of_C l f || ~~ NFeval R_of_C l g. | Hypothesis | 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",
"is_tauto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
eval_or_cnf | := (eval_or_cnf (is_tautoT _)). | Notation | eval_or_cnf | 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",
... | [
"is_tautoT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hold pol k : eKind k -> Prop | :=
if k is isProp then if pol then id else not else fun x => x = pol. | Definition | hold | 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",
... | [
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_bool_spec T k (f : BFormula T k) :
GFormula k -> option bool -> Type | :=
| IsBoolT : is_bool_spec (TT k) (Some true)
| IsBoolF : is_bool_spec (FF k) (Some false)
| IsBoolNone : is_bool_spec f None. | Variant | is_bool_spec | 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 | |
is_boolP T k (f : BFormula T k) : is_bool_spec f f (is_bool f). | Proof.
case: f => {}k => [||x|f u|f g|f g|f|f u g|f g|f] /=; try exact: IsBoolNone.
- exact: IsBoolT.
- exact: IsBoolF.
Qed. | Lemma | is_boolP | 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",
... | [
"is_bool_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_cnf_of_GFormula l pol k (f : GFormula k) :
eval_cnf (fun f => ~~ NFeval R_of_C l f) (cnf_of_GFormula pol f) ->
hold pol (RBFeval R_of_C l f). | Proof.
elim: f pol => {}k /=.
- by move: k => -[] [].
- by move: k => -[] [] //= _.
- by move=> x [] /=.
- move=> f u [].
+ by rewrite eval_cnf_normalise//; case: k => [/RFevalP|//].
+ by rewrite eval_cnf_negate//; case: k => [/RFevalP|/negbTE].
- case: k => /= f IHf g IHg [].
+ by rewrite eval_and_cnf => /andP[/... | Lemma | eval_cnf_of_GFormula | 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",
"RBFeval",
"RFevalP",
"apply",
"cnf_of_GFormula",
"eval_and_cnf",
"eval_cnf",
"eval_cnf_negate",
"eval_cnf_normalise",
"eval_or_cnf",
"ff",
"hold",
"is_boolP",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NFeval | := (NFeval R_of_C). | Notation | NFeval | 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 | |
pexpr_times_nformula | := (pexpr_times_nformula
0 1 +%R *%R eq_op). | Notation | pexpr_times_nformula | 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 | |
nformula_times_nformula | := (nformula_times_nformula
0 1 +%R *%R eq_op). | Notation | nformula_times_nformula | 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 | |
nformula_plus_nformula | := (nformula_plus_nformula
0 +%R eq_op). | Notation | nformula_plus_nformula | 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 | |
eval_Psatz | := (eval_Psatz 0 1 +%R *%R eq_op <=%R). | Notation | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deduce | := nformula_plus_nformula. | Notation | deduce | 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",
... | [
"nformula_plus_nformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unsat | := (check_inconsistent 0 eq_op <=%R). | Notation | unsat | 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_inconsistent"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
check_normalised_formulas | := (check_normalised_formulas
0 1 +%R *%R eq_op <=%R). | Notation | check_normalised_formulas | 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 | |
eval_OpMult o o' (x x' : R) : eval_op1 o x -> eval_op1 o' x' ->
forall o'', OpMult o o' = Some o'' -> eval_op1 o'' (x * x'). | Proof.
case: o => /=[/eqP->| ro | ro | ro].
- by rewrite mul0r; case: o' => /= + + [<-]/=.
- by case: o' => //= /eqP-> + [<-]/=; rewrite mulr0.
- case: o' => /=[/eqP->|//| ro' | ro'] o'' [<-]/=.
+ by rewrite mulr0.
+ exact: mulr_gt0.
+ by rewrite mulr_ge0// ltW.
- case: o' => /=[/eqP->|//| ro' | ro'] o'' [<-]/=.
... | Lemma | eval_OpMult | 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",
... | [
"eval_op1",
"ltW",
"mul0r",
"mulr0",
"mulr_ge0",
"mulr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_OpAdd o o' (x x' : R) : eval_op1 o x -> eval_op1 o' x' ->
forall o'', OpAdd o o' = Some o'' -> eval_op1 o'' (x + x'). | Proof.
case: o => /=[/eqP->| ro | ro | ro].
- by rewrite add0r => + + [<-].
- by case: o' => //= /eqP-> + [<-]; rewrite addr0.
- case: o' => /=[/eqP->|//| ro' | ro'] o'' [<-]/=.
+ by rewrite addr0.
+ exact: addr_gt0.
+ exact: ltr_wpDr.
- case: o' => /=[/eqP->|//| ro' | ro'] o'' [<-]/=.
+ by rewrite addr0.
+ e... | Lemma | eval_OpAdd | 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",
... | [
"add0r",
"addr0",
"addr_ge0",
"addr_gt0",
"eval_op1",
"ltr_wpDl",
"ltr_wpDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_pexpr_times_nformula l e f f' :
NFeval l f -> pexpr_times_nformula e f = Some f' -> NFeval l f'. | Proof. by case: f => e' []//= /eqP ee' [<-]/=; rewrite PevalM ee' mulr0. Qed. | Lemma | eval_pexpr_times_nformula | 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",
"PevalM",
"e'",
"ee'",
"mulr0",
"pexpr_times_nformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_nformula_times_nformula l f f' f'' : NFeval l f -> NFeval l f' ->
nformula_times_nformula f f' = Some f'' -> NFeval l f''. | Proof.
case: f f' => [e o] [e' o'] /= ee ee'.
case: OpMult (eval_OpMult ee ee') => [o'' /(_ _ erefl) + [<-]/=|//].
by rewrite PevalM.
Qed. | Lemma | eval_nformula_times_nformula | 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",
"PevalM",
"e'",
"ee'",
"eval_OpMult",
"nformula_times_nformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_nformula_plus_nformula l f f' f'' : NFeval l f -> NFeval l f' ->
nformula_plus_nformula f f' = Some f'' -> NFeval l f''. | Proof.
case: f f' => [e o] [e' o'] /= ee ee'.
case: OpAdd (eval_OpAdd ee ee') => [o'' /(_ _ erefl) + [<-]/=|//].
by rewrite PevalD.
Qed. | Lemma | eval_nformula_plus_nformula | 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",
"PevalD",
"e'",
"ee'",
"eval_OpAdd",
"nformula_plus_nformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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