fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
prodrV(I : eqType) (r : seq I) (P : pred I) (E : I -> R) :
(forall i, P i -> E i \is a GRing.unit) ->
\prod_(i <- r | P i) (E i)^-1 = (\prod_(i <- r | P i) E i)^-1.
Proof.
by move=> /rev_prodrV->; rewrite rev_prodr (perm_big r)// perm_rev.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | prodrV | |
scaler_injl: {in unit, @right_injective R A A *:%R}.
Proof.
move=> k Uk x1 x2 Hx1x2.
by rewrite -[x1]scale1r -(mulVr Uk) -scalerA Hx1x2 scalerA mulVr // scale1r.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_injl | |
scaler_unitk x : k \in unit -> (k *: x \in unit) = (x \in unit).
Proof.
move=> Uk; apply/idP/idP=> [Ukx | Ux]; apply/unitrP; last first.
exists (k^-1 *: x^-1).
by rewrite -!scalerAl -!scalerAr !scalerA !mulVr // !mulrV // scale1r.
exists (k *: (k *: x)^-1); split.
apply: (mulrI Ukx).
by rewrite mulr1 mulrA -sca... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_unit | |
invrZk x : k \in unit -> x \in unit -> (k *: x)^-1 = k^-1 *: x^-1.
Proof.
move=> Uk Ux; have Ukx: (k *: x \in unit) by rewrite scaler_unit.
apply: (mulIr Ukx).
by rewrite mulVr // -scalerAl -scalerAr scalerA !mulVr // scale1r.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invrZ | |
divalg_closed:= [/\ 1 \in S, linear_closed S & divr_2closed S]. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closed | |
divalg_closedBdiv: divalg_closed -> divring_closed S.
Proof. by case=> S1 /linear_closedB. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closedBdiv | |
divalg_closedZ: divalg_closed -> subalg_closed S.
Proof. by case=> S1 Slin Sdiv; split=> //; have [] := @divr_closedM A S. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closedZ | |
addr_closed:= nmod_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addr_closed | |
oppr_closed:= oppr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | oppr_closed | |
zmod_closed:= zmod_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | zmod_closed | |
mulr_closed:= mulr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr_closed | |
semiring_closed:= semiring_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | semiring_closed | |
smulr_closed:= smulr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | smulr_closed | |
subring_closed:= subring_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subring_closed | |
scaler_closed:= scaler_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_closed | |
subsemimod_closed:= subsemimod_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subsemimod_closed | |
linear_closed:= linear_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | linear_closed | |
submod_closed:= submod_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | submod_closed | |
subalg_closed:= subalg_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subalg_closed | |
invr_closed:= invr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_closed | |
divr_2closed:= divr_2closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_2closed | |
divr_closed:= divr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closed | |
sdivr_closed:= sdivr_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closed | |
divring_closed:= divring_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closed | |
divalg_closed:= divalg_closed. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closed | |
zmod_closedD: zmod_closed >-> nmod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | zmod_closedD | |
zmod_closedN: zmod_closed >-> oppr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | zmod_closedN | |
semiring_closedD: semiring_closed >-> addr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | semiring_closedD | |
semiring_closedM: semiring_closed >-> mulr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | semiring_closedM | |
smulr_closedM: smulr_closed >-> mulr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | smulr_closedM | |
smulr_closedN: smulr_closed >-> oppr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | smulr_closedN | |
subring_closedB: subring_closed >-> zmod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subring_closedB | |
subring_closedM: subring_closed >-> smulr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subring_closedM | |
subring_closed_semi: subring_closed >-> semiring_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subring_closed_semi | |
subsemimod_closedD: subsemimod_closed >-> addr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subsemimod_closedD | |
subsemimod_closedZ: subsemimod_closed >-> scaler_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subsemimod_closedZ | |
linear_closedB: linear_closed >-> subr_2closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | linear_closedB | |
submod_closedB: submod_closed >-> zmod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | submod_closedB | |
submod_closed_semi: submod_closed >-> subsemimod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | submod_closed_semi | |
subsemialg_closedZ: subsemialg_closed >-> subsemimod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subsemialg_closedZ | |
subsemialg_closedM: subsemialg_closed >-> semiring_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subsemialg_closedM | |
subalg_closedZ: subalg_closed >-> submod_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subalg_closedZ | |
subalg_closedBM: subalg_closed >-> subring_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subalg_closedBM | |
subalg_closed_semi: subalg_closed >-> subsemialg_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subalg_closed_semi | |
divr_closedV: divr_closed >-> invr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closedV | |
divr_closedM: divr_closed >-> mulr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closedM | |
sdivr_closed_div: sdivr_closed >-> divr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closed_div | |
sdivr_closedM: sdivr_closed >-> smulr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closedM | |
divring_closedBM: divring_closed >-> subring_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closedBM | |
divring_closed_div: divring_closed >-> sdivr_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closed_div | |
divalg_closedBdiv: divalg_closed >-> divring_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closedBdiv | |
divalg_closedZ: divalg_closed >-> subalg_closed. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divalg_closedZ | |
term: Type :=
| Var of nat
| Const of R
| NatConst of nat
| Add of term & term
| Opp of term
| NatMul of term & nat
| Mul of term & term
| Inv of term
| Exp of term & nat. | Inductive | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | term | |
formula: Type :=
| Bool of bool
| Equal of term & term
| Unit of term
| And of formula & formula
| Or of formula & formula
| Implies of formula & formula
| Not of formula
| Exists of nat & formula
| Forall of nat & formula. | Inductive | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | formula | |
True:= (Bool true). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | True | |
False:= (Bool false).
Local Notation "''X_' i" := (Var _ i) : term_scope.
Local Notation "n %:R" := (NatConst _ n) : term_scope.
Local Notation "x %:T" := (Const x) : term_scope.
Local Notation "0" := 0%:R%T : term_scope.
Local Notation "1" := 1%:R%T : term_scope.
Local Infix "+" := Add : term_scope.
Local Notation "- ... | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | False | |
tsubst(t : term R) (s : nat * term R) :=
match t with
| 'X_i => if i == s.1 then s.2 else t
| _%:T | _%:R => t
| t1 + t2 => tsubst t1 s + tsubst t2 s
| - t1 => - tsubst t1 s
| t1 *+ n => tsubst t1 s *+ n
| t1 * t2 => tsubst t1 s * tsubst t2 s
| t1^-1 => (tsubst t1 s)^-1
| t1 ^+ n => tsubst t1 s ^+ n
... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | tsubst | |
fsubst(f : formula R) (s : nat * term R) :=
match f with
| Bool _ => f
| t1 == t2 => tsubst t1 s == tsubst t2 s
| Unit t1 => Unit (tsubst t1 s)
| f1 /\ f2 => fsubst f1 s /\ fsubst f2 s
| f1 \/ f2 => fsubst f1 s \/ fsubst f2 s
| f1 ==> f2 => fsubst f1 s ==> fsubst f2 s
| ~ f1 => ~ fsubst f1 s
| ('exist... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | fsubst | |
eval(e : seq R) (t : term R) {struct t} : R :=
match t with
| ('X_i)%T => e`_i
| (x%:T)%T => x
| (n%:R)%T => n%:R
| (t1 + t2)%T => eval e t1 + eval e t2
| (- t1)%T => - eval e t1
| (t1 *+ n)%T => eval e t1 *+ n
| (t1 * t2)%T => eval e t1 * eval e t2
| t1^-1%T => (eval e t1)^-1
| (t1 ^+ n)%T => eval ... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eval | |
same_env(e e' : seq R) := nth 0 e =1 nth 0 e'. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | same_env | |
eq_evale e' t : same_env e e' -> eval e t = eval e' t.
Proof. by move=> eq_e; elim: t => //= t1 -> // t2 ->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eq_eval | |
eval_tsubste t s :
eval e (tsubst t s) = eval (set_nth 0 e s.1 (eval e s.2)) t.
Proof.
case: s => i u; elim: t => //=; do 2?[move=> ? -> //] => j.
by rewrite nth_set_nth /=; case: (_ == _).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eval_tsubst | |
holds(e : seq R) (f : formula R) {struct f} : Prop :=
match f with
| Bool b => b
| (t1 == t2)%T => eval e t1 = eval e t2
| Unit t1 => eval e t1 \in unit
| (f1 /\ f2)%T => holds e f1 /\ holds e f2
| (f1 \/ f2)%T => holds e f1 \/ holds e f2
| (f1 ==> f2)%T => holds e f1 -> holds e f2
| (~ f1)%T => ~ holds... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | holds | |
same_env_syme e' : same_env e e' -> same_env e' e.
Proof. exact: fsym. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | same_env_sym | |
eq_holdse e' f : same_env e e' -> holds e f -> holds e' f.
Proof.
pose sv := set_nth (0 : R).
have eq_i i v e1 e2: same_env e1 e2 -> same_env (sv e1 i v) (sv e2 i v).
by move=> eq_e j; rewrite !nth_set_nth /= eq_e.
elim: f e e' => //=.
- by move=> t1 t2 e e' eq_e; rewrite !(eq_eval _ eq_e).
- by move=> t e e' eq_e; r... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eq_holds | |
holds_fsubste f i v :
holds e (fsubst f (i, v%:T)%T) <-> holds (set_nth 0 e i v) f.
Proof.
elim: f e => //=; do [
by move=> *; rewrite !eval_tsubst
| move=> f1 IHf1 f2 IHf2 e; move: (IHf1 e) (IHf2 e); tauto
| move=> f IHf e; move: (IHf e); tauto
| move=> j f IHf e].
- case eq_ji: (j == i); first rewrite (eqP eq_ji)... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | holds_fsubst | |
rterm(t : term R) :=
match t with
| _^-1 => false
| t1 + t2 | t1 * t2 => rterm t1 && rterm t2
| - t1 | t1 *+ _ | t1 ^+ _ => rterm t1
| _ => true
end%T. | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rterm | |
rformula(f : formula R) :=
match f with
| Bool _ => true
| t1 == t2 => rterm t1 && rterm t2
| Unit t1 => false
| f1 /\ f2 | f1 \/ f2 | f1 ==> f2 => rformula f1 && rformula f2
| ~ f1 | ('exists 'X__, f1) | ('forall 'X__, f1) => rformula f1
end%T. | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rformula | |
ub_var(t : term R) :=
match t with
| 'X_i => i.+1
| t1 + t2 | t1 * t2 => maxn (ub_var t1) (ub_var t2)
| - t1 | t1 *+ _ | t1 ^+ _ | t1^-1 => ub_var t1
| _ => 0%N
end%T. | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | ub_var | |
to_rterm(t : term R) (r : seq (term R)) (n : nat) {struct t} :=
match t with
| t1^-1 =>
let: (t1', r1) := to_rterm t1 r n in
('X_(n + size r1), rcons r1 t1')
| t1 + t2 =>
let: (t1', r1) := to_rterm t1 r n in
let: (t2', r2) := to_rterm t2 r1 n in
(t1' + t2', r2)
| - t1 =>
let: (t1', r1... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | to_rterm | |
to_rterm_idt r n : rterm t -> to_rterm t r n = (t, r).
Proof.
elim: t r n => //.
- by move=> t1 IHt1 t2 IHt2 r n /= /andP[rt1 rt2]; rewrite {}IHt1 // IHt2.
- by move=> t IHt r n /= rt; rewrite {}IHt.
- by move=> t IHt r n m /= rt; rewrite {}IHt.
- by move=> t1 IHt1 t2 IHt2 r n /= /andP[rt1 rt2]; rewrite {}IHt1 // IHt2.... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | to_rterm_id | |
eq0_rformt1 :=
let m := ub_var t1 in
let: (t1', r1) := to_rterm t1 [::] m in
let fix loop r i := match r with
| [::] => t1' == 0
| t :: r' =>
let f := 'X_i * t == 1 /\ t * 'X_i == 1 in
'forall 'X_i, (f \/ 'X_i == t /\ ~ ('exists 'X_i, f)) ==> loop r' i.+1
end%T
in loop r1 m. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eq0_rform | |
to_rformf :=
match f with
| Bool b => f
| t1 == t2 => eq0_rform (t1 - t2)
| Unit t1 => eq0_rform (t1 * t1^-1 - 1)
| f1 /\ f2 => to_rform f1 /\ to_rform f2
| f1 \/ f2 => to_rform f1 \/ to_rform f2
| f1 ==> f2 => to_rform f1 ==> to_rform f2
| ~ f1 => ~ to_rform f1
| ('exists 'X_i, f1) => 'exists 'X_i, ... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | to_rform | |
to_rform_rformulaf : rformula (to_rform f).
Proof.
suffices eq0_ring t1: rformula (eq0_rform t1) by elim: f => //= => f1 ->.
rewrite /eq0_rform; move: (ub_var t1) => m; set tr := _ m.
suffices: all rterm (tr.1 :: tr.2).
case: tr => {}t1 r /= /andP[t1_r].
by elim: r m => [|t r IHr] m; rewrite /= ?andbT // => /andP[-... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | to_rform_rformula | |
to_rformPe f : holds e (to_rform f) <-> holds e f.
Proof.
suffices{e f} equal0_equiv e t1 t2:
holds e (eq0_rform (t1 - t2)) <-> (eval e t1 == eval e t2).
- elim: f e => /=; try tauto.
+ move=> t1 t2 e.
by split; [move/equal0_equiv/eqP | move/eqP/equal0_equiv].
+ by move=> t1 e; rewrite unitrE; apply: equal0_e... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | to_rformP | |
qf_form(f : formula R) :=
match f with
| Bool _ | _ == _ | Unit _ => true
| f1 /\ f2 | f1 \/ f2 | f1 ==> f2 => qf_form f1 && qf_form f2
| ~ f1 => qf_form f1
| _ => false
end%T. | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_form | |
qf_evale := fix loop (f : formula R) : bool :=
match f with
| Bool b => b
| t1 == t2 => (eval e t1 == eval e t2)%bool
| Unit t1 => eval e t1 \in unit
| f1 /\ f2 => loop f1 && loop f2
| f1 \/ f2 => loop f1 || loop f2
| f1 ==> f2 => (loop f1 ==> loop f2)%bool
| ~ f1 => ~~ loop f1
|_ => false
end%T. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_eval | |
qf_evalPe f : qf_form f -> reflect (holds e f) (qf_eval e f).
Proof.
elim: f => //=; try by move=> *; apply: idP.
- by move=> t1 t2 _; apply: eqP.
- move=> f1 IHf1 f2 IHf2 /= /andP[/IHf1[] f1T]; last by right; case.
by case/IHf2; [left | right; case].
- move=> f1 IHf1 f2 IHf2 /= /andP[/IHf1[] f1F]; first by do 2 left... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_evalP | |
and_dnfbcs1 bcs2 :=
\big[cat/nil]_(bc1 <- bcs1)
map (fun bc2 => (bc1.1 ++ bc2.1, bc1.2 ++ bc2.2)) bcs2. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | and_dnf | |
qf_to_dnf(f : formula R) (neg : bool) {struct f} :=
match f with
| Bool b => if b (+) neg then [:: ([::], [::])] else [::]
| t1 == t2 => [:: if neg then ([::], [:: t1 - t2]) else ([:: t1 - t2], [::])]
| f1 /\ f2 => (if neg then cat else and_dnf) [rec f1, neg] [rec f2, neg]
| f1 \/ f2 => (if neg then and_dnf e... | Fixpoint | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_to_dnf | |
dnf_to_form:=
let pos_lit t := And (t == 0) in let neg_lit t := And (t != 0) in
let cls bc := Or (foldr pos_lit True bc.1 /\ foldr neg_lit True bc.2) in
foldr cls False. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | dnf_to_form | |
cat_dnfPe bcs1 bcs2 :
qf_eval e (dnf_to_form (bcs1 ++ bcs2))
= qf_eval e (dnf_to_form bcs1 \/ dnf_to_form bcs2).
Proof.
by elim: bcs1 => //= bc1 bcs1 IH1; rewrite -orbA; congr orb; rewrite IH1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | cat_dnfP | |
and_dnfPe bcs1 bcs2 :
qf_eval e (dnf_to_form (and_dnf bcs1 bcs2))
= qf_eval e (dnf_to_form bcs1 /\ dnf_to_form bcs2).
Proof.
elim: bcs1 => [|bc1 bcs1 IH1] /=; first by rewrite /and_dnf big_nil.
rewrite /and_dnf big_cons -/(and_dnf bcs1 bcs2) cat_dnfP /=.
rewrite {}IH1 /= andb_orl; congr orb.
elim: bcs2 bc1 {bcs1}... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | and_dnfP | |
qf_to_dnfPe :
let qev f b := qf_eval e (dnf_to_form (qf_to_dnf f b)) in
forall f, qf_form f && rformula f -> qev f false = qf_eval e f.
Proof.
move=> qev; have qevT f: qev f true = ~~ qev f false.
rewrite {}/qev; elim: f => //=; do [by case | move=> f1 IH1 f2 IH2 | ].
- by move=> t1 t2; rewrite !andbT !orbF.
... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_to_dnfP | |
dnf_to_form_qfbcs : qf_form (dnf_to_form bcs).
Proof.
by elim: bcs => //= [[clT clF] _ ->] /=; elim: clT => //=; elim: clF.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | dnf_to_form_qf | |
dnf_rtermcl := all rterm cl.1 && all rterm cl.2. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | dnf_rterm | |
qf_to_dnf_rtermf b : rformula f -> all dnf_rterm (qf_to_dnf f b).
Proof.
set ok := all dnf_rterm.
have cat_ok bcs1 bcs2: ok bcs1 -> ok bcs2 -> ok (bcs1 ++ bcs2).
by move=> ok1 ok2; rewrite [ok _]all_cat; apply/andP.
have and_ok bcs1 bcs2: ok bcs1 -> ok bcs2 -> ok (and_dnf bcs1 bcs2).
rewrite /and_dnf unlock; elim: ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | qf_to_dnf_rterm | |
dnf_to_rformbcs : rformula (dnf_to_form bcs) = all dnf_rterm bcs.
Proof.
elim: bcs => //= [[cl1 cl2] bcs ->]; rewrite {2}/dnf_rterm /=; congr (_ && _).
by (congr andb; [elim: cl1 | elim: cl2]) => //= t cl ->; rewrite andbT.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | dnf_to_rform | |
If:= (pred_f /\ then_f \/ ~ pred_f /\ else_f)%T. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | If | |
If_form_qf:
qf_form pred_f -> qf_form then_f -> qf_form else_f -> qf_form If.
Proof. by move=> /= -> -> ->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | If_form_qf | |
If_form_rf:
rformula pred_f -> rformula then_f -> rformula else_f -> rformula If.
Proof. by move=> /= -> -> ->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | If_form_rf | |
eval_Ife :
let ev := qf_eval e in ev If = (if ev pred_f then ev then_f else ev else_f).
Proof. by rewrite /=; case: ifP => _; rewrite ?orbF. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eval_If | |
Pick:=
\big[Or/False]_(p : {ffun pred I})
((\big[And/True]_i (if p i then pred_f i else ~ pred_f i))
/\ (if pick p is Some i then then_f i else else_f))%T. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Pick | |
Pick_form_qf:
(forall i, qf_form (pred_f i)) ->
(forall i, qf_form (then_f i)) ->
qf_form else_f ->
qf_form Pick.
Proof.
move=> qfp qft qfe; have mA := (big_morph qf_form) true andb.
rewrite mA // big1 //= => p _.
rewrite mA // big1 => [|i _]; first by case: pick.
by rewrite fun_if if_same /= qfp.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Pick_form_qf | |
eval_Picke (qev := qf_eval e) :
let P i := qev (pred_f i) in
qev Pick = (if pick P is Some i then qev (then_f i) else qev else_f).
Proof.
move=> P; rewrite ((big_morph qev) false orb) //= big_orE /=.
apply/existsP/idP=> [[p] | true_at_P].
rewrite ((big_morph qev) true andb) //= big_andE /=.
case/andP=> /forallP... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eval_Pick | |
foldExistsPI e :
(exists2 e', {in [predC I], same_env e e'} & holds e' f)
<-> holds e (foldr Exists f I).
Proof.
elim: I e => /= [|i I IHi] e.
by split=> [[e' eq_e] |]; [apply: eq_holds => i; rewrite eq_e | exists e].
split=> [[e' eq_e f_e'] | [x]]; last set e_x := set_nth 0 e i x.
exists e'`_i; apply/IHi; ex... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | foldExistsP | |
foldForallPI e :
(forall e', {in [predC I], same_env e e'} -> holds e' f)
<-> holds e (foldr Forall f I).
Proof.
elim: I e => /= [|i I IHi] e.
by split=> [|f_e e' eq_e]; [apply | apply: eq_holds f_e => i; rewrite eq_e].
split=> [f_e' x | f_e e' eq_e]; first set e_x := set_nth 0 e i x.
apply/IHi=> e' eq_e; app... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | foldForallP | |
integral_domain_axiom(R : pzRingType) :=
forall x y : R, x * y = 0 -> (x == 0) || (y == 0).
HB.mixin Record ComUnitRing_isIntegral R of ComUnitRing R := {
mulf_eq0_subproof : integral_domain_axiom R;
}.
#[mathcomp(axiom="integral_domain_axiom"), short(type="idomainType")]
HB.structure Definition IntegralDomain :=
... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | integral_domain_axiom | |
mulf_eq0x y : (x * y == 0) = (x == 0) || (y == 0).
Proof.
apply/eqP/idP; first exact: mulf_eq0_subproof.
by case/pred2P=> ->; rewrite (mulr0, mul0r).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulf_eq0 | |
prodf_eq0(I : finType) (P : pred I) (F : I -> R) :
reflect (exists2 i, P i & (F i == 0)) (\prod_(i | P i) F i == 0).
Proof.
apply: (iffP idP) => [|[i Pi /eqP Fi0]]; last first.
by rewrite (bigD1 i) //= Fi0 mul0r.
elim: (index_enum _) => [|i r IHr]; first by rewrite big_nil oner_eq0.
rewrite big_cons /=; have [Pi | ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | prodf_eq0 |
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