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 |
|---|---|---|---|---|---|---|---|---|---|---|
RExpr : pzSemiRingType -> Type | :=
(* 0 *)
| R0 R : RExpr R
(* addition: x + y and (n + m)%N *)
| RAdd R : RExpr R -> RExpr R -> RExpr R
| RnatAdd : RExpr nat -> RExpr nat -> RExpr nat
(* natmul: x *+ n, including n%:R = 1 *+ n *)
| RMuln R : RExpr R -> RExpr nat -> RExpr R
(* opposite *)
| ROpp (R : pzRingType) : RExpr R -> RExpr R... | Inductive | RExpr | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"R1",
"additive",
"int",
"large_nat",
"nat"
] | Type for reified expressions | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
RExpr_ind' | := Induction for RExpr Sort Prop
with MExpr_ind' := Induction for MExpr Sort Prop. | Scheme | RExpr_ind' | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"RExpr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Reval R (e : RExpr R) : R | :=
match e with
| R0 _ => 0
| RAdd _ e1 e2 => Reval e1 + Reval e2
| RnatAdd e1 e2 => addn (Reval e1) (Reval e2)
| RMuln _ e1 e2 => Reval e1 *+ Reval e2
| ROpp _ e1 => - Reval e1
| RMulz _ e1 e2 => Reval e1 *~ Reval e2
| R1 _ => 1
| RMul _ e1 e2 => Reval e1 * Reval e2
| RnatMul e1 e2 => muln (Reval e... | Fixpoint | Reval | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Meval",
"Posz",
"R1",
"RExpr",
"add_pos_nat",
"addn",
"expn",
"muln",
"nat_of_large_nat"
] | Evaluating result of reification should be convertible to input expr. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Reval_eqs (lpe : list ((RExpr R * RExpr R) * (PExpr C * PExpr C))) :
Prop | :=
if lpe isn't ((lhs, rhs), _) :: lpe then True
else Reval lhs = Reval rhs /\ Reval_eqs lpe. | Fixpoint | Reval_eqs | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"RExpr",
"Reval",
"True",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm (i : bool) R (f : R -> F) (e : RExpr R) : F | :=
let popp v := if opp_intr is Some (opp, _) then opp v
else f (Reval e) in (* should never happen *)
let wintr v := if opp_intr is Some (_, intr) then v intr
else f (Reval e) in (* should never happen *)
let pinv v := if inv isn't Some i then f (Reval e) (* should never happen *)
else if push_inv th... | Fixpoint | Rnorm | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"F_of_N",
"Meval",
"Mnorm",
"N_of_large_nat",
"R1",
"RExpr",
"Reval",
"add",
"exp",
"id",
"intr",
"inv",
"inv_id",
"invi",
"mul",
"natmul",
"one",
"opp",
"zero"
] | i means "currently pushing an inv" | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_Rnorm i R (f f' : R -> F) (e : RExpr R) :
f =1 f' -> Rnorm i f e = Rnorm i f' e. | Proof.
pose P R e :=
forall i (f f' : R -> F), f =1 f' -> Rnorm i f e = Rnorm i f' e.
pose P0 V e :=
forall i (f f' : V -> F), f =1 f' -> Mnorm i f e = Mnorm i f' e.
move: i f f'; elim e using (@RExpr_ind' P P0); rewrite {R e}/P {}/P0 //=.
- by move=> R e1 IHe1 e2 IHe2 i f f' feq; rewrite -(IHe1 _ f) -?(IHe2 _ f) ?... | Lemma | eq_Rnorm | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Mnorm",
"P0",
"RExpr",
"RExpr_ind'",
"Rnorm",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_eq_F_of_N (F : pzSemiRingType) (f f' : bool -> N -> F) (ff' : f =2 f')
zero one add mul opp_intr exp inv pi i :
forall (R : pzSemiRingType) (env : R -> F) e,
Rnorm f zero one add mul opp_intr exp inv pi i env e =
Rnorm f' zero one add mul opp_intr exp inv pi i env e. | Proof.
move=> R m e.
pose P R e := forall f f' pi i (m : R -> F), f =2 f' ->
Rnorm f zero one add mul opp_intr exp inv pi i m e =
Rnorm f' zero one add mul opp_intr exp inv pi i m e.
pose P0 V e := forall f f' pi i (m : V -> F), f =2 f' ->
Mnorm f zero one add mul opp_intr exp inv pi i m e =
Mnorm f' zero one a... | Lemma | Rnorm_eq_F_of_N | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Mnorm",
"P0",
"RExpr_ind'",
"Rnorm",
"add",
"apply",
"env",
"exp",
"intr",
"inv",
"mul",
"one",
"opp",
"pi",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_or_ring | :=
| Field of fieldType | Ring of pzRingType | SemiRing of pzSemiRingType. | Variant | field_or_ring | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Ring",
"SemiRing"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semiring_of_field_or_ring (RF : field_or_ring) : pzSemiRingType | :=
match RF with Field R => R | Ring R => R | SemiRing R => R end. | Coercion | semiring_of_field_or_ring | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Ring",
"SemiRing",
"field_or_ring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ring_opp_intr (R : field_or_ring) : option ((R -> R) * (int -> R)) | :=
match R with Field R' | Ring R' => Some (@GRing.opp R', intr) | _ => None end. | Definition | ring_opp_intr | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Ring",
"field_or_ring",
"int",
"intr",
"opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_inv (R : field_or_ring) : option (R -> R) | :=
if R is Field F then Some (@GRing.inv F) else None. | Definition | field_inv | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"field_or_ring",
"inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_id {R : field_or_ring} | := if field_inv R is Some i then i else id. | Definition | inv_id | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"field_inv",
"field_or_ring",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invi {R : field_or_ring} i (r : R) | := if i then inv_id r else r. | Definition | invi | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"field_or_ring",
"inv_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_of_N | := (fun b n => invi b (N.to_nat n)%:R). | Notation | F_of_N | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"invi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expN | := (fun x n => x ^+ N.to_nat n). | Notation | expN | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm | := (Rnorm F_of_N 0 1 +%R *%R (ring_opp_intr F) expN
(field_inv F)). | Notation | Rnorm | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"F_of_N",
"expN",
"field_inv",
"ring_opp_intr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Mnorm | := (Mnorm F_of_N 0 1 +%R *%R (ring_opp_intr F) expN
(field_inv F)). | Notation | Mnorm | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"F_of_N",
"expN",
"field_inv",
"ring_opp_intr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rnorm_correct pi (e : RExpr F) :
pi = (if F is SemiRing _ then false else true) && pi ->
Reval e = Rnorm pi false id e. | Proof.
move=> piF.
suff: forall (i : bool) R (f : {rmorphism R -> F}) (e' : RExpr R),
invi (pi && i) (f (Reval e')) = Rnorm pi i f e'.
by move/(_ false _ idfun e); rewrite andbF.
move=> i R f {}e.
have invi0 b : invi b 0 = 0 :> F.
by rewrite /invi /inv_id /field_inv; case: b F => -[]// F' /[!invr0].
have inviM ... | Lemma | Rnorm_correct | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Meval",
"Mnorm",
"N_to_natS",
"NegzE",
"P0",
"Posz",
"RExpr",
"RExpr_ind'",
"Reval",
"Rnorm",
"SemiRing",
"add_pos_natE",
"additive",
"apply",
"comp",
"e'",
"eq_Rnorm",
"exprVn",
"exprnN",
"field_inv",
"fmorphV",
"id",
"intmul",
"intz",
"inv_id",
"invfM",
"invi",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semiring_correct (R : comPzSemiRingType) n env
(lpe : seq ((RExpr R * RExpr R) * (PExpr N * PExpr N)))
(re1 re2 : RExpr R) (pe1 pe2 : PExpr N) :
Reval_eqs lpe ->
(forall R_of_N zero one add mul exp natr,
let R_of_N := R_of_N (natr : nat -> R) in
let norm := Rnorm (fun=> R_of_N) zero one ad... | Proof.
have R_of_NE : (fun=> R_of_N R (GRing.natmul 1))
=2 fun b n => @invi (SemiRing R) b (N.to_nat n)%:R by case=> [] [].
rewrite !(@Rnorm_correct (SemiRing R) false _ erefl).
rewrite -!(Rnorm_eq_F_of_N R_of_NE) => elpe.
move=> /(_ (R_of_N R) 0 1 +%R *%R (@GRing.exp R) (GRing.natmul 1))[-> ->]/=.
move=> nelpe; appl... | Lemma | semiring_correct | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Nsemiring_correct",
"PEeval",
"RExpr",
"R_of_N",
"Reval",
"Reval_eqs",
"Rnorm",
"Rnorm_correct",
"Rnorm_eq_F_of_N",
"SemiRing",
"add",
"apply",
"env",
"env_nth",
"eqb",
"eqxx",
"eval",
"exp",
"id",
"invi",
"map",
"mul",
"nat",
"natmul",
"norm",
"one",
"ring_check... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ring_correct (R : comPzRingType) n env
(lpe : seq ((RExpr R * RExpr R) * (PExpr Z * PExpr Z)))
(re1 re2 : RExpr R) (pe1 pe2 : PExpr Z) :
Reval_eqs lpe ->
(forall R_of_Z zero one add opp mul exp,
let R_of_N _ n := R_of_Z (Z.of_N n) in
let opp_intr := Some (opp, intr) in
let norm := R... | Proof.
pose R_of_N (b : bool) n : Ring R := R_of_Z R (Z.of_N n).
have R_of_NE : R_of_N =2 fun b n => @invi (Ring R) b (N.to_nat n)%:R.
by case=> [] [].
rewrite !(@Rnorm_correct (Ring R) false _ erefl).
rewrite -!(Rnorm_eq_F_of_N R_of_NE) => elpe.
move=> /( _ (R_of_Z R) 0 1 +%R -%R *%R (@GRing.exp R))[-> ->]/=.
move=>... | Lemma | ring_correct | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"PEeval",
"RExpr",
"R_of_N",
"R_of_Z",
"Reval",
"Reval_eqs",
"Ring",
"Rnorm",
"Rnorm_correct",
"Rnorm_eq_F_of_N",
"Zring_correct",
"add",
"apply",
"env",
"env_nth",
"eqb",
"eqxx",
"eval",
"exp",
"id",
"intr",
"invi",
"map",
"mul",
"norm",
"one",
"opp",
"ring_che... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
z_const_helper n | :=
match n with
| Posz n => (true, Nat.to_num_hex_uint n)
| Negz n => (false, Nat.to_num_hex_uint n)
end. | Definition | z_const_helper | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ring_reflection Lem R VarMap Lpe RE1 RE2 PE1 PE2 LpeProofs | :=
exact (Lem R 100%N VarMap Lpe RE1 RE2 PE1 PE2 LpeProofs
ltac:(reflexivity) ltac:(vm_compute; reflexivity)). | Ltac | ring_reflection | algebra | algebra/ring_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"preorder",
"ssralg",
"ssrint",
"binnums",
"mathcomp.algebra",
"Extra",
... | [] | Main tactics, called from the elpi parser (c.f., ring.elpi) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"''e_' j" | := (delta_mx 0 j)
(format "''e_' j", at level 8, j at level 2) : ring_scope. | Notation | ''e_' j | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"delta_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"M ^ phi" | := (map_mx phi M) : sesquilinear_scope. | Notation | M ^ phi | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"M ^t phi" | := ((M ^T) ^ phi) : sesquilinear_scope. | Notation | M ^t phi | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_map_mx_id (R : nzRingType) m n (M : 'M[R]_(m, n)) (f : R -> R) :
f =1 id -> M ^ f = M. | Proof. by move=> /eq_map_mx->; rewrite map_mx_id. Qed. | Lemma | eq_map_mx_id | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"eq_map_mx",
"id",
"map_mx_id"
] | TODO: move? | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
idfunK : involutive (@idfun R). | Proof. by []. Qed. | Let | idfunK | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorphK (f : involutive_rmorphism R) : involutive f. | Proof. by move: f => [? [? ? []]]. Qed. | Lemma | rmorphK | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjCfun_involutive : involutive (@conjC C). | Proof. exact: conjCK. Qed. | Let | conjCfun_involutive | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"conjC",
"conjCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxCK {C : numClosedFieldType} m n (A : 'M[C]_(m, n)) :
(A ^ conjC) ^ conjC = A. | Proof. by apply/matrixP=> i j; rewrite !mxE conjCK. Qed. | Lemma | map_mxCK | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"conjC",
"conjCK",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bilinear_for (R : nzRingType) (U U' : lmodType R) (V : zmodType)
(s : GRing.Scale.law R V) (s' : GRing.Scale.law R V) (f : U -> U' -> V) | :=
((forall u', GRing.linear_for (s : R -> V -> V) (f ^~ u'))
* (forall u, GRing.linear_for s' (f u)))%type. | Definition | bilinear_for | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"law",
"linear_for",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bilinear f | := (bilinear_for *:%R *:%R f). | Notation | bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
biscalar f | := (bilinear_for *%R *%R f). | Notation | biscalar | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mapUUV | := (@Bilinear.type R U U' V s s'). | Notation | mapUUV | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"type"
] | Support for right-to-left rewriting with the generic linearZ rule. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
map_class | := mapUUV. | Definition | map_class | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_at_left (a : R) | := mapUUV. | Definition | map_at_left | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_at_right (b : R) | := mapUUV. | Definition | map_at_right | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_at_both (a b : R) | := mapUUV. | Definition | map_at_both | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_for_left a s_a | :=
MapForLeft {map_for_left_map : mapUUV; _ : s a = s_a }. | Structure | map_for_left | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_for_right b s'_b | :=
MapForRight {map_for_right_map : mapUUV; _ : s' b = s'_b }. | Structure | map_for_right | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_for_both a b s_a s'_b | :=
MapForBoth {map_for_both_map : mapUUV; _ : s a = s_a ; _ : s' b = s'_b }. | Structure | map_for_both | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unify_map_at_left a (f : map_at_left a) | :=
MapForLeft f (erefl (s a)). | Definition | unify_map_at_left | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_left"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unify_map_at_right b (f : map_at_right b) | :=
MapForRight f (erefl (s' b)). | Definition | unify_map_at_right | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_right"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unify_map_at_both a b (f : map_at_both a b) | :=
MapForBoth f (erefl (s a)) (erefl (s' b)). | Definition | unify_map_at_both | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_both"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
wrapped | := Wrap {unwrap : mapUUV}. | Structure | wrapped | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mapUUV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
wrap (f : map_class) | := Wrap f. | Definition | wrap | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_class"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'bilinear' U -> V -> W | s & t }" | :=
(@Bilinear.type _ U%type V%type W%type s t)
(U at level 98, V at level 98, W at level 99,
format "{ 'bilinear' U -> V -> W | s & t }") : ring_scope. | Notation | { 'bilinear' U -> V -> W | s & t } | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'bilinear' U -> V -> W | s }" | :=
({bilinear U -> V -> W | s.1 & s.2})
(U at level 98, V at level 98, W at level 99,
format "{ 'bilinear' U -> V -> W | s }") : ring_scope. | Notation | { 'bilinear' U -> V -> W | s } | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'bilinear' U -> V -> W }" | := {bilinear U -> V -> W | *:%R & *:%R}
(U at level 98, V at level 98, W at level 99,
format "{ 'bilinear' U -> V -> W }") : ring_scope. | Notation | { 'bilinear' U -> V -> W } | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'biscalar' U }" | := {bilinear U%type -> U%type -> _ | *%R & *%R}
(format "{ 'biscalar' U }") : ring_scope. | Notation | { 'biscalar' U } | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
applyr_head t (f : U -> U' -> V) u v | := let: tt := t in f v u. | Definition | applyr_head | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
applyr | := (applyr_head tt). | Notation | applyr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyr_head"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bilinear.map_for_left_map : Bilinear.map_for_left >-> Bilinear.type. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_for_left",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.map_for_right_map : Bilinear.map_for_right >-> Bilinear.type. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_for_right",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.map_for_both_map : Bilinear.map_for_both >-> Bilinear.type. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_for_both",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_left : Bilinear.map_at_left >-> Bilinear.map_for_left. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_left",
"map_for_left",
"unify_map_at_left"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_right : Bilinear.map_at_right >-> Bilinear.map_for_right. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_right",
"map_for_right",
"unify_map_at_right"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_both : Bilinear.map_at_both >-> Bilinear.map_for_both. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_at_both",
"map_for_both",
"unify_map_at_both"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_left. | Canonical | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"unify_map_at_left"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_right. | Canonical | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"unify_map_at_right"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unify_map_at_both. | Canonical | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"unify_map_at_both"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.unwrap : Bilinear.wrapped >-> Bilinear.type. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"type",
"wrapped"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.wrap : Bilinear.map_class >-> Bilinear.wrapped. | Coercion | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_class",
"wrap",
"wrapped"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Bilinear.wrap. | Canonical | Bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
linear0r : f z 0 = 0. | Proof. by rewrite raddf0. Qed. | Lemma | linear0r | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearNr : {morph f z : x / - x}. | Proof. exact: raddfN. Qed. | Lemma | linearNr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearDr : {morph f z : x y / x + y}. | Proof. exact: raddfD. Qed. | Lemma | linearDr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddfD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearBr : {morph f z : x y / x - y}. | Proof. exact: raddfB. Qed. | Lemma | linearBr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearMnr n : {morph f z : x / x *+ n}. | Proof. exact: raddfMn. Qed. | Lemma | linearMnr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddfMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearMNnr n : {morph f z : x / x *- n}. | Proof. exact: raddfMNn. Qed. | Lemma | linearMNnr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddfMNn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_sumr I r (P : pred I) E :
f z (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f z (E i). | Proof. exact: raddf_sum. Qed. | Lemma | linear_sumr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"raddf_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZr_LR : scalable_for s' (f z). | Proof. exact: linearZ_LR. Qed. | Lemma | linearZr_LR | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"linearZ_LR",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearPr a : {morph f z : u v / a *: u + v >-> s' a u + v}. | Proof. exact: linearP. Qed. | Lemma | linearPr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"linearP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
applyrE x : applyr f x =1 f^~ x. | Proof. by []. Qed. | Lemma | applyrE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear0l : f 0 z = 0. | Proof. by rewrite -applyrE raddf0. Qed. | Lemma | linear0l | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearNl : {morph f^~ z : x / - x}. | Proof. by move=> ?; rewrite -applyrE raddfN. Qed. | Lemma | linearNl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearDl : {morph f^~ z : x y / x + y}. | Proof. by move=> ? ?; rewrite -applyrE raddfD. Qed. | Lemma | linearDl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddfD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearBl : {morph f^~ z : x y / x - y}. | Proof. by move=> ? ?; rewrite -applyrE raddfB. Qed. | Lemma | linearBl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearMnl n : {morph f^~ z : x / x *+ n}. | Proof. by move=> ?; rewrite -applyrE raddfMn. Qed. | Lemma | linearMnl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddfMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearMNnl n : {morph f^~ z : x / x *- n}. | Proof. by move=> ?; rewrite -applyrE raddfMNn. Qed. | Lemma | linearMNnl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddfMNn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_sumlz I r (P : pred I) E :
f (\sum_(i <- r | P i) E i) z = \sum_(i <- r | P i) f (E i) z. | Proof. by rewrite -applyrE raddf_sum. Qed. | Lemma | linear_sumlz | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"raddf_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZl_LR : scalable_for s (f ^~ z). | Proof. by move=> ? ?; rewrite -applyrE linearZ_LR. Qed. | Lemma | linearZl_LR | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"linearZ_LR",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearPl a : {morph f^~ z : u v / a *: u + v >-> s a u + v}. | Proof. by move=> ? ?; rewrite -applyrE linearP. Qed. | Lemma | linearPl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"applyrE",
"linearP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZl z (c : S) (a : R) (h_c := h c)
(f : Bilinear.map_for_left U U' s s' a h_c) u :
f (a *: u) z = h_c (Bilinear.wrap f u z). | Proof. by rewrite linearZl_LR; case: f => f /= ->. Qed. | Lemma | linearZl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"linearZl_LR",
"map_for_left",
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZr z c' b (h'_c' := h' c')
(f : Bilinear.map_for_right U U' s s' b h'_c') u :
f z (b *: u) = h'_c' (Bilinear.wrap f z u). | Proof. by rewrite linearZr_LR; case: f => f /= ->. Qed. | Lemma | linearZr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"linearZr_LR",
"map_for_right",
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZlr c c' a b (h_c := h c) (h'_c' := h' c')
(f : Bilinear.map_for_both U U' s s' a b h_c h'_c') u v :
f (a *: u) (b *: v) = h_c (h'_c' (Bilinear.wrap f u v)). | Proof. by rewrite linearZl_LR linearZ_LR; case: f => f /= -> ->. Qed. | Lemma | linearZlr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"linearZ_LR",
"linearZl_LR",
"map_for_both",
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linearZrl c c' a b (h_c := h c) (h'_c' := h' c')
(f : Bilinear.map_for_both U U' s s' a b h_c h'_c') u v :
f (a *: u) (b *: v) = h'_c' (h_c (Bilinear.wrap f u v)). | Proof. by rewrite linearZ_LR/= linearZl_LR; case: f => f /= -> ->. Qed. | Lemma | linearZrl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"Bilinear",
"linearZ_LR",
"linearZl_LR",
"map_for_both",
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_is_bilinear (R : comNzRingType) m n p : bilinear_for
(GRing.Scale.Law.clone _ _ *:%R _) (GRing.Scale.Law.clone _ _ *:%R _)
(@mulmx R m n p). | Proof.
split=> [u'|u] a x y /=.
- by rewrite mulmxDl scalemxAl.
- by rewrite mulmxDr scalemxAr.
Qed. | Lemma | mulmx_is_bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear_for",
"clone",
"mulmx",
"mulmxDl",
"mulmxDr",
"scalemxAl",
"scalemxAr",
"split"
] | Canonical mulmx_bilinear (R : comNzRingType) m n p := [bilinear of @mulmx R m n p]. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
form u v | := (u *m M *m (v ^t theta)) 0 0. | Definition | form | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u , v ]" | := (form u%R v%R) : ring_scope. | Notation | ''[' u , v ] | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u ]" | := '[u, u] : ring_scope. | Notation | ''[' u ] | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form0l u : '[0, u] = 0. | Proof. by rewrite /form !mul0mx mxE. Qed. | Lemma | form0l | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"mul0mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form0r u : '[u, 0] = 0. | Proof. by rewrite /form trmx0 map_mx0 mulmx0 mxE. Qed. | Lemma | form0r | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"map_mx0",
"mulmx0",
"mxE",
"trmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formDl u v w : '[u + v, w] = '[u, w] + '[v, w]. | Proof. by rewrite /form !mulmxDl mxE. Qed. | Lemma | formDl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"mulmxDl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formDr u v w : '[u, v + w] = '[u, v] + '[u, w]. | Proof. by rewrite /form linearD !map_mxD !mulmxDr mxE. Qed. | Lemma | formDr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"linearD",
"map_mxD",
"mulmxDr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formZr a u v : '[u, a *: v] = theta a * '[u, v]. | Proof. by rewrite /form !(linearZ, map_mxZ) /= mxE. Qed. | Lemma | formZr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"linearZ",
"map_mxZ",
"mxE",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formZl a u v : '[a *: u, v] = a * '[u, v]. | Proof.
by do !rewrite /form -[_ *: _ *m _]/(mulmxr _ _) linearZ /=; rewrite mxE.
Qed. | Lemma | formZl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"linearZ",
"mulmxr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formNl u v : '[- u, v] = - '[u, v]. | Proof. by rewrite -scaleN1r formZl mulN1r. Qed. | Lemma | formNl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"formZl",
"mulN1r",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formNr u v : '[u, - v] = - '[u, v]. | Proof. by rewrite -scaleN1r formZr rmorphN1 mulN1r. Qed. | Lemma | formNr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"formZr",
"mulN1r",
"rmorphN1",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
formee i j : '['e_i, 'e_j] = M i j. | Proof.
rewrite /form -rowE -map_trmx map_delta_mx -[M in LHS]trmxK.
by rewrite -tr_col -trmx_mul -rowE !mxE.
Qed. | Lemma | formee | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"map_delta_mx",
"map_trmx",
"mxE",
"rowE",
"tr_col",
"trmxK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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