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 |
|---|---|---|---|---|---|---|---|---|---|---|
R043 | := [:: F4; F2; F5; F0; F3; F1]. | Definition | R043 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R043f (sc : cube) : cube | := tnth [tuple of R043] sc. | Definition | R043f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R043",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R034 | := [:: F3; F5; F1; F4; F0; F2]. | Definition | R034 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R034f (sc : cube) : cube | := tnth [tuple of R034] sc. | Definition | R034f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R034",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S1 | := [:: F5; F2; F1; F4; F3; F0]. | Definition | S1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | last symmetries | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
S1f (sc : cube) : cube | := tnth [tuple of S1] sc. | Definition | S1f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S1",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S2 | := [:: F5; F3; F4; F1; F2; F0]. | Definition | S2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S2f (sc : cube) : cube | := tnth [tuple of S2] sc. | Definition | S2f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S2",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S3 | := [:: F1; F0; F3; F2; F5; F4]. | Definition | S3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S3f (sc : cube) : cube | := tnth [tuple of S3] sc. | Definition | S3f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S3",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S4 | := [:: F4; F5; F3; F2; F0; F1]. | Definition | S4 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S4f (sc : cube) : cube | := tnth [tuple of S4] sc. | Definition | S4f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S4",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S5 | := [:: F2; F4; F0; F5; F1; F3]. | Definition | S5 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S5f (sc : cube) : cube | := tnth [tuple of S5] sc. | Definition | S5f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S5",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S6 | := [::F3; F4; F5; F0; F1; F2]. | Definition | S6 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S6f (sc : cube) : cube | := tnth [tuple of S6] sc. | Definition | S6f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S6",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S1_inv : involutive S1f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S1_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S1f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S2_inv : involutive S2f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S2_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S2f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S3_inv : involutive S3f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S3_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S3f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S4_inv : involutive S4f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S4_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S4f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S5_inv : involutive S5f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S5_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S5f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S6_inv : involutive S6f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S6_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S6f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S05_inj : injective S05f. | Proof. by apply: can_inj S05f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S05_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S05f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S14_inj : injective S14f. | Proof. by apply: can_inj S14f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S14_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S14f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S23_inv : involutive S23f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S23_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S23f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R05_inj : injective R05f. | Proof. by apply: can_inj R50f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R05_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R05f",
"R50f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R14_inj : injective R14f. | Proof. by apply: can_inj R41f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R14_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R14f",
"R41f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R23_inj : injective R23f. | Proof. by apply: can_inj R32f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R23_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R23f",
"R32f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R50_inj : injective R50f. | Proof. by apply: can_inj R05f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R50_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R05f",
"R50f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R41_inj : injective R41f. | Proof. by apply: can_inj R14f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R41_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R14f",
"R41f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R32_inj : injective R32f. | Proof. by apply: can_inj R23f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R32_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R23f",
"R32f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R024_inj : injective R024f. | Proof. by apply: can_inj R042f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R024_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R024f",
"R042f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R042_inj : injective R042f. | Proof. by apply: can_inj R024f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R042_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R024f",
"R042f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R012_inj : injective R012f. | Proof. by apply: can_inj R021f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R012_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R012f",
"R021f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R021_inj : injective R021f. | Proof. by apply: can_inj R012f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R021_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R012f",
"R021f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R031_inj : injective R031f. | Proof. by apply: can_inj R013f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R031_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R013f",
"R031f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R013_inj : injective R013f. | Proof. by apply: can_inj R031f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R013_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R013f",
"R031f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R043_inj : injective R043f. | Proof. by apply: can_inj R034f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R043_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R034f",
"R043f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R034_inj : injective R034f. | Proof. by apply: can_inj R043f _ => z; apply/eqP; case: z; do 6?case. Qed. | Lemma | R034_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R034f",
"R043f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
id3 | := 1 : {perm cube}. | Definition | id3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"cube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s05 | := (perm S05_inj). | Definition | s05 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S05_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s14 : {perm cube}. | Proof.
apply: (@perm _ S14f); apply: can_inj S14f _ => z.
by apply/eqP; case: z; do 6?case.
Defined. | Definition | s14 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S14f",
"apply",
"cube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s23 | := (perm (inv_inj S23_inv)). | Definition | s23 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S23_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r05 | := (perm R05_inj). | Definition | r05 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R05_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r14 | := (perm R14_inj). | Definition | r14 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R14_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r23 | := (perm R23_inj). | Definition | r23 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R23_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r50 | := (perm R50_inj). | Definition | r50 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R50_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r41 | := (perm R41_inj). | Definition | r41 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R41_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r32 | := (perm R32_inj). | Definition | r32 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R32_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r024 | := (perm R024_inj). | Definition | r024 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R024_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r042 | := (perm R042_inj). | Definition | r042 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R042_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r012 | := (perm R012_inj). | Definition | r012 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R012_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r021 | := (perm R021_inj). | Definition | r021 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R021_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r031 | := (perm R031_inj). | Definition | r031 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R031_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r013 | := (perm R013_inj). | Definition | r013 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R013_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r043 | := (perm R043_inj). | Definition | r043 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R043_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r034 | := (perm R034_inj). | Definition | r034 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R034_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s1 | := (perm (inv_inj S1_inv)). | Definition | s1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S1_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s2 | := (perm (inv_inj S2_inv)). | Definition | s2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S2_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s3 | := (perm (inv_inj S3_inv)). | Definition | s3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S3_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s4 | := (perm (inv_inj S4_inv)). | Definition | s4 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S4_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s5 | := (perm (inv_inj S5_inv)). | Definition | s5 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S5_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s6 | := (perm (inv_inj S6_inv)). | Definition | s6 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S6_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dir_iso3 | := [set p |
[|| id3 == p, s05 == p, s14 == p, s23 == p, r05 == p, r14 == p, r23 == p,
r50 == p, r41 == p, r32 == p, r024 == p, r042 == p, r012 == p, r021 == p,
r031 == p, r013 == p, r043 == p, r034 == p,
s1 == p, s2 == p, s3 == p, s4 == p, s5 == p | s6 == p]]. | Definition | dir_iso3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"id3",
"r012",
"r013",
"r021",
"r024",
"r031",
"r034",
"r042",
"r043",
"r05",
"r14",
"r23",
"r32",
"r41",
"r50",
"s05",
"s1",
"s14",
"s2",
"s23",
"s3",
"s4",
"s5",
"s6"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dir_iso3l | := [:: id3; s05; s14; s23; r05; r14; r23; r50; r41;
r32; r024; r042; r012; r021; r031; r013; r043; r034;
s1; s2; s3; s4; s5; s6]. | Definition | dir_iso3l | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"id3",
"r012",
"r013",
"r021",
"r024",
"r031",
"r034",
"r042",
"r043",
"r05",
"r14",
"r23",
"r32",
"r41",
"r50",
"s05",
"s1",
"s14",
"s2",
"s23",
"s3",
"s4",
"s5",
"s6"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S0 | := [:: F5; F4; F3; F2; F1; F0]. | Definition | S0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S0f (sc : cube) : cube | := tnth [tuple of S0] sc. | Definition | S0f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S0",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S0_inv : involutive S0f. | Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | S0_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S0f",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s0 | := (perm (inv_inj S0_inv)). | Definition | s0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S0_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_iso3 (p : {perm cube}) | := forall fi, p (s0 fi) = s0 (p fi). | Definition | is_iso3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"cube",
"s0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dir_iso_iso3 : forall p, p \in dir_iso3 -> is_iso3 p. | Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> <- a; rewrite !permE; case: a; do 6?case.
Qed. | Lemma | dir_iso_iso3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3",
"inE",
"is_iso3",
"permE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso3_ndir : forall p, p \in dir_iso3 -> is_iso3 (s0 * p). | Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> <- a; rewrite !(permM, permE); case: a; do 6?case.
Qed. | Lemma | iso3_ndir | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3",
"inE",
"is_iso3",
"permE",
"permM",
"s0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sop (p : {perm cube}) : seq cube | := fgraph (val p). | Definition | sop | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"cube",
"fgraph",
"seq",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sop_inj : injective sop. | Proof. by move=> p1 p2 /val_inj/(can_inj fgraphK)/val_inj. Qed. | Lemma | sop_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"fgraphK",
"sop",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_tuple (t1 t2 : seq cube) | :=
map (fun n : 'I_6 => nth F0 t2 n) t1. | Definition | prod_tuple | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"cube",
"map",
"nth",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sop_spec x (n0 : 'I_6): nth F0 (sop x) n0 = x n0. | Proof. by rewrite nth_fgraph_ord pvalE. Qed. | Lemma | sop_spec | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"nth",
"nth_fgraph_ord",
"pvalE",
"sop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_t_correct : forall (x y : {perm cube}) (i : cube),
(x * y) i = nth F0 (prod_tuple (sop x) (sop y)) i. | Proof.
move=> x y i; rewrite permM -!sop_spec [RHS](nth_map F0) // size_tuple /=.
by rewrite card_ord ltn_ord.
Qed. | Lemma | prod_t_correct | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"card_ord",
"cube",
"ltn_ord",
"nth",
"nth_map",
"permM",
"prod_tuple",
"size_tuple",
"sop",
"sop_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sop_morph : {morph sop : x y / x * y >-> prod_tuple x y}. | Proof.
move=> x y; apply: (@eq_from_nth _ F0) => [|/= i].
by rewrite size_map !size_tuple.
rewrite size_tuple card_ord => lti6.
by rewrite -[i]/(val (Ordinal lti6)) sop_spec -prod_t_correct.
Qed. | Lemma | sop_morph | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"apply",
"card_ord",
"eq_from_nth",
"prod_t_correct",
"prod_tuple",
"size_map",
"size_tuple",
"sop",
"sop_spec",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ecubes : seq cube | := [:: F0; F1; F2; F3; F4; F5]. | Definition | ecubes | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5",
"cube",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ecubes_def : ecubes = enum (@predT cube). | Proof. by apply: (inj_map val_inj); rewrite val_enum_ord. Qed. | Lemma | ecubes_def | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"cube",
"ecubes",
"enum",
"inj_map",
"val_enum_ord",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_iso_L | := [::
[:: F0; F1; F2; F3; F4; F5];
S05; S14; S23; R05; R14; R23; R50; R41; R32;
R024; R042; R012; R021; R031; R013; R043; R034;
S1; S2; S3; S4; S5; S6]. | Definition | seq_iso_L | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5",
"R012",
"R013",
"R021",
"R024",
"R031",
"R034",
"R042",
"R043",
"R05",
"R14",
"R23",
"R32",
"R41",
"R50",
"S05",
"S1",
"S14",
"S2",
"S23",
"S3",
"S4",
"S5",
"S6"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seqs1 : forall f injf, sop (@perm _ f injf) = map f ecubes. | Proof.
move=> f ?; rewrite ecubes_def /sop /= -codom_ffun pvalE.
by apply: eq_codom; apply: permE.
Qed. | Lemma | seqs1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"codom_ffun",
"ecubes",
"ecubes_def",
"eq_codom",
"injf",
"map",
"permE",
"pvalE",
"sop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Lcorrect : seq_iso_L == map sop [:: id3; s05; s14; s23; r05; r14; r23;
r50; r41; r32; r024; r042; r012; r021; r031; r013; r043; r034;
s1; s2; s3; s4; s5; s6]. | Proof. by rewrite /= !seqs1. Qed. | Lemma | Lcorrect | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"id3",
"map",
"r012",
"r013",
"r021",
"r024",
"r031",
"r034",
"r042",
"r043",
"r05",
"r14",
"r23",
"r32",
"r41",
"r50",
"s05",
"s1",
"s14",
"s2",
"s23",
"s3",
"s4",
"s5",
"s6",
"seq_iso_L",
"seqs1",
"sop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso0_1 : dir_iso3 =i dir_iso3l. | Proof. by move=> p; rewrite /= !inE /= -!(eq_sym p). Qed. | Lemma | iso0_1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3",
"dir_iso3l",
"eq_sym",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
L_iso : forall p, (p \in dir_iso3) = (sop p \in seq_iso_L). | Proof.
by move=> p; rewrite (eqP Lcorrect) mem_map ?iso0_1 //; apply: sop_inj.
Qed. | Lemma | L_iso | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Lcorrect",
"apply",
"dir_iso3",
"iso0_1",
"mem_map",
"seq_iso_L",
"sop",
"sop_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stable : forall x y,
x \in dir_iso3 -> y \in dir_iso3 -> x * y \in dir_iso3. | Proof.
move=> x y; rewrite !L_iso sop_morph => Hx Hy.
by move/sop: y Hy; apply/allP; move/sop: x Hx; apply/allP; vm_compute.
Qed. | Lemma | stable | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"L_iso",
"allP",
"apply",
"dir_iso3",
"sop",
"sop_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso_eq_F0_F1 : forall r s : {perm cube}, r \in dir_iso3 ->
s \in dir_iso3 -> r F0 = s F0 -> r F1 = s F1 -> r = s. | Proof.
move=> r s; rewrite !L_iso => hr hs hrs0 hrs1; apply: sop_inj; apply/eqP.
move/eqP: hrs0; apply/implyP; move/eqP: hrs1; apply/implyP; rewrite -!sop_spec.
by move/sop: r hr; apply/allP; move/sop: s hs; apply/allP; vm_compute.
Qed. | Lemma | iso_eq_F0_F1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"L_iso",
"allP",
"apply",
"cube",
"dir_iso3",
"sop",
"sop_inj",
"sop_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ndir_s0p : forall p, p \in dir_iso3 -> s0 * p \notin dir_iso3. | Proof.
move=> p; rewrite !L_iso sop_morph seqs1.
by move/sop: p; apply/allP; vm_compute.
Qed. | Lemma | ndir_s0p | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"L_iso",
"allP",
"apply",
"dir_iso3",
"s0",
"seqs1",
"sop",
"sop_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
indir_iso3l | := map (mul s0) dir_iso3l. | Definition | indir_iso3l | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3l",
"map",
"mul",
"s0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso3l | := dir_iso3l ++ indir_iso3l. | Definition | iso3l | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3l",
"indir_iso3l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_iso3_L | := map sop iso3l. | Definition | seq_iso3_L | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"iso3l",
"map",
"sop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqperm : forall p1 p2 : {perm cube},
(p1 == p2) = all (fun s => p1 s == p2 s) ecubes. | Proof.
move=> p1 p2; apply/eqP/allP=> [-> // | Ep12]; apply/permP=> x.
by apply/eqP; rewrite Ep12 // ecubes_def mem_enum.
Qed. | Lemma | eqperm | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"all",
"allP",
"apply",
"cube",
"ecubes",
"ecubes_def",
"mem_enum",
"permP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso_eq_F0_F1_F2 : forall r s : {perm cube}, is_iso3 r ->
is_iso3 s -> r F0 = s F0 -> r F1 = s F1 -> r F2 = s F2 -> r = s. | Proof.
move=> r s hr hs hrs0 hrs1 hrs2.
have:= hrs0; have:= hrs1; have:= hrs2.
have e23: F2 = s0 F3 by apply/eqP; rewrite permE /S0f (tnth_nth F0).
have e14: F1 = s0 F4 by apply/eqP; rewrite permE /S0f (tnth_nth F0).
have e05: F0 = s0 F5 by apply/eqP; rewrite permE /S0f (tnth_nth F0).
rewrite e23 e14 e05; rewrite !hr !... | Lemma | iso_eq_F0_F1_F2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5",
"S0f",
"apply",
"cube",
"eqperm",
"eqxx",
"is_iso3",
"permE",
"perm_inj",
"s0",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso_tac | :=
let a := fresh "a" in apply/permP => a;
apply/eqP; rewrite !permM !permE; case: a; do 6?case. | Ltac | iso_tac | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"permE",
"permM",
"permP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_tac | :=
apply: esym (etrans _ (mul1g _)); apply: canRL (mulgK _) _; iso_tac. | Ltac | inv_tac | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"iso_tac",
"mul1g",
"mulgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dir_s0p : forall p, (s0 * p) \in dir_iso3 -> p \notin dir_iso3. | Proof.
move=> p Hs0p; move: (ndir_s0p Hs0p); rewrite mulgA.
have e: (s0^-1=s0) by inv_tac.
by rewrite -{1}e mulVg mul1g.
Qed. | Lemma | dir_s0p | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"dir_iso3",
"inv_tac",
"mul1g",
"mulVg",
"mulgA",
"ndir_s0p",
"s0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_iso3b p | := (p * s0 == s0 * p). | Definition | is_iso3b | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"s0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso3 | := [set p | is_iso3b p]. | Definition | iso3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"is_iso3b"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_iso3P : forall p, reflect (is_iso3 p) (p \in iso3). | Proof.
move=> p; apply: (iffP idP); rewrite inE /iso3 /is_iso3b /is_iso3 => e.
by move=> fi; rewrite -!permM (eqP e).
by apply/eqP; apply/permP=> z; rewrite !permM (e z).
Qed. | Lemma | is_iso3P | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"inE",
"is_iso3",
"is_iso3b",
"iso3",
"permM",
"permP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_iso3 : group_set iso3. | Proof.
apply/group_setP; split.
by apply/is_iso3P => fi; rewrite -!permM mulg1 mul1g.
move=> x1 y; rewrite /iso3 !inE /= /is_iso3.
rewrite /is_iso3b.
rewrite -mulgA.
move/eqP => hx1; move/eqP => hy.
rewrite hy !mulgA. by rewrite -hx1.
Qed. | Lemma | group_set_iso3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"group_set",
"group_setP",
"inE",
"is_iso3",
"is_iso3P",
"is_iso3b",
"iso3",
"mul1g",
"mulg1",
"mulgA",
"permM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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