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 |
|---|---|---|---|---|---|---|---|---|---|---|
onet_neq0 {R : nzSemiRingType} : (1%R : 'T[R]) != 0%R. | Proof.
apply/eqP; case=> /matrixP.
set i := \prod_(i < k) (u_ i)%:num; set j := \prod_(i < l) (d_ i)%:num.
have: 0 < i by apply/prodn_gt0 => ?; rewrite gtn0.
have: 0 < j by apply/prodn_gt0 => ?; rewrite gtn0.
case: i j => [|i] [|j]// _ _ /(_ ord0 ord0).
by apply/eqP; rewrite !mxE oner_neq0.
Qed. | Let | onet_neq0 | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"gtn0",
"matrixP",
"mxE",
"num",
"oner_neq0",
"ord0",
"prodn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitt {R : unitRingType} (t : 'T[R]) | :=
[forall ij, (\val t ij.1 ij.2) \is a GRing.unit]. | Definition | unitt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"unit",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invt {R : unitRingType} (t : 'T[R]) | :=
if t \in unitt then Tensor (map_mx GRing.inv (\val t)) else t. | Definition | invt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"inv",
"map_mx",
"unitt",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVt {R : unitRingType} : {in @unitt R, left_inverse 1%R invt *%R}. | Proof.
move=> t t_unit; apply/val_inj/matrixP=> i j/=.
rewrite /invt t_unit !mxE mulVr//=.
by move: t_unit; rewrite /unitt=> /forallP /(_ (i, j)).
Qed. | Let | mulVt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"forallP",
"invt",
"matrixP",
"mulVr",
"mxE",
"unitt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divtt {R : unitRingType} : {in @unitt R, right_inverse 1%R invt *%R}. | Proof.
move=> t t_unit; apply/val_inj/matrixP=> i j/=.
rewrite /invt t_unit !mxE divrr//.
by move: t_unit; rewrite /unitt=> /forallP /(_ (i, j)).
Qed. | Let | divtt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"divrr",
"forallP",
"invt",
"matrixP",
"mxE",
"unitt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unittP {R : unitRingType}
: forall x y, y * x = 1%R /\ x * y = 1 -> @unitt R x. | Proof.
move=> x y [/eqP + /eqP]; rewrite /eq_op/==> /eqP/matrixP yx1 /eqP/matrixP xy1.
apply/forallP=> ij; apply/unitrP; exists (\val y ij.1 ij.2).
move: (conj (yx1 ij.1 ij.2) (xy1 ij.1 ij.2)).
by rewrite !mxE.
Qed. | Let | unittP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"conj",
"forallP",
"matrixP",
"mxE",
"unitrP",
"unitt",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invt_out {R : unitRingType} : {in [predC @unitt R], invt =1 id}. | Proof. by move=> t /negbTE t_not_unit; rewrite /invt t_not_unit. Qed. | Let | invt_out | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"id",
"invt",
"unitt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"*h%R" | := hmul : function_scope. | Notation | *h%R | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul"
] | Notations for Hadamard product | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"x *h y" | := (hmul x y) : ring_scope. | Notation | x *h y | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_nil : 1%N = \prod_(i < 0) (([tuple] : 0.-tuple {posnum nat}) i)%:num. | Proof. by rewrite big_ord0. Qed. | Lemma | prod_nil | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"big_ord0",
"nat",
"num",
"posnum",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ord_prod_nil : all_equal_to (cast_ord prod_nil ord0). | Proof.
case=> [[?|n n_ord]]; apply: val_inj=>//=.
by move: n_ord; rewrite -prod_nil.
Qed. | Lemma | ord_prod_nil | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"cast_ord",
"ord0",
"prod_nil",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nil (t : 'sT[R]) : R | :=
\val t (cast_ord prod_nil ord0) (cast_ord prod_nil ord0). | Definition | tensor_nil | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"ord0",
"prod_nil",
"sT",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_tK : cancel const_t tensor_nil. | Proof. by move=> t; rewrite /tensor_nil mxE. Qed. | Lemma | const_tK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"mxE",
"tensor_nil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nilK : cancel tensor_nil const_t. | Proof.
by move=> t; apply/val_inj/matrixP => i j; rewrite mxE !ord_prod_nil.
Qed. | Lemma | tensor_nilK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_t",
"matrixP",
"mxE",
"ord_prod_nil",
"tensor_nil",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"t .[::]" | := (tensor_nil t) : tensor_scope. | Notation | t .[::] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor_nil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nil_eqP {R : Type} t u : t.[::] = u.[::] :> R <-> t = u. | Proof.
by split=> [?|->//]; apply/val_inj/matrixP=> i j; rewrite !ord_prod_nil.
Qed. | Lemma | tensor_nil_eqP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"ord_prod_nil",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nil_is_nmod_morphism {U : nmodType} : nmod_morphism (@tensor_nil U). | Proof. by split=> [|? ?]; rewrite /tensor_nil mxE. Qed. | Fact | tensor_nil_is_nmod_morphism | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"mxE",
"nmod_morphism",
"split",
"tensor_nil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nil_is_monoid_morphism {R : pzSemiRingType} :
monoid_morphism (@tensor_nil R). | Proof. by split=> [|? ?]; rewrite /tensor_nil mxE. Qed. | Fact | tensor_nil_is_monoid_morphism | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"monoid_morphism",
"mxE",
"split",
"tensor_nil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_nilV {R : unitRingType} t
: (t^-1).[::] = t.[::]^-1 :> R. | Proof.
rewrite /tensor_nil {1}/GRing.inv/=/invt.
case (t \in @unitt _ _ [tuple] [tuple] R) eqn:t_unit; rewrite t_unit.
by rewrite mxE.
apply/esym/invr_out; move: t_unit=> /negbT /forallP not_all_unit.
apply/negP=> ?; apply: not_all_unit=> ij.
by rewrite !ord_prod_nil.
Qed. | Lemma | tensor_nilV | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"eqn",
"forallP",
"inv",
"invr_out",
"invt",
"mxE",
"ord_prod_nil",
"tensor_nil",
"tuple",
"unitt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_t_is_nmod_morphism {U : nmodType} :
nmod_morphism (@const_t U _ _ u_ d_). | Proof. by split=> [|? ?]; apply/val_inj/matrixP=> i j; rewrite !mxE. Qed. | Fact | const_t_is_nmod_morphism | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_t",
"matrixP",
"mxE",
"nmod_morphism",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_t_is_monoid_morphism {R : pzSemiRingType} :
monoid_morphism (@const_t R _ _ u_ d_). | Proof. by split=> [|? ?]; apply/val_inj/matrixP=> i j; rewrite !mxE. Qed. | Fact | const_t_is_monoid_morphism | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_t",
"matrixP",
"monoid_morphism",
"mxE",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_tV {R : unitRingType} x
: @const_t R _ _ u_ d_ x^-1 = (const_t x)^-1. | Proof.
apply/val_inj/matrixP=> i j.
rewrite {2}/GRing.inv/=/invt.
case (const_t x \in @unitt _ _ u_ d_ R) eqn:t_unit; rewrite !mxE=>//.
apply invr_out; move: t_unit=> /negbT /forallP not_all_unit.
apply/negP=> ?.
apply: not_all_unit=> ?.
by rewrite mxE.
Qed. | Lemma | const_tV | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_t",
"eqn",
"forallP",
"inv",
"invr_out",
"invt",
"matrixP",
"mxE",
"unitt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_u | := (fprod (fun i : 'I_k => 'I_(u_ i)%:num)). | Notation | fprod_u | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"fprod",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_fprod_u : #|fprod_u| = \prod_(i < k) (u_ i)%:num. | Proof. by rewrite card_fprod; apply: eq_bigr => i _; rewrite card_ord. Qed. | Lemma | card_fprod_u | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"card_fprod",
"card_ord",
"eq_bigr",
"fprod_u",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_index (f : fprod_u) : 'I_(\prod_(i < k) (u_ i)%:num) | :=
cast_ord card_fprod_u (enum_rank f). | Definition | tensor_index | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"card_fprod_u",
"cast_ord",
"enum_rank",
"fprod_u",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_unindex (i : 'I_(\prod_(i < k) (u_ i)%:num)) : fprod_u | :=
enum_val (cast_ord (esym card_fprod_u) i). | Definition | tensor_unindex | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"card_fprod_u",
"cast_ord",
"enum_val",
"fprod_u",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_indexK : cancel tensor_index tensor_unindex. | Proof.
by move=> f; rewrite /tensor_index /tensor_unindex cast_ordK enum_rankK.
Qed. | Lemma | tensor_indexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ordK",
"enum_rankK",
"tensor_index",
"tensor_unindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_unindexK : cancel tensor_unindex tensor_index. | Proof.
by move=> i; rewrite /tensor_index /tensor_unindex enum_valK cast_ordKV.
Qed. | Lemma | tensor_unindexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ordKV",
"enum_valK",
"tensor_index",
"tensor_unindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_index_bij : bijective tensor_index. | Proof.
by exists tensor_unindex; [exact: tensor_indexK | exact: tensor_unindexK].
Qed. | Lemma | tensor_index_bij | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor_index",
"tensor_indexK",
"tensor_unindex",
"tensor_unindexK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_dffun_index : 'I_(\prod_(i < k) (u_ i)%:num) ->
{dffun forall i : 'I_k, 'I_(u_ i)%:num} | :=
@dffun_of_fprod _ _ \o tensor_unindex. | Definition | tensor_dffun_index | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"dffun_of_fprod",
"num",
"tensor_unindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_dffun_unindex : {dffun forall i : 'I_k, 'I_(u_ i)%:num} ->
'I_(\prod_(i < k) (u_ i)%:num) | :=
tensor_index \o @fprod_of_dffun _ _. | Definition | tensor_dffun_unindex | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"fprod_of_dffun",
"num",
"tensor_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_dffun_indexK : cancel tensor_dffun_index tensor_dffun_unindex. | Proof.
by move=> i; rewrite /tensor_dffun_index /tensor_dffun_unindex/=
dffun_of_fprodK tensor_unindexK.
Qed. | Lemma | tensor_dffun_indexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"dffun_of_fprodK",
"tensor_dffun_index",
"tensor_dffun_unindex",
"tensor_unindexK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_dffun_unindexK : cancel tensor_dffun_unindex tensor_dffun_index. | Proof.
by move=> f; rewrite /tensor_dffun_index /tensor_dffun_unindex/=
tensor_indexK fprod_of_dffunK.
Qed. | Lemma | tensor_dffun_unindexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"fprod_of_dffunK",
"tensor_dffun_index",
"tensor_dffun_unindex",
"tensor_indexK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_dffun_index_bij : bijective tensor_dffun_index. | Proof.
by exists tensor_dffun_unindex;
[exact: tensor_dffun_indexK | exact: tensor_dffun_unindexK].
Qed. | Lemma | tensor_dffun_index_bij | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor_dffun_index",
"tensor_dffun_indexK",
"tensor_dffun_unindex",
"tensor_dffun_unindexK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
u_cons | := [tuple of u :: u_]. | Notation | u_cons | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensormx_cast : #|{:'I_u%:num * 'I_(\prod_(i < k) (u_ i)%:num)}|
= \prod_(i < k.+1) (u_cons i)%:num. | Proof.
rewrite card_prod !card_ord big_ord_recl ffunE [tnth _ _]tnth0.
by congr (_ * _); apply: eq_bigr => i _; rewrite !ffunE tnthS.
Qed. | Lemma | tensormx_cast | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"big_ord_recl",
"card_ord",
"card_prod",
"eq_bigr",
"ffunE",
"num",
"tnth",
"tnth0",
"tnthS",
"u_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensormx_index (ij : 'I_u%:num * 'I_(\prod_(i < k) (u_ i)%:num))
: 'I_(\prod_(i < k.+1) (u_cons i)%:num) | :=
cast_ord tensormx_cast (enum_rank ij). | Definition | tensormx_index | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"enum_rank",
"num",
"tensormx_cast",
"u_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensormx_unindex (i : 'I_(\prod_(i < k.+1) (u_cons i)%:num))
: 'I_u%:num * 'I_(\prod_(i < k) (u_ i)%:num) | :=
enum_val (cast_ord (esym tensormx_cast) i). | Definition | tensormx_unindex | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"enum_val",
"num",
"tensormx_cast",
"u_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensormx_indexK : cancel tensormx_index tensormx_unindex. | Proof. by move=> ij; rewrite /tensormx_unindex cast_ordK enum_rankK. Qed. | Lemma | tensormx_indexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ordK",
"enum_rankK",
"tensormx_index",
"tensormx_unindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensormx_unindexK : cancel tensormx_unindex tensormx_index. | Proof. by move=> i; rewrite /tensormx_index enum_valK cast_ordKV. Qed. | Lemma | tensormx_unindexK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ordKV",
"enum_valK",
"tensormx_index",
"tensormx_unindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
d_cons | := [tuple of d :: d_]. | Notation | d_cons | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nindex (t : 'T[R]_(u_cons, d_)) (i : 'I_u%:num) : 'T[R]_(u_, d_) | :=
Tensor (\matrix_(i', j) (\val t) (tensormx_index (i, i')) j). | Definition | nindex | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensormx_index",
"u_cons",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oindex (t : 'T[R]_(u_, d_cons)) (j : 'I_d%:num) : 'T[R]_(u_, d_) | :=
Tensor (\matrix_(i, j') (\val t) i (tensormx_index (j, j'))). | Definition | oindex | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"d_cons",
"num",
"tensormx_index",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nstack (f : 'I_u%:num -> 'T[R]_(u_, d_)) : 'T[R]_(u_cons, d_) | :=
Tensor (
\matrix_(i, j) \val (f (tensormx_unindex i).1) (tensormx_unindex i).2 j). | Definition | nstack | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensormx_unindex",
"u_cons",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ostack (f : 'I_d%:num -> 'T[R]_(u_, d_)) : 'T[R]_(u_, d_cons) | :=
Tensor (
\matrix_(i, j) \val (f (tensormx_unindex j).1) i (tensormx_unindex j).2). | Definition | ostack | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"d_cons",
"num",
"tensormx_unindex",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"t ^^ i" | := (nindex t i) : tensor_scope. | Notation | t ^^ i | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"nindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"t `_ i" | := (oindex t i) : tensor_scope. | Notation | t `_ i | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"oindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"t ^^= i" | := ((t^^i).[::]) : tensor_scope. | Notation | t ^^= i | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"t `_= i" | := ((t`_i).[::]) : tensor_scope. | Notation | t `_= i | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor ^^ ( i < u ) E" | := (nstack (fun i : 'I_u => E))
(only parsing) : tensor_scope. | Notation | \tensor ^^ ( i < u ) E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"nstack"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor `_ ( i < d ) E" | := (ostack (fun i : 'I_d => E))
(only parsing) : tensor_scope. | Notation | \tensor `_ ( i < d ) E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"ostack"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor ^^ i E" | := (\tensor^^(i < _) E) : tensor_scope. | Notation | \tensor ^^ i E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor `_ i E" | := (\tensor`_(i < _) E) : tensor_scope. | Notation | \tensor `_ i E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor ^^ ( i < u ) => E" | := (\tensor^^(i < u) const_t E)
(only parsing) : tensor_scope. | Notation | \tensor ^^ ( i < u ) => E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor `_ ( i < d ) => E" | := (\tensor`_(i < d) const_t E)
(only parsing) : tensor_scope. | Notation | \tensor `_ ( i < d ) => E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor ^^ i => E" | := (\tensor^^i const_t E) : tensor_scope. | Notation | \tensor ^^ i => E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tensor `_ i => E" | := (\tensor`_i const_t E) : tensor_scope. | Notation | \tensor `_ i => E | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntensorP (u : {posnum nat}) (k l : nat)
(u_ : k.-tuple {posnum nat}) (d_ : l.-tuple {posnum nat})
(t v : 'T[R]_([tuple of u :: u_], d_))
: t = v <-> forall i, t^^i = v^^i. | Proof.
split=> [->//|eq_i]; apply/val_inj/matrixP=> i j.
move: (eq_i (tensormx_unindex i).1)=> [/matrixP] /(_ (tensormx_unindex i).2 j).
by rewrite !mxE -surjective_pairing tensormx_unindexK.
Qed. | Lemma | ntensorP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mxE",
"nat",
"posnum",
"split",
"tensormx_unindex",
"tensormx_unindexK",
"tuple",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
otensorP (d : {posnum nat}) (k l : nat)
(u_ : k.-tuple {posnum nat}) (d_ : l.-tuple {posnum nat})
(t v : 'T[R]_(u_, [tuple of d :: d_]))
: t = v <-> forall i, t`_i = v`_i. | Proof.
split=> [->//|eq_i]; apply/val_inj/matrixP=> i j.
move: (eq_i (tensormx_unindex j).1)=> [/matrixP] /(_ i (tensormx_unindex j).2).
by rewrite !mxE -surjective_pairing tensormx_unindexK.
Qed. | Lemma | otensorP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mxE",
"nat",
"posnum",
"split",
"tensormx_unindex",
"tensormx_unindexK",
"tuple",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntensor_eqP (u : {posnum nat}) (t v : 'nT[R]_([tuple u]))
: t = v <-> forall i, t^^=i = v^^=i. | Proof.
split=> [->//|eq_i]; apply/ntensorP=> i.
by move: (eq_i i)=> /tensor_nil_eqP.
Qed. | Lemma | ntensor_eqP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"nat",
"ntensorP",
"posnum",
"split",
"tensor_nil_eqP",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
otensor_eqP (d : {posnum nat}) (t v : 'oT[R]_([tuple d]))
: t = v <-> forall i, t`_=i = v`_=i. | Proof.
split=> [->//|eq_i]; apply/otensorP=> i.
by move: (eq_i i)=> /tensor_nil_eqP.
Qed. | Lemma | otensor_eqP | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"nat",
"otensorP",
"posnum",
"split",
"tensor_nil_eqP",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nstackE {u : {posnum nat}} {k l}
{u_ : k.-tuple {posnum nat}} {d_ : l.-tuple {posnum nat}}
(f : 'I_u%:num%R -> 'T[R]_(u_, d_)) i
: (nstack f)^^i = f i. | Proof. by apply/val_inj/matrixP => x y; rewrite !mxE tensormx_indexK. Qed. | Lemma | nstackE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mxE",
"nat",
"nstack",
"num",
"posnum",
"tensormx_indexK",
"tuple",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ostackE {d : {posnum nat}} {k l}
{u_ : k.-tuple {posnum nat}} {d_ : l.-tuple {posnum nat}}
(f : 'I_d%:num%R -> 'T[R]_(u_, d_)) i
: (ostack f)`_i = f i. | Proof. by apply/val_inj/matrixP => x y; rewrite !mxE tensormx_indexK. Qed. | Lemma | ostackE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mxE",
"nat",
"num",
"ostack",
"posnum",
"tensormx_indexK",
"tuple",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nstack_eqE {u : {posnum nat}} (f : 'I_u%:num%R -> R) i
: (\tensor^^i0 => f i0)^^=i = f i. | Proof. by rewrite nstackE const_tK. Qed. | Lemma | nstack_eqE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_tK",
"i0",
"nat",
"nstackE",
"num",
"posnum",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ostack_eqE {d : {posnum nat}} (f : 'I_d%:num%R -> R) i
: (\tensor`_i0 => f i0)`_=i = f i. | Proof. by rewrite ostackE const_tK. Qed. | Lemma | ostack_eqE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_tK",
"i0",
"nat",
"num",
"ostackE",
"posnum",
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntensor_of_tuple (t : x%:num%R.-tuple R)
: 'nT[R]_([tuple x]) | := \tensor^^i => (tnth t i). | Definition | ntensor_of_tuple | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensor",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
otensor_of_tuple (t : x%:num%R.-tuple R)
: 'oT[R]_([tuple x]) | := \tensor`_i => (tnth t i). | Definition | otensor_of_tuple | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensor",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_ntensor (t : 'nT[R]_([tuple x])) | :=
[tuple t^^=i | i < x%:num%R]. | Definition | tuple_of_ntensor | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_otensor (t : 'oT[R]_([tuple x])) | :=
[tuple t`_=i | i < x%:num%R]. | Definition | tuple_of_otensor | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntensor_of_tupleE t i : (ntensor_of_tuple t)^^=i = tnth t i. | Proof. exact: nstack_eqE. Qed. | Lemma | ntensor_of_tupleE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"nstack_eqE",
"ntensor_of_tuple",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
otensor_of_tupleE t i : (otensor_of_tuple t)`_=i = tnth t i. | Proof. exact: ostack_eqE. Qed. | Lemma | otensor_of_tupleE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"ostack_eqE",
"otensor_of_tuple",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nstack_tuple (t : x%:num%R.-tuple 'T[R]_(u_, d_)) | :=
\tensor^^i tnth t i. | Definition | nstack_tuple | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensor",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ostack_tuple (t : x%:num%R.-tuple 'T[R]_(u_, d_)) | :=
\tensor`_i tnth t i. | Definition | ostack_tuple | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num",
"tensor",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nstack_tupleE t i : (nstack_tuple t)^^i = tnth t i. | Proof. exact: nstackE. Qed. | Lemma | nstack_tupleE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"nstackE",
"nstack_tuple",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ostack_tupleE t i : (ostack_tuple t)`_i = tnth t i. | Proof. exact: ostackE. Qed. | Lemma | ostack_tupleE | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"ostackE",
"ostack_tuple",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntensor_of_tupleK : cancel ntensor_of_tuple tuple_of_ntensor. | Proof.
by move=> t; apply/eq_from_tnth=> i; rewrite tnth_mktuple nstack_eqE.
Qed. | Lemma | ntensor_of_tupleK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"eq_from_tnth",
"nstack_eqE",
"ntensor_of_tuple",
"tnth_mktuple",
"tuple_of_ntensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_ntensorK : cancel tuple_of_ntensor ntensor_of_tuple. | Proof.
by move=> t; apply/ntensor_eqP=> i; rewrite nstackE const_tK tnth_mktuple.
Qed. | Lemma | tuple_of_ntensorK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_tK",
"nstackE",
"ntensor_eqP",
"ntensor_of_tuple",
"tnth_mktuple",
"tuple_of_ntensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
otensor_of_tupleK : cancel otensor_of_tuple tuple_of_otensor. | Proof.
by move=> t; apply/eq_from_tnth=> i; rewrite tnth_mktuple ostack_eqE.
Qed. | Lemma | otensor_of_tupleK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"eq_from_tnth",
"ostack_eqE",
"otensor_of_tuple",
"tnth_mktuple",
"tuple_of_otensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_otensorK : cancel tuple_of_otensor otensor_of_tuple. | Proof.
by move=> t; apply/otensor_eqP=> i; rewrite ostackE const_tK tnth_mktuple.
Qed. | Lemma | tuple_of_otensorK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_tK",
"ostackE",
"otensor_eqP",
"otensor_of_tuple",
"tnth_mktuple",
"tuple_of_otensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tensor' ^^ t ; .. ; tn ]" | :=
(nstack_tuple [tuple of t :: .. [:: tn] ..]) : tensor_scope. | Notation | [ 'tensor' ^^ t ; .. ; tn ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"nstack_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tensor' ^^= x ; .. ; xn ]" | :=
(ntensor_of_tuple [tuple of x :: .. [:: xn] ..]) : tensor_scope. | Notation | [ 'tensor' ^^= x ; .. ; xn ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"ntensor_of_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tensor' `_ t ; .. ; tn ]" | :=
(ostack_tuple [tuple of t :: .. [:: tn] ..]) : tensor_scope. | Notation | [ 'tensor' `_ t ; .. ; tn ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"ostack_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tensor' `_= x ; .. ; xn ]" | :=
(otensor_of_tuple [tuple of x :: .. [:: xn] ..]) : tensor_scope. | Notation | [ 'tensor' `_= x ; .. ; xn ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"otensor_of_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_of_matrix (M : 'M_(_, _)) :
'T[R]_([tuple n], [tuple m]) | :=
\tensor^^i \tensor`_j => M i j. | Definition | tensor_of_matrix | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_of_tensor t : 'M[R]_(n%:num%R, m%:num%R) | :=
\matrix_(i, j) (t^^i)`_=j. | Definition | matrix_of_tensor | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_of_matrixK : cancel tensor_of_matrix matrix_of_tensor. | Proof. by move=> M; apply/matrixP=> i j; rewrite mxE nstackE ostack_eqE. Qed. | Lemma | tensor_of_matrixK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"matrix_of_tensor",
"mxE",
"nstackE",
"ostack_eqE",
"tensor_of_matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_of_tensorK : cancel matrix_of_tensor tensor_of_matrix. | Proof.
move=> T; apply/ntensorP=> i; apply/otensor_eqP=> j.
by rewrite nstackE ostack_eqE mxE.
Qed. | Lemma | matrix_of_tensorK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrix_of_tensor",
"mxE",
"nstackE",
"ntensorP",
"ostack_eqE",
"otensor_eqP",
"tensor_of_matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_fcat {n m : nat} (f : {posnum nat} ^ n) (g : {posnum nat} ^ m) :
(\prod_(i < n) (f i)%:num%R) * (\prod_(i < m) (g i)%:num%R)
= \prod_(i < n + m) ((f +++ g) i)%:num%R. | Proof.
rewrite big_split_ord; congr (_ * _); apply: eq_bigr => i _.
by rewrite fcat_lshift.
by rewrite fcat_rshift.
Qed. | Lemma | prod_fcat | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"big_split_ord",
"eq_bigr",
"fcat_lshift",
"fcat_rshift",
"nat",
"num",
"posnum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_card (m n : nat) : #|{:'I_m * 'I_n}| = (m * n)%N. | Proof. by rewrite card_prod !card_ord. Qed. | Lemma | prod_card | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"card_ord",
"card_prod",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_split (m n : nat) (i : 'I_(m * n)) : 'I_m * 'I_n | :=
enum_val (cast_ord (esym (prod_card m n)) i). | Let | prod_split | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"enum_val",
"nat",
"prod_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_unsplit (m n : nat) (ij : 'I_m * 'I_n) : 'I_(m * n) | :=
cast_ord (prod_card m n) (enum_rank ij). | Definition | prod_unsplit | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"enum_rank",
"nat",
"prod_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mults (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_))
: 'T[R]_((u1_ +++ u2_), (d1_ +++ d2_)) | :=
Tensor (\matrix_(i, j)
let ii := prod_split (cast_ord (esym (prod_fcat _ _)) i) in
let jj := prod_split (cast_ord (esym (prod_fcat _ _)) j) in
(\val t ii.1 jj.1) * (\val u ii.2 jj.2))%R. | Definition | mults | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"cast_ord",
"prod_fcat",
"prod_split",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
multsDl (t u : 'T[R]_(u1_, d1_)) (v : 'T[R]_(u2_, d2_)) :
mults (t + u) v = (mults t v + mults u v). | Proof.
by apply/val_inj/matrixP => i j; rewrite /tensor_val/= !mxE/= mulrDl.
Qed. | Lemma | multsDl | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mulrDl",
"mults",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
multsDr (t : 'T[R]_(u1_, d1_)) (u v : 'T[R]_(u2_, d2_)) :
mults t (u + v) = mults t u + mults t v. | Proof.
by apply/val_inj/matrixP => i j; rewrite /tensor_val/= !mxE /= mulrDr.
Qed. | Lemma | multsDr | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mulrDr",
"mults",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mults0l (t : 'T[R]_(u2_, d2_)) :
mults (0 : 'T[R]_(u1_, d1_)) t = 0. | Proof.
by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mul0r.
Qed. | Lemma | mults0l | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mul0r",
"mults",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mults0r (t : 'T[R]_(u1_, d1_)) :
mults t (0 : 'T[R]_(u2_, d2_)) = 0. | Proof.
by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE mulr0.
Qed. | Lemma | mults0r | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"matrixP",
"mulr0",
"mults",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mults_const (a b : R) :
mults (@const_t R _ _ u1_ d1_ a) (@const_t R _ _ u2_ d2_ b) =
const_t (a * b)%R. | Proof.
by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE.
Qed. | Lemma | mults_const | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"const_t",
"matrixP",
"mults",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"*t%R" | := mults : ring_scope. | Notation | *t%R | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"mults"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x *t y" | := (mults x y) : tensor_scope. | Notation | x *t y | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"mults"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mults_linear_l (u : 'T[R]_(u2_, d2_)) :
GRing.linear_for *:%R (fun t : 'T[R]_(u1_, d1_) => t *t u). | Proof.
move=> a x y; apply/val_inj/matrixP=> i j; rewrite /tensor_val/= !mxE/=.
rewrite mulrDl; congr (_ + _);
by rewrite mulrA.
Qed. | Let | mults_linear_l | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"linear_for",
"matrixP",
"mulrA",
"mulrDl",
"mxE",
"tensor_val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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