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 |
|---|---|---|---|---|---|---|---|---|---|---|
and_dnf bcs1 bcs2 | :=
\big[cat/nil]_(bc1 <- bcs1)
map (fun bc2 => (bc1.1 ++ bc2.1, bc1.2 ++ bc2.2)) bcs2. | Definition | and_dnf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"cat",
"map"
] | DNF preserving conjunction | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 else cat) [rec f1, neg] [rec f2, neg]
| f1 ==> f... | Fixpoint | qf_to_dnf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"Bool",
"and_dnf",
"cat",
"f1",
"f2",
"formula",
"neg"
] | Computes a DNF from a qf ring formula | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 | dnf_to_form | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"And",
"False",
"True",
"foldr"
] | Conversely, transforms a DNF into a formula | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cat_dnfP e 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 | cat_dnfP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"dnf_to_form",
"qf_eval"
] | Catenation of dnf is the Or of formulas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
and_dnfP e 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} => [|bc2 bcs2 IH] bc1 /=; first by rewrite andbF.
rewrite {}IH /= andb_orr; congr orb => {bcs2}.
suffices aux (l1 l2 : seq... | Lemma | and_dnfP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"And",
"True",
"and_dnf",
"big_cons",
"big_nil",
"cat_dnfP",
"dnf_to_form",
"foldr",
"l1",
"l2",
"qf_eval",
"seq",
"term"
] | and_dnf is the And of formulas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
qf_to_dnfP e :
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.
- by rewrite and_dnfP cat_dnfP negb_and -IH1 -IH2.
- by rewrite and_dnfP cat_dnfP negb_or -IH1 -IH2.
- by rewrite and_dnfP cat_dnfP /= n... | Lemma | qf_to_dnfP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"and_dnfP",
"cat_dnfP",
"dnf_to_form",
"f1",
"f2",
"qf_eval",
"qf_form",
"qf_to_dnf",
"rformula",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnf_to_form_qf bcs : qf_form (dnf_to_form bcs). | Proof.
by elim: bcs => //= [[clT clF] _ ->] /=; elim: clT => //=; elim: clF.
Qed. | Lemma | dnf_to_form_qf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"dnf_to_form",
"qf_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnf_rterm cl | := all rterm cl.1 && all rterm cl.2. | Definition | dnf_rterm | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"all",
"rterm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_to_dnf_rterm f 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: bcs1 => //= cl1 bcs1 IH1; rewrite -andbA.
case/and3P=> ok11 ok12... | Lemma | qf_to_dnf_rterm | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"all",
"all_cat",
"and_dnf",
"apply",
"dnf_rterm",
"qf_to_dnf",
"rformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnf_to_rform bcs : 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 | dnf_to_rform | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"all",
"dnf_rterm",
"dnf_to_form",
"rformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
If | := (pred_f /\ then_f \/ ~ pred_f /\ else_f)%T. | Definition | If | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
If_form_qf :
qf_form pred_f -> qf_form then_f -> qf_form else_f -> qf_form If. | Proof. by move=> /= -> -> ->. Qed. | Lemma | If_form_qf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"If",
"qf_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
If_form_rf :
rformula pred_f -> rformula then_f -> rformula else_f -> rformula If. | Proof. by move=> /= -> -> ->. Qed. | Lemma | If_form_rf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"If",
"rformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_If e :
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 | eval_If | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"If",
"qf_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Pick | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"And",
"False",
"True",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 _|]; last by case: pick.
by rewrite fun_if if_same /= qfp.
Qed. | Lemma | Pick_form_qf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"Pick",
"big1",
"big_morph",
"last",
"pick",
"qf_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_Pick e (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-eq_p_P.
rewrite (@eq_pick _ _ P) => [i|]; last by case: pick.
by move/(_ i): eq_p_P => /=; case: (p i) => //= /negPf.
exists [ffun... | Lemma | eval_Pick | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"Pick",
"apply",
"big_andE",
"big_morph",
"big_orE",
"eq_pick",
"existsP",
"ffunE",
"forallP",
"last",
"pick",
"qf_eval",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldExistsP I 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; exists e' => // j.
by have:= eq_e j; rewrite nth_set_nth /= !inE; case: eqP => // ->.
case/IHi=> e' eq_e f_e'; e... | Lemma | foldExistsP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"e'",
"eq_holds",
"foldr",
"holds",
"inE",
"last",
"nth_set_nth",
"same_env",
"set_nth",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldForallP I 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; apply: f_e' => j.
by have:= eq_e j; rewrite nth_set_nth /= !inE; case: eqP.
move/IHi: (f_e e'`_i); apply=> j.
by ... | Lemma | foldForallP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"e'",
"eq_holds",
"foldr",
"holds",
"inE",
"nth_set_nth",
"same_env",
"set_nth",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decidable_field_axiom (R : unitRingType)
(s : seq R -> pred (formula R)) | :=
forall e f, reflect (holds e f) (s e f). | Definition | decidable_field_axiom | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"formula",
"holds",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sol_subproof n f :
reflect (exists s, (size s == n) && sat s f)
(sat [::] (foldr Exists f (iota 0 n))). | Proof.
apply: (iffP (satP _ _)) => [|[s]]; last first.
case/andP=> /eqP sz_s /satP f_s; apply/foldExistsP.
exists s => // i; rewrite !inE mem_iota -leqNgt add0n => le_n_i.
by rewrite !nth_default ?sz_s.
case/foldExistsP=> e e0 f_e; set s := take n (set_nth 0 e n 0).
have sz_s: size s = n by rewrite size_take size... | Fact | sol_subproof | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"add0n",
"apply",
"e0",
"eq_holds",
"eq_sym",
"eqn_leq",
"eqxx",
"foldExistsP",
"foldr",
"inE",
"iota",
"last",
"leqNgt",
"leqP",
"leq_max",
"leqnn",
"mem_iota",
"nth_default",
"nth_set_nth",
"nth_take",
"sat",
"satP",
"set_nth",
"size",
"size_set_nth",
"size_take",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sol n f | :=
if sol_subproof n f is ReflectT sP then xchoose sP else nseq n 0. | Definition | sol | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"nseq",
"sol_subproof",
"xchoose"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_sol n f : size (sol n f) = n. | Proof.
rewrite /sol; case: sol_subproof => [sP | _]; last exact: size_nseq.
by case/andP: (xchooseP sP) => /eqP.
Qed. | Lemma | size_sol | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"last",
"size",
"size_nseq",
"sol",
"sol_subproof",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solP n f : reflect (exists2 s, size s = n & holds s f) (sat (sol n f) f). | Proof.
rewrite /sol; case: sol_subproof => [sP | sPn].
case/andP: (xchooseP sP) => _ ->; left.
by case: sP => s; case/andP; move/eqP=> <-; move/satP; exists s.
apply: (iffP (satP _ _)); first by exists (nseq n 0); rewrite ?size_nseq.
by case=> s sz_s; move/satP=> f_s; case: sPn; exists s; rewrite sz_s eqxx.
Qed. | Lemma | solP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"eqxx",
"holds",
"nseq",
"sat",
"satP",
"size",
"size_nseq",
"sol",
"sol_subproof",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sat f1 f2 :
(forall e, holds e f1 <-> holds e f2) -> sat^~ f1 =1 sat^~ f2. | Proof. by move=> eqf12 e; apply/satP/satP; case: (eqf12 e). Qed. | Lemma | eq_sat | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"f1",
"f2",
"holds",
"sat",
"satP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sol f1 f2 :
(forall e, holds e f1 <-> holds e f2) -> sol^~ f1 =1 sol^~ f2. | Proof.
rewrite /sol => /eq_sat eqf12 n.
do 2![case: sol_subproof] => //= [f1s f2s | ns1 [s f2s] | [s f1s] []].
- by apply: eq_xchoose => s; rewrite eqf12.
- by case: ns1; exists s; rewrite -eqf12.
by exists s; rewrite eqf12.
Qed. | Lemma | eq_sol | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"eq_sat",
"eq_xchoose",
"f1",
"f2",
"holds",
"sol",
"sol_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
wf_QE_proj | :=
forall i bc (bc_i := proj i bc),
dnf_rterm bc -> qf_form bc_i && rformula bc_i. | Definition | wf_QE_proj | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"dnf_rterm",
"qf_form",
"rformula"
] | The elimination projector is well_formed. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
valid_QE_proj | :=
forall i bc (ex_i_bc := ('exists 'X_i, dnf_to_form [:: bc])%T) e,
dnf_rterm bc -> reflect (holds e ex_i_bc) (qf_eval e (proj i bc)). | Definition | valid_QE_proj | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"dnf_rterm",
"dnf_to_form",
"holds",
"qf_eval"
] | The elimination projector is valid | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
(wf_proj : wf_QE_proj) (ok_proj : valid_QE_proj). | Hypotheses | wf_proj | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"valid_QE_proj",
"wf_QE_proj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
elim_aux f n | := foldr Or False (map (proj n) (qf_to_dnf f false)). | Let | elim_aux | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"False",
"foldr",
"map",
"qf_to_dnf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quantifier_elim f | :=
match f with
| f1 /\ f2 => (quantifier_elim f1) /\ (quantifier_elim f2)
| f1 \/ f2 => (quantifier_elim f1) \/ (quantifier_elim f2)
| f1 ==> f2 => (~ quantifier_elim f1) \/ (quantifier_elim f2)
| ~ f => ~ quantifier_elim f
| ('exists 'X_n, f) => elim_aux (quantifier_elim f) n
| ('forall 'X_n, f) => ~ el... | Fixpoint | quantifier_elim | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"elim_aux",
"f1",
"f2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quantifier_elim_wf f :
let qf := quantifier_elim f in rformula f -> qf_form qf && rformula qf. | Proof.
suffices aux_wf f0 n : let qf := elim_aux f0 n in
rformula f0 -> qf_form qf && rformula qf.
- by elim: f => //=; do ?[ move=> f1 IH1 f2 IH2;
case/andP=> rf1 rf2;
case/andP:(IH1 rf1)=> -> ->;
case/andP:(IH2 rf2)=> -> -> //
| move... | Lemma | quantifier_elim_wf | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"False",
"all",
"apply",
"dnf_rterm",
"elim_aux",
"f1",
"f2",
"foldr",
"map",
"qf",
"qf_form",
"qf_to_dnf_rterm",
"quantifier_elim",
"rformula",
"wf_proj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quantifier_elim_rformP e f :
rformula f -> reflect (holds e f) (qf_eval e (quantifier_elim f)). | Proof.
pose rc e n f := exists x, qf_eval (set_nth 0 e n x) f.
have auxP f0 e0 n0: qf_form f0 && rformula f0 ->
reflect (rc e0 n0 f0) (qf_eval e0 (elim_aux f0 n0)).
+ rewrite /elim_aux => cf; set bcs := qf_to_dnf f0 false.
apply: (@iffP (rc e0 n0 (dnf_to_form bcs))); last first.
- by case=> x; rewrite -qf_to_dnfP... | Lemma | quantifier_elim_rformP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"all",
"apply",
"dnf_rterm",
"dnf_to_form",
"dnf_to_form_qf",
"e0",
"elim_aux",
"f1",
"f2",
"holds",
"last",
"qf_eval",
"qf_evalP",
"qf_form",
"qf_to_dnf",
"qf_to_dnfP",
"qf_to_dnf_rterm",
"quantifier_elim",
"quantifier_elim_wf",
"rformula",
"set_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_sat e f | := qf_eval e (quantifier_elim (to_rform f)). | Definition | proj_sat | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"qf_eval",
"quantifier_elim",
"to_rform"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_satP : DecidableField.axiom proj_sat. | Proof.
move=> e f; have fP := quantifier_elim_rformP e (to_rform_rformula f).
by apply: (iffP fP); move/to_rformP.
Qed. | Lemma | proj_satP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"apply",
"axiom",
"fP",
"proj_sat",
"quantifier_elim_rformP",
"to_rformP",
"to_rform_rformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closed_field_axiom (R : pzRingType) | :=
forall n (P : nat -> R), n > 0 ->
exists x : R, x ^+ n = \sum_(i < n) P i * (x ^+ i). | Definition | closed_field_axiom | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"nat"
] | Axiom == all non-constant monic polynomials have a root | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
imaginary_exists : {i : F | i ^+ 2 = -1}. | Proof.
have /sig_eqW[i Di2] := @solve_monicpoly F 2 (nth 0 [:: -1]) isT.
by exists i; rewrite Di2 !big_ord_recl big_ord0 mul0r mulr1 !addr0.
Qed. | Lemma | imaginary_exists | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"addr0",
"big_ord0",
"big_ord_recl",
"mul0r",
"mulr1",
"nth",
"sig_eqW",
"solve_monicpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_eval | := eq_eval. | Definition | eq_eval | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_tsubst | := eval_tsubst. | Definition | eval_tsubst | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_holds | := eq_holds. | Definition | eq_holds | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
holds_fsubst | := holds_fsubst. | Definition | holds_fsubst | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
satP {F e f} | := @satP F e f. | Definition | satP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sat | := eq_sat. | Definition | eq_sat | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solP {F n f} | := @solP F n f. | Definition | solP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sol | := eq_sol. | Definition | eq_sol | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_sol | := size_sol. | Definition | size_sol | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solve_monicpoly | := @solve_monicpoly. | Definition | solve_monicpoly | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imaginary_exists | := imaginary_exists. | Definition | imaginary_exists | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x != y" | := (GRing.Not (x == y)) : term_scope. | Notation | x != y | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/decfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_pred {R : unitRingType} | :=
Eval cbv [ unit_subdef NzRing_hasMulInverse.unit_subdef ] in
(fun u : R => unit_subdef u). | Definition | unit_pred | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit {R : unitRingType} | := [qualify a u : R | unit_pred u]. | Definition | unit | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"unit_pred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^-1" | := (inv x). | Notation | x ^-1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x / y" | := (x * y^-1). | Notation | x / y | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^- n" | := ((x ^+ n)^-1). | Notation | x ^- n | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divrr : {in unit, right_inverse 1 (@inv R) *%R}. | Proof. exact: divrr_subproof. Qed. | Lemma | divrr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrV | := divrr. | Definition | mulrV | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"divrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVr : {in unit, left_inverse 1 (@inv R) *%R}. | Proof. exact: mulVr_subproof. Qed. | Lemma | mulVr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_out x : x \isn't a unit -> x^-1 = x. | Proof. exact: invr_out_subproof. Qed. | Lemma | invr_out | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrP x : reflect (exists y, y * x = 1 /\ x * y = 1) (x \is a unit). | Proof.
apply: (iffP idP) => [Ux | []]; last exact: unitrP_subproof.
by exists x^-1; rewrite divrr ?mulVr.
Qed. | Lemma | unitrP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"divrr",
"last",
"mulVr",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKr : {in unit, left_loop (@inv R) *%R}. | Proof. by move=> x Ux y; rewrite mulrA mulVr ?mul1r. Qed. | Lemma | mulKr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"mul1r",
"mulVr",
"mulrA",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVKr : {in unit, rev_left_loop (@inv R) *%R}. | Proof. by move=> x Ux y; rewrite mulrA mulrV ?mul1r. Qed. | Lemma | mulVKr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"mul1r",
"mulrA",
"mulrV",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrK : {in unit, right_loop (@inv R) *%R}. | Proof. by move=> x Ux y; rewrite -mulrA divrr ?mulr1. Qed. | Lemma | mulrK | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"divrr",
"inv",
"mulr1",
"mulrA",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrVK : {in unit, rev_right_loop (@inv R) *%R}. | Proof. by move=> x Ux y; rewrite -mulrA mulVr ?mulr1. Qed. | Lemma | mulrVK | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"mulVr",
"mulr1",
"mulrA",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divrK | := mulrVK. | Definition | divrK | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"mulrVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrI : {in @unit R, right_injective *%R}. | Proof. by move=> x Ux; apply: can_inj (mulKr Ux). Qed. | Lemma | mulrI | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"mulKr",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulIr : {in @unit R, left_injective *%R}. | Proof. by move=> x Ux; apply: can_inj (mulrK Ux). Qed. | Lemma | mulIr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"mulrK",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
telescope_prodr n m (f : nat -> R) :
(forall k, n < k < m -> f k \is a unit) -> n < m ->
\prod_(n <= k < m) (f k / f k.+1) = f n / f m. | Proof.
move=> Uf ltnm; rewrite (telescope_big (fun i j => f i / f j)) ?ltnm//.
by move=> k ltnkm /=; rewrite mulrA divrK// Uf.
Qed. | Lemma | telescope_prodr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"divrK",
"mulrA",
"nat",
"telescope_big",
"unit"
] | Due to noncommutativity, fractions are inverted. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
telescope_prodr_eq n m (f u : nat -> R) : n < m ->
(forall k, n < k < m -> f k \is a unit) ->
(forall k, (n <= k < m)%N -> u k = f k / f k.+1) ->
\prod_(n <= k < m) u k = f n / f m. | Proof.
by move=> ? ? uE; under eq_big_nat do rewrite uE //=; exact: telescope_prodr.
Qed. | Lemma | telescope_prodr_eq | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"eq_big_nat",
"nat",
"telescope_prodr",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commrV x y : comm x y -> comm x y^-1. | Proof.
have [Uy cxy | /invr_out-> //] := boolP (y \in unit).
by apply: (canLR (mulrK Uy)); rewrite -mulrA cxy mulKr.
Qed. | Lemma | commrV | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"comm",
"invr_out",
"mulKr",
"mulrA",
"mulrK",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrE x : (x \is a unit) = (x / x == 1). | Proof.
apply/idP/eqP=> [Ux | xx1]; first exact: divrr.
by apply/unitrP; exists x^-1; rewrite -commrV.
Qed. | Lemma | unitrE | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"commrV",
"divrr",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invrK : involutive (@inv R). | Proof.
move=> x; case Ux: (x \in unit); last by rewrite !invr_out ?Ux.
rewrite -(mulrK Ux _^-1) -mulrA commrV ?mulKr //.
by apply/unitrP; exists x; rewrite divrr ?mulVr.
Qed. | Lemma | invrK | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"commrV",
"divrr",
"inv",
"invr_out",
"last",
"mulKr",
"mulVr",
"mulrA",
"mulrK",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_inj : injective (@inv R). | Proof. exact: inv_inj invrK. Qed. | Lemma | invr_inj | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv",
"invrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrV x : (x^-1 \in unit) = (x \in unit). | Proof. by rewrite !unitrE invrK commrV. Qed. | Lemma | unitrV | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"commrV",
"invrK",
"unit",
"unitrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitr1 : 1 \in @unit R. | Proof. by apply/unitrP; exists 1; rewrite mulr1. Qed. | Lemma | unitr1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"mulr1",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr1 : 1^-1 = 1 :> R. | Proof. by rewrite -{2}(mulVr unitr1) mulr1. Qed. | Lemma | invr1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"mulVr",
"mulr1",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
div1r x : 1 / x = x^-1. | Proof. by rewrite mul1r. Qed. | Lemma | div1r | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divr1 x : x / 1 = x. | Proof. by rewrite invr1 mulr1. Qed. | Lemma | divr1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"invr1",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_div m d :
d %| m -> d%:R \is a @unit R -> (m %/ d)%:R = m%:R / d%:R :> R. | Proof.
by rewrite dvdn_eq => /eqP def_m unit_d; rewrite -{2}def_m natrM mulrK.
Qed. | Lemma | natr_div | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"dvdn_eq",
"mulrK",
"natrM",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divrI : {in unit, right_injective (fun x y => x / y)}. | Proof. by move=> x /mulrI/inj_comp; apply; apply: invr_inj. Qed. | Lemma | divrI | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"invr_inj",
"mulrI",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divIr : {in unit, left_injective (fun x y => x / y)}. | Proof. by move=> x; rewrite -unitrV => /mulIr. Qed. | Lemma | divIr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"mulIr",
"unit",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitr0 : (0 \is a @unit R) = false. | Proof. by apply/unitrP=> [[x [_ /esym/eqP]]]; rewrite mul0r oner_eq0. Qed. | Lemma | unitr0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"mul0r",
"oner_eq0",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr0 : 0^-1 = 0 :> R. | Proof. by rewrite invr_out ?unitr0. Qed. | Lemma | invr0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"invr_out",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrN1 : -1 \is a @unit R. | Proof. by apply/unitrP; exists (-1); rewrite mulrNN mulr1. Qed. | Lemma | unitrN1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"mulr1",
"mulrNN",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invrN1 : (-1)^-1 = -1 :> R. | Proof. by rewrite -{2}(divrr unitrN1) mulN1r opprK. Qed. | Lemma | invrN1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"divrr",
"mulN1r",
"opprK",
"unitrN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_sign n : ((-1) ^- n) = (-1) ^+ n :> R. | Proof. by rewrite -signr_odd; case: (odd n); rewrite (invr1, invrN1). Qed. | Lemma | invr_sign | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"invr1",
"invrN1",
"odd",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrMl x y : y \is a unit -> (x * y \is a unit) = (x \is a unit). | Proof.
move=> Uy; wlog Ux: x y Uy / x \is a unit => [WHxy|].
by apply/idP/idP=> Ux; first rewrite -(mulrK Uy x); rewrite WHxy ?unitrV.
rewrite Ux; apply/unitrP; exists (y^-1 * x^-1).
by rewrite -!mulrA mulKr ?mulrA ?mulrK ?divrr ?mulVr.
Qed. | Lemma | unitrMl | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"divrr",
"mulKr",
"mulVr",
"mulrA",
"mulrK",
"unit",
"unitrP",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrMr x y : x \is a unit -> (x * y \is a unit) = (y \is a unit). | Proof.
move=> Ux; apply/idP/idP=> [Uxy | Uy]; last by rewrite unitrMl.
by rewrite -(mulKr Ux y) unitrMl ?unitrV.
Qed. | Lemma | unitrMr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"last",
"mulKr",
"unit",
"unitrMl",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitr_prod {I : Type} (P : pred I) (E : I -> R) (r : seq I) :
(forall i, P i -> E i \is a GRing.unit) ->
(\prod_(i <- r | P i) E i \is a GRing.unit). | Proof.
by move=> Eunit; elim/big_rec: _ => [/[!unitr1] |i x /Eunit/unitrMr->].
Qed. | Lemma | unitr_prod | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"big_rec",
"seq",
"unit",
"unitr1",
"unitrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitr_prod_in {I : eqType} (P : pred I) (E : I -> R) (r : seq I) :
{in r, forall i, P i -> E i \is a GRing.unit} ->
(\prod_(i <- r | P i) E i \is a GRing.unit). | Proof.
by rewrite big_seq_cond => H; apply: unitr_prod => i /andP[]; exact: H.
Qed. | Lemma | unitr_prod_in | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"big_seq_cond",
"seq",
"unit",
"unitr_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invrM : {in unit &, forall x y, (x * y)^-1 = y^-1 * x^-1}. | Proof.
move=> x y Ux Uy; have Uxy: (x * y \in unit) by rewrite unitrMl.
by apply: (mulrI Uxy); rewrite divrr ?mulrA ?mulrK ?divrr.
Qed. | Lemma | invrM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"divrr",
"mulrA",
"mulrI",
"mulrK",
"unit",
"unitrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrM_comm x y :
comm x y -> (x * y \is a unit) = (x \is a unit) && (y \is a unit). | Proof.
move=> cxy; apply/idP/andP=> [Uxy | [Ux Uy]]; last by rewrite unitrMl.
suffices Ux: x \in unit by rewrite unitrMr in Uxy.
apply/unitrP; case/unitrP: Uxy => z [zxy xyz]; exists (y * z).
rewrite mulrA xyz -{1}[y]mul1r -{1}zxy cxy -!mulrA (mulrA x) (mulrA _ z) xyz.
by rewrite mul1r -cxy.
Qed. | Lemma | unitrM_comm | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"comm",
"last",
"mul1r",
"mulrA",
"unit",
"unitrMl",
"unitrMr",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrX x n : x \is a unit -> x ^+ n \is a unit. | Proof.
by move=> Ux; elim: n => [|n IHn]; rewrite ?unitr1 // exprS unitrMl.
Qed. | Lemma | unitrX | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"exprS",
"unit",
"unitr1",
"unitrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrX_pos x n : n > 0 -> (x ^+ n \in unit) = (x \in unit). | Proof.
case: n => // n _; rewrite exprS unitrM_comm; first exact: commrX.
by case Ux: (x \is a unit); rewrite // unitrX.
Qed. | Lemma | unitrX_pos | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"commrX",
"exprS",
"unit",
"unitrM_comm",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprVn x n : x^-1 ^+ n = x ^- n. | Proof.
elim: n => [|n IHn]; first by rewrite !expr0 ?invr1.
case Ux: (x \is a unit); first by rewrite exprSr exprS IHn -invrM // unitrX.
by rewrite !invr_out ?unitrX_pos ?Ux.
Qed. | Lemma | exprVn | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"expr0",
"exprS",
"exprSr",
"invr1",
"invrM",
"invr_out",
"unit",
"unitrX",
"unitrX_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprB m n x : n <= m -> x \is a unit -> x ^+ (m - n) = x ^+ m / x ^+ n. | Proof. by move/subnK=> {2}<- Ux; rewrite exprD mulrK ?unitrX. Qed. | Lemma | exprB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"exprD",
"mulrK",
"subnK",
"unit",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_neq0 x : x != 0 -> x^-1 != 0. | Proof.
move=> nx0; case Ux: (x \is a unit); last by rewrite invr_out ?Ux.
by apply/eqP=> x'0; rewrite -unitrV x'0 unitr0 in Ux.
Qed. | Lemma | invr_neq0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"invr_out",
"last",
"unit",
"unitr0",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_eq0 x : (x^-1 == 0) = (x == 0). | Proof. by apply: negb_inj; apply/idP/idP; move/invr_neq0; rewrite ?invrK. Qed. | Lemma | invr_eq0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"invrK",
"invr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_eq1 x : (x^-1 == 1) = (x == 1). | Proof. by rewrite (inv_eq invrK) invr1. Qed. | Lemma | invr_eq1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"inv_eq",
"invr1",
"invrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_unitrP (x y : R^c) : y * x = 1 /\ x * y = 1 -> x \is a unit. | Proof. by case=> [yx1 xy1]; apply/unitrP; exists y. Qed. | Lemma | rev_unitrP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_prodrV (R : unitRingType)
(I : Type) (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 : R^c))^-1). | Proof.
move=> Eunit; symmetry.
apply: (big_morph_in GRing.unit _ _ (unitr1 R^c) (@invrM _) (invr1 _)) Eunit.
by move=> x y xunit; rewrite unitrMr.
Qed. | Lemma | rev_prodrV | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/divalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"prime",
"nmodule",
"rings_modules_and_algebras",
"GRing",
"GRing.Theory",
"AllExports",
"ClosedExports"
] | [
"apply",
"big_morph_in",
"invr1",
"invrM",
"seq",
"unit",
"unitr1",
"unitrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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