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 |
|---|---|---|---|---|---|---|---|---|---|---|
ffact_fact n m : m <= n -> n ^_ m * (n - m)`! = n`!. | Proof.
by elim: n m => [|n IHn] [|m] //= le_m_n; rewrite ?mul1n // -mulnA IHn.
Qed. | Lemma | ffact_fact | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"mul1n",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffact_factd n m : m <= n -> n ^_ m = n`! %/ (n - m)`!. | Proof. by move/ffact_fact <-; rewrite mulnK ?fact_gt0. Qed. | Lemma | ffact_factd | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"fact_gt0",
"ffact_fact",
"mulnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binomial n m | :=
match n, m with
| n'.+1, m'.+1 => binomial n' m + binomial n' m'
| _, 0 => 1
| 0, _.+1 => 0
end. | Fixpoint | binomial | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"n'"
] | Binomial coefficients | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"''C' ( n , m )" | := (binomial n m) : nat_scope. | Notation | ''C' ( n , m ) | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"binomial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binE n m : binomial n m =
match n, m with
| n'.+1, m'.+1 => binomial n' m + binomial n' m'
| _, 0 => 1
| 0, _.+1 => 0
end. | Proof. by case: n. Qed. | Lemma | binE | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"binomial",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin0 n : 'C(n, 0) = 1. | Proof. by case: n. Qed. | Lemma | bin0 | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin0n m : 'C(0, m) = (m == 0). | Proof. by case: m. Qed. | Lemma | bin0n | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binS n m : 'C(n.+1, m.+1) = 'C(n, m.+1) + 'C(n, m). | Proof. by []. Qed. | Lemma | binS | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin1 n : 'C(n, 1) = n. | Proof. by elim: n => //= n IHn; rewrite binS bin0 IHn addn1. Qed. | Lemma | bin1 | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addn1",
"bin0",
"binS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_gt0 n m : (0 < 'C(n, m)) = (m <= n). | Proof.
by elim: n m => [|n IHn] [|m] //; rewrite addn_gt0 !IHn orbC ltn_neqAle andKb.
Qed. | Lemma | bin_gt0 | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addn_gt0",
"andKb",
"ltn_neqAle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_bin2l n1 n2 m : n1 <= n2 -> 'C(n1, m) <= 'C(n2, m). | Proof.
by elim: n1 n2 m => [|n1 IHn] [|n2] [|n] le_n12 //; rewrite leq_add ?IHn.
Qed. | Lemma | leq_bin2l | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"leq_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_small n m : n < m -> 'C(n, m) = 0. | Proof. by rewrite ltnNge -bin_gt0; case: posnP. Qed. | Lemma | bin_small | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_gt0",
"ltnNge",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binn n : 'C(n, n) = 1. | Proof. by elim: n => [|n IHn] //; rewrite binS bin_small. Qed. | Lemma | binn | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"binS",
"bin_small"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_bin_diag n m : n * 'C(n.-1, m) = m.+1 * 'C(n, m.+1). | Proof.
rewrite [RHS]mulnC; elim: n m => [|[|n] IHn] [|m] //=; first by rewrite bin1.
by rewrite mulSn [in _ * _]binS mulnDr addnCA !IHn -mulnS -mulnDl -binS.
Qed. | Lemma | mul_bin_diag | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addnCA",
"bin1",
"binS",
"mulSn",
"mulnC",
"mulnDl",
"mulnDr",
"mulnS"
] | Multiply to move diagonally down and right in the Pascal triangle. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
bin_fact n m : m <= n -> 'C(n, m) * (m`! * (n - m)`!) = n`!. | Proof.
elim: n m => [|n IHn] [|m] // le_m_n; first by rewrite bin0 !mul1n.
by rewrite !factS -!mulnA mulnCA mulnA -mul_bin_diag -mulnA IHn.
Qed. | Lemma | bin_fact | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin0",
"factS",
"mul1n",
"mul_bin_diag",
"mulnA",
"mulnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_factd n m : 0 < n -> 'C(n, m) = n`! %/ (m`! * (n - m)`!). | Proof.
have [/bin_fact<-|*] := leqP m n; first by rewrite mulnK ?muln_gt0 ?fact_gt0.
by rewrite divnMA bin_small ?divn_small ?fact_gt0 ?ltn_fact.
Qed. | Lemma | bin_factd | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_fact",
"bin_small",
"divnMA",
"divn_small",
"fact_gt0",
"leqP",
"ltn_fact",
"mulnK",
"muln_gt0"
] | In fact the only exception for bin_factd is n = 0 and m = 1 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
bin_ffact n m : 'C(n, m) * m`! = n ^_ m. | Proof.
have [lt_n_m | le_m_n] := ltnP n m; first by rewrite bin_small ?ffact_small.
by rewrite ffact_factd // -(bin_fact le_m_n) mulnA mulnK ?fact_gt0.
Qed. | Lemma | bin_ffact | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_fact",
"bin_small",
"fact_gt0",
"ffact_factd",
"ffact_small",
"ltnP",
"mulnA",
"mulnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_ffactd n m : 'C(n, m) = n ^_ m %/ m`!. | Proof. by rewrite -bin_ffact mulnK ?fact_gt0. Qed. | Lemma | bin_ffactd | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_ffact",
"fact_gt0",
"mulnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_sub n m : m <= n -> 'C(n, n - m) = 'C(n, m). | Proof.
by move=> le_m_n; rewrite !bin_ffactd !ffact_factd ?leq_subr // divnAC subKn.
Qed. | Lemma | bin_sub | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_ffactd",
"divnAC",
"ffact_factd",
"leq_subr",
"subKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_bin_down n m : n * 'C(n.-1, m) = (n - m) * 'C(n, m). | Proof.
case: n => //= n; have [lt_n_m | le_m_n] := ltnP n m.
by rewrite (eqnP lt_n_m) mulnC bin_small.
by rewrite -!['C(_, m)]bin_sub ?leqW ?subSn ?mul_bin_diag.
Qed. | Lemma | mul_bin_down | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin_small",
"bin_sub",
"eqnP",
"leqW",
"ltnP",
"mul_bin_diag",
"mulnC",
"subSn"
] | Multiply to move down in the Pascal triangle. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_bin_left n m : m.+1 * 'C(n, m.+1) = (n - m) * 'C(n, m). | Proof. by rewrite -mul_bin_diag mul_bin_down. Qed. | Lemma | mul_bin_left | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"mul_bin_diag",
"mul_bin_down"
] | Multiply to move left in the Pascal triangle. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
binSn n : 'C(n.+1, n) = n.+1. | Proof. by rewrite -bin_sub ?leqnSn // subSnn bin1. Qed. | Lemma | binSn | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin1",
"bin_sub",
"leqnSn",
"subSnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin2 n : 'C(n, 2) = (n * n.-1)./2. | Proof. by rewrite -[n.-1]bin1 mul_bin_diag -divn2 mulKn. Qed. | Lemma | bin2 | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin1",
"divn2",
"mulKn",
"mul_bin_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin2odd n : odd n -> 'C(n, 2) = n * n.-1./2. | Proof. by case: n => // n oddn; rewrite bin2 -!divn2 muln_divA ?dvdn2. Qed. | Lemma | bin2odd | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin2",
"divn2",
"dvdn2",
"muln_divA",
"odd",
"oddn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prime_dvd_bin k p : prime p -> 0 < k < p -> p %| 'C(p, k). | Proof.
move=> p_pr /andP[k_gt0 lt_k_p].
suffices /Gauss_dvdr<-: coprime p (p - k) by rewrite -mul_bin_down dvdn_mulr.
by rewrite prime_coprime // dvdn_subr 1?ltnW // gtnNdvd.
Qed. | Lemma | prime_dvd_bin | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"Gauss_dvdr",
"coprime",
"dvdn_mulr",
"dvdn_subr",
"gtnNdvd",
"ltnW",
"mul_bin_down",
"p_pr",
"prime",
"prime_coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin2_sum n : \sum_(0 <= i < n) i = 'C(n, 2). | Proof.
elim: n => [|n IHn]; first by rewrite big_geq.
by rewrite big_nat_recr // IHn binS bin1.
Qed. | Lemma | bin2_sum | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"big_geq",
"big_nat_recr",
"bin1",
"binS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
triangular_sum | := bin2_sum (only parsing). | Notation | triangular_sum | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"bin2_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnDn a b n :
(a + b) ^ n = \sum_(i < n.+1) 'C(n, i) * (a ^ (n - i) * b ^ i). | Proof.
elim: n => [|n IHn]; rewrite big_ord_recl muln1 ?big_ord0 //.
rewrite expnS {}IHn /= mulnDl !big_distrr /= big_ord_recl muln1 subn0.
rewrite !big_ord_recr /= !binn !subnn bin0 !subn0 !mul1n -!expnS -addnA.
congr (_ + _); rewrite addnA -big_split /=; congr (_ + _).
apply: eq_bigr => i _; rewrite mulnCA (mulnA a) ... | Theorem | expnDn | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addnA",
"apply",
"big_distrr",
"big_ord0",
"big_ord_recl",
"big_ord_recr",
"big_split",
"bin0",
"binn",
"eq_bigr",
"expnS",
"expnSr",
"mul1n",
"muln1",
"mulnA",
"mulnC",
"mulnCA",
"mulnDl",
"subn0",
"subnSK",
"subnn"
] | textbook proof of `bin2_sum`. Should be moved out of the main
library, to a dedicated "showcase" library.
Lemma textbook_triangular_sum n : \sum_(0 <= i < n) i = 'C(n, 2).
Proof.
rewrite bin2; apply: canRL half_double _.
rewrite -addnn {1}big_nat_rev -big_split big_mkord /= ?add0n.
rewrite (eq_bigr (fun _ => n.-1)); ... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Pascal | := expnDn. | Definition | Pascal | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"expnDn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Vandermonde k l i :
\sum_(j < i.+1) 'C(k, j) * 'C(l, i - j) = 'C(k + l , i). | Proof.
pose f k i := \sum_(j < i.+1) 'C(k, j) * 'C(l, i - j).
suffices{k i} fxx k i: f k.+1 i.+1 = f k i.+1 + f k i.
elim: k i => [i | k IHk [|i]]; last by rewrite -/(f _ _) fxx /f !IHk -binS.
by rewrite big_ord_recl big1_eq addn0 mul1n subn0.
by rewrite big_ord_recl big_ord0 addn0 !bin0 muln1.
rewrite {}/f big... | Lemma | Vandermonde | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addn0",
"addnA",
"apply",
"big1_eq",
"big_ord0",
"big_ord_recl",
"big_split",
"bin0",
"binS",
"eq_bigr",
"last",
"mul1n",
"muln1",
"mulnDl",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn_exp m n k :
m ^ k - n ^ k = (m - n) * (\sum_(i < k) m ^ (k.-1 -i) * n ^ i). | Proof.
case: k => [|k]; first by rewrite big_ord0 muln0.
rewrite mulnBl !big_distrr big_ord_recl big_ord_recr /= subn0 muln1.
rewrite subnn mul1n -!expnS subnDA; congr (_ - _); apply: canRL (addnK _) _.
congr (_ + _); apply: eq_bigr => i _.
by rewrite (mulnCA n) -expnS mulnA -expnS subnSK /=.
Qed. | Lemma | subn_exp | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addnK",
"apply",
"big_distrr",
"big_ord0",
"big_ord_recl",
"big_ord_recr",
"eq_bigr",
"expnS",
"mul1n",
"muln0",
"muln1",
"mulnA",
"mulnBl",
"mulnCA",
"subn0",
"subnDA",
"subnSK",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predn_exp m k : (m ^ k).-1 = m.-1 * (\sum_(i < k) m ^ i). | Proof.
rewrite -!subn1 -[in LHS](exp1n k) subn_exp; congr (_ * _).
symmetry; rewrite (reindex_inj rev_ord_inj); apply: eq_bigr => i _ /=.
by rewrite -subn1 -subnDA exp1n muln1.
Qed. | Lemma | predn_exp | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"eq_bigr",
"exp1n",
"muln1",
"reindex_inj",
"rev_ord_inj",
"subn1",
"subnDA",
"subn_exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_pred_predX n e : (n.-1 %| (n ^ e).-1)%N. | Proof. by rewrite predn_exp dvdn_mulr. Qed. | Lemma | dvdn_pred_predX | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"dvdn_mulr",
"predn_exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modn_summ I r (P : pred I) F d :
\sum_(i <- r | P i) F i %% d = \sum_(i <- r | P i) F i %[mod d]. | Proof.
by apply/eqP; elim/big_rec2: _ => // i m n _; rewrite modnDml eqn_modDl.
Qed. | Lemma | modn_summ | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"big_rec2",
"eqn_modDl",
"modnDml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prime_modn_expSn p n : prime p -> n.+1 ^ p = (n ^ p).+1 %[mod p]. | Proof.
case: p => // p pP.
rewrite -[(_ ^ _).+1]addn0 (expnDn 1) big_ord_recr big_ord_recl /=.
rewrite subnn binn exp1n !mul1n addnAC -modnDmr; congr ((_ + _) %% _).
apply/eqP/dvdn_sum => -[i ?] _; exact/dvdn_mulr/prime_dvd_bin.
Qed. | Lemma | prime_modn_expSn | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addn0",
"addnAC",
"apply",
"big_ord_recl",
"big_ord_recr",
"binn",
"dvdn_mulr",
"dvdn_sum",
"exp1n",
"expnDn",
"modnDmr",
"mul1n",
"pP",
"prime",
"prime_dvd_bin",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fermat_little a p : prime p -> a ^ p = a %[mod p]. | Proof.
move=> pP.
elim: a => [|a IH]; first by rewrite exp0n // prime_gt0.
by rewrite prime_modn_expSn // -addn1 -modnDml IH modnDml addn1.
Qed. | Lemma | fermat_little | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addn1",
"exp0n",
"modnDml",
"pP",
"prime",
"prime_gt0",
"prime_modn_expSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_uniq_tuples T n (A : pred T) :
#|[set t : n.-tuple T | all A t & uniq t]| = #|A| ^_ n. | Proof.
elim: n A => [|n IHn] A.
by rewrite (@eq_card1 _ [tuple]) // => t; rewrite [t]tuple0 inE.
rewrite -sum1dep_card (partition_big (@thead _ _) A) /= => [t|].
by case/tupleP: t => x t; do 2!case/andP.
rewrite ffactnS -sum_nat_const; apply: eq_bigr => x Ax.
rewrite (cardD1 x) [x \in A]Ax /= -(IHn [predD1 A & x]) ... | Lemma | card_uniq_tuples | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"all",
"all_predC",
"all_predI",
"apply",
"behead",
"cardD1",
"eq_bigl",
"eq_bigr",
"eq_card1",
"eqxx",
"ffactnS",
"has_pred1",
"inE",
"partition_big",
"predD1",
"reindex",
"sum1dep_card",
"sum_nat_const",
"thead",
"theadE",
"tuple",
"tuple0",
"tupleP",
"tuple_eta",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_inj_ffuns_on D T (R : pred T) :
#|[set f : {ffun D -> T} in ffun_on R | injectiveb f]| = #|R| ^_ #|D|. | Proof.
rewrite -card_uniq_tuples.
have bijFF: {on (_ : pred _), bijective (@Finfun D T)}.
by exists fgraph => x _; [apply: FinfunK | apply: fgraphK].
rewrite -(on_card_preimset (bijFF _)); apply: eq_card => /= t.
rewrite !inE -(big_andE predT) -big_image /= big_all.
by rewrite -[t in RHS]FinfunK -codom_ffun.
Qed. | Lemma | card_inj_ffuns_on | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"Finfun",
"FinfunK",
"apply",
"big_all",
"big_andE",
"big_image",
"card_uniq_tuples",
"codom_ffun",
"eq_card",
"ffun_on",
"fgraph",
"fgraphK",
"inE",
"injectiveb",
"on",
"on_card_preimset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_inj_ffuns D T :
#|[set f : {ffun D -> T} | injectiveb f]| = #|T| ^_ #|D|. | Proof.
rewrite -card_inj_ffuns_on; apply: eq_card => f.
by rewrite 2!inE; case: ffun_onP.
Qed. | Lemma | card_inj_ffuns | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"card_inj_ffuns_on",
"eq_card",
"ffun_onP",
"inE",
"injectiveb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cards_draws T (B : {set T}) k :
#|[set A : {set T} | A \subset B & #|A| == k]| = 'C(#|B|, k). | Proof.
have [ltTk | lekT] := ltnP #|B| k.
rewrite bin_small // eq_card0 // => A.
rewrite inE eqn_leq [k <= _]leqNgt.
have [AsubB /=|//] := boolP (A \subset B).
by rewrite (leq_ltn_trans (subset_leq_card AsubB)) ?andbF.
apply/eqP; rewrite -(eqn_pmul2r (fact_gt0 k)) bin_ffact // eq_sym.
rewrite -sum_nat_cond_cons... | Lemma | cards_draws | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"bin_ffact",
"bin_small",
"card_imset",
"card_inj_ffuns",
"card_inj_ffuns_on",
"card_ord",
"enum_rankK_in",
"enum_rank_in",
"enum_val",
"enum_valP",
"enum_val_inj",
"eqEcard",
"eq_bigl",
"eq_bigr",
"eq_card0",
"eq_sym",
"eqn_leq",
"eqn_pmul2r",
"eqxx",
"fact_gt0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_draws T k : #|[set A : {set T} | #|A| == k]| = 'C(#|T|, k). | Proof.
by rewrite -cardsT -cards_draws; apply: eq_card => A; rewrite !inE subsetT.
Qed. | Lemma | card_draws | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"cardsT",
"cards_draws",
"eq_card",
"inE",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_ltn_sorted_tuples m n :
#|[set t : m.-tuple 'I_n | sorted ltn (map val t)]| = 'C(n, m). | Proof.
have [-> | n_gt0] := posnP n; last pose i0 := Ordinal n_gt0.
case: m => [|m]; last by apply: eq_card0; case/tupleP=> [[]].
by apply: (@eq_card1 _ [tuple]) => t; rewrite [t]tuple0 inE.
rewrite -[n in RHS]card_ord -card_draws.
pose f_t (t : m.-tuple 'I_n) := [set i in t].
pose f_A (A : {set 'I_n}) := [tuple of... | Lemma | card_ltn_sorted_tuples | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"apply",
"cardE",
"card_draws",
"card_ord",
"card_uniqP",
"cardsE",
"enum",
"eq_bigl",
"eq_card0",
"eq_card1",
"eq_filter",
"eqxx",
"filter_map",
"i0",
"inE",
"inj_eq",
"inj_map",
"iota_ltn_sorted",
"irr_sorted_eq",
"last",
"ltn",
"ltn_trans",
"ltnn",
"map",
"mapP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_sorted_tuples m n :
#|[set t : m.-tuple 'I_n.+1 | sorted leq (map val t)]| = 'C(m + n, m). | Proof.
set In1 := 'I_n.+1; pose x0 : In1 := ord0.
have add_mnP (i : 'I_m) (x : In1) : i + x < m + n.
by rewrite -ltnS -addSn -!addnS leq_add.
pose add_mn t i := Ordinal (add_mnP i (tnth t i)).
pose add_mn_nat (t : m.-tuple In1) i := i + nth x0 t i.
have add_mnC t: val \o add_mn t =1 add_mn_nat t \o val.
by move=> i... | Lemma | card_sorted_tuples | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"add0n",
"addKn",
"addSn",
"addSnnS",
"addn1",
"addnS",
"apply",
"card_ltn_sorted_tuples",
"drop0",
"drop_nth",
"drop_oversize",
"drop_size",
"enumT",
"eq_bigl",
"eq_from_tnth",
"eq_map",
"inE",
"inord",
"inordK",
"inord_val",
"last",
"leq",
"leq_add",
"leq_add2l",
"l... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_partial_ord_partitions m n :
#|[set t : m.-tuple 'I_n.+1 | \sum_(i <- t) i <= n]| = 'C(m + n, m). | Proof.
symmetry; set In1 := 'I_n.+1; pose x0 : In1 := ord0.
pose add_mn (i j : In1) : In1 := inord (i + j).
pose f_add (t : m.-tuple In1) := [tuple of scanl add_mn x0 t].
rewrite -card_sorted_tuples -!sum1dep_card (reindex f_add) /=; last first.
apply: eq_bigl => t; rewrite -[\sum_(i <- t) i]add0n.
transitivity (pa... | Lemma | card_partial_ord_partitions | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"add0n",
"addKn",
"addn0",
"addnA",
"apply",
"big_cons",
"big_nil",
"card_sorted_tuples",
"eq_bigl",
"inE",
"inord",
"inord_val",
"last",
"leq",
"leqNgt",
"leqP",
"leq_add2r",
"leq_addr",
"leq_ltn_trans",
"leq_subr",
"leq_trans",
"ltnS",
"ltn_ord",
"map",
"ord0",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_ord_partitions m n :
#|[set t : m.+1.-tuple 'I_n.+1 | \sum_(i <- t) i == n]| = 'C(m + n, m). | Proof.
symmetry; set In1 := 'I_n.+1; pose x0 : In1 := ord0.
pose f_add (t : m.-tuple In1) := [tuple of sub_ord (\sum_(x <- t) x) :: t].
rewrite -card_partial_ord_partitions -!sum1dep_card (reindex f_add) /=;
last first.
by apply: eq_bigl => t; rewrite big_cons /= addnC (sameP maxn_idPr eqP) maxnE.
exists (fun t :... | Lemma | card_ord_partitions | boot | boot/binomial.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"finset"
] | [
"addnC",
"addnK",
"apply",
"behead",
"big_cons",
"card_partial_ord_partitions",
"def_n",
"eq_bigl",
"inE",
"last",
"maxnE",
"maxn_idPr",
"ord0",
"reindex",
"sub_ord",
"sum1dep_card",
"tuple",
"tupleP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
code | := foldr (fun n m => 2 ^ n * m.*2.+1) 0. | Definition | code | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"foldr"
] | nk 0s n2 0s n1 0s | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
decode_rec (v q r : nat) {struct q} | :=
match q, r with
| 0, _ => [:: v]
| q'.+1, 0 => v :: [rec 0, q', q']
| q'.+1, 1 => [rec v.+1, q', q']
| q'.+1, r'.+2 => [rec v, q', r']
end where "[ 'rec' v , q , r ]" := (decode_rec v q r). | Fixpoint | decode_rec | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decode n | := if n is 0 then [::] else [rec 0, n.-1, n.-1]. | Definition | decode | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decodeK : cancel decode code. | Proof.
have m2s: forall n, n.*2 - n = n by move=> n; rewrite -addnn addnK.
case=> //= n; rewrite -[n.+1]mul1n -(expn0 2) -[n in RHS]m2s.
elim: n {2 4}n {1 3}0 => [|q IHq] [|[|r]] v //=; rewrite {}IHq ?mul1n ?m2s //.
by rewrite expnSr -mulnA mul2n.
Qed. | Lemma | decodeK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"addnK",
"addnn",
"code",
"decode",
"expn0",
"expnSr",
"mul1n",
"mul2n",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codeK : cancel code decode. | Proof.
elim=> //= v s IHs; rewrite -[_ * _]prednK ?muln_gt0 ?expn_gt0 //=.
set two := 2; rewrite -[v in RHS]addn0; elim: v 0 => [|v IHv {IHs}] q.
rewrite mul1n add0n /= -{}[in RHS]IHs; case: (code s) => // u; pose n := u.+1.
by transitivity [rec q, n + u.+1, n.*2]; [rewrite addnn | elim: n => //=].
rewrite expnS -m... | Lemma | codeK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"add0n",
"addSnnS",
"addn0",
"addnn",
"code",
"decode",
"expnS",
"expn_gt0",
"mul1n",
"mul2n",
"mulnA",
"muln_gt0",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_code s : all (fun j => j < code s) s. | Proof.
elim: s => //= i s IHs; rewrite -[_.+1]muln1 leq_mul 1?ltn_expl //=.
apply: sub_all IHs => j /leqW lejs; rewrite -[j.+1]mul1n leq_mul ?expn_gt0 //.
by rewrite ltnS -[j]mul1n -mul2n leq_mul.
Qed. | Lemma | ltn_code | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"all",
"apply",
"code",
"expn_gt0",
"leqW",
"leq_mul",
"ltnS",
"ltn_expl",
"mul1n",
"mul2n",
"muln1",
"sub_all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn_decode n : all (ltn^~ n) (decode n). | Proof. by rewrite -{1}[n]decodeK ltn_code. Qed. | Lemma | gtn_decode | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"all",
"decode",
"decodeK",
"ltn",
"ltn_code"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_of_opt | := @oapp T _ (nseq 1) [::]. | Definition | seq_of_opt | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_of_optK : cancel seq_of_opt ohead. | Proof. by case. Qed. | Lemma | seq_of_optK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"ohead",
"seq_of_opt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_of_pair (p : T1 * T2) | := @Tagged T1 p.1 (fun _ => T2) p.2. | Definition | tag_of_pair | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_of_tag (u : {i : T1 & T2}) | := (tag u, tagged u). | Definition | pair_of_tag | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_of_pairK : cancel tag_of_pair pair_of_tag. | Proof. by case. Qed. | Lemma | tag_of_pairK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"pair_of_tag",
"tag_of_pair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_of_tagK : cancel pair_of_tag tag_of_pair. | Proof. by case. Qed. | Lemma | pair_of_tagK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"pair_of_tag",
"tag_of_pair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opair_of_sum (s : T1 + T2) | :=
match s with inl x => (Some x, None) | inr y => (None, Some y) end. | Definition | opair_of_sum | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_of_opair p | :=
oapp (some \o @inr T1 T2) (omap (@inl _ T2) p.1) p.2. | Definition | sum_of_opair | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opair_of_sumK : pcancel opair_of_sum sum_of_opair. | Proof. by case. Qed. | Lemma | opair_of_sumK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"opair_of_sum",
"sum_of_opair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bool_of_unitK : cancel (fun _ => true) (fun _ => tt). | Proof. by case. Qed. | Lemma | bool_of_unitK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree | := Leaf of T | Node of nat & seq tree. | Inductive | tree | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Leaf",
"nat",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree_rect K IH_leaf IH_node | :=
fix loop t : K t := match t with
| Leaf x => IH_leaf x
| Node n f0 =>
let fix iter_pair f : foldr (fun t => prod (K t)) unit f :=
if f is t :: f' then (loop t, iter_pair f') else tt in
IH_node n f0 (iter_pair f0)
end. | Definition | tree_rect | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Leaf",
"foldr",
"prod",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree_rec (K : tree -> Set) | := @tree_rect K. | Definition | tree_rec | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"tree",
"tree_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree_ind K IH_leaf IH_node | :=
fix loop t : K t : Prop := match t with
| Leaf x => IH_leaf x
| Node n f0 =>
let fix iter_conj f : foldr (fun t => and (K t)) True f :=
if f is t :: f' then conj (loop t) (iter_conj f') else Logic.I
in IH_node n f0 (iter_conj f0)
end. | Definition | tree_ind | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Leaf",
"True",
"conj",
"foldr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encode t : seq (nat + T) | :=
match t with
| Leaf x => [:: inr _ x]
| Node n f => inl _ n.+1 :: rcons (flatten (map encode f)) (inl _ 0)
end. | Fixpoint | encode | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Leaf",
"flatten",
"map",
"nat",
"rcons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decode_step c fs | :=
match c with
| inr x => (Leaf x :: fs.1, fs.2)
| inl 0 => ([::], fs.1 :: fs.2)
| inl n.+1 => (Node n fs.1 :: head [::] fs.2, behead fs.2)
end. | Definition | decode_step | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Leaf",
"behead",
"head"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decode c | := ohead (foldr decode_step ([::], [::]) c).1. | Definition | decode | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"decode_step",
"foldr",
"ohead"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codeK : pcancel encode decode. | Proof.
move=> t; rewrite /decode; set fs := (_, _).
suffices ->: foldr decode_step fs (encode t) = (t :: fs.1, fs.2) by [].
elim: t => //= n f IHt in (fs) *; elim: f IHt => //= t f IHf [].
by rewrite rcons_cat foldr_cat => -> /= /IHf[-> -> ->].
Qed. | Lemma | codeK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"decode",
"decode_step",
"encode",
"foldr",
"foldr_cat",
"rcons_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find | := find_subdef. | Notation | find | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
correct | := choice_correct_subdef. | Notation | correct | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
complete | := choice_complete_subdef. | Notation | complete | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extensional | := choice_extensional_subdef. | Notation | extensional | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xchoose_subproof P exP :
{x | find P (ex_minn (@choice_complete_subdef _ P exP)) = Some x}. | Proof.
case: (ex_minnP (complete exP)) => n.
by case: (find P n) => // x; exists x.
Qed. | Fact | xchoose_subproof | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"complete",
"exP",
"ex_minn",
"ex_minnP",
"find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dc | := decode. | Notation | dc | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"decode"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xchoose P exP | := sval (@xchoose_subproof T P exP). | Definition | xchoose | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"exP",
"xchoose_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xchooseP P exP : P (@xchoose P exP). | Proof. by rewrite /xchoose; case: (xchoose_subproof exP) => x /= /correct. Qed. | Lemma | xchooseP | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"correct",
"exP",
"xchoose",
"xchoose_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_xchoose P Q exP exQ : P =1 Q -> @xchoose P exP = @xchoose Q exQ. | Proof.
rewrite /xchoose => eqPQ.
case: (xchoose_subproof exP) => x; case: (xchoose_subproof exQ) => y /=.
case: ex_minnP => n; rewrite -(extensional eqPQ) => Pn minQn.
case: ex_minnP => m; rewrite !(extensional eqPQ) => Qm minPm.
by case: (eqVneq m n) => [-> -> [] //|]; rewrite eqn_leq minQn ?minPm.
Qed. | Lemma | eq_xchoose | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"eqVneq",
"eqn_leq",
"exP",
"ex_minnP",
"extensional",
"xchoose",
"xchoose_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sigW P : (exists x, P x) -> {x | P x}. | Proof. by move=> exP; exists (xchoose exP); apply: xchooseP. Qed. | Lemma | sigW | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"exP",
"xchoose",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig2W P Q : (exists2 x, P x & Q x) -> {x | P x & Q x}. | Proof.
move=> exPQ; have [|x /andP[]] := @sigW (predI P Q); last by exists x.
by have [x Px Qx] := exPQ; exists x; apply/andP.
Qed. | Lemma | sig2W | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Px",
"apply",
"last",
"sigW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig_eqW (vT : eqType) (lhs rhs : T -> vT) :
(exists x, lhs x = rhs x) -> {x | lhs x = rhs x}. | Proof.
move=> exP; suffices [x /eqP Ex]: {x | lhs x == rhs x} by exists x.
by apply: sigW; have [x /eqP Ex] := exP; exists x.
Qed. | Lemma | sig_eqW | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"exP",
"rhs",
"sigW",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig2_eqW (vT : eqType) (P : pred T) (lhs rhs : T -> vT) :
(exists2 x, P x & lhs x = rhs x) -> {x | P x & lhs x = rhs x}. | Proof.
move=> exP; suffices [x Px /eqP Ex]: {x | P x & lhs x == rhs x} by exists x.
by apply: sig2W; have [x Px /eqP Ex] := exP; exists x.
Qed. | Lemma | sig2_eqW | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Px",
"apply",
"exP",
"rhs",
"sig2W",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig_eq2W (vT vT' : eqType) (lhs rhs : T -> vT) (lhs' rhs' : T -> vT') :
(exists2 x : T, lhs x = rhs x & lhs' x = rhs' x) ->
{x : T | lhs x = rhs x & lhs' x = rhs' x}. | Proof.
move=> e; suff [x /eqP]: {x : T | lhs x == rhs x & lhs' x = rhs' x} by exists x.
by apply: sig2_eqW; case: e => x /eqP; exists x.
Qed. | Lemma | sig_eq2W | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"rhs",
"sig2_eqW",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
choose P x0 | :=
if insub x0 : {? x | P x} is Some (exist x Px) then
xchoose (ex_intro [eta P] x Px)
else x0. | Definition | choose | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Px",
"insub",
"xchoose"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
chooseP P x0 : P x0 -> P (choose P x0). | Proof. by move=> Px0; rewrite /choose insubT xchooseP. Qed. | Lemma | chooseP | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"choose",
"insubT",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
choose_id P x0 y0 : P x0 -> P y0 -> choose P x0 = choose P y0. | Proof. by move=> Px0 Py0; rewrite /choose !insubT /=; apply: eq_xchoose. Qed. | Lemma | choose_id | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"choose",
"eq_xchoose",
"insubT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_choose P Q : P =1 Q -> choose P =1 choose Q. | Proof.
rewrite /choose => eqPQ x0.
do [case: insubP; rewrite eqPQ] => [[x Px] Qx0 _| ?]; last by rewrite insubN.
by rewrite insubT; apply: eq_xchoose.
Qed. | Lemma | eq_choose | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Px",
"apply",
"choose",
"eq_xchoose",
"insubN",
"insubP",
"insubT",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCanHasChoice f' : pcancel f f' -> hasChoice sT. | Proof.
move=> fK; pose liftP sP := [pred x | oapp sP false (f' x)].
pose sf sP := [fun n => obind f' (find (liftP sP) n)].
exists sf => [sP n x | sP [y sPy] | sP sQ eqPQ n] /=.
- by case Df: (find _ n) => //= [?] Dx; have:= correct Df; rewrite /= Dx.
- have [|n Pn] := @complete T (liftP sP); first by exists (f y); rewr... | Lemma | PCanHasChoice | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Dx",
"apply",
"complete",
"correct",
"extensional",
"fK",
"find",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CanHasChoice f' (fK : cancel f f') | :=
PCanHasChoice (can_pcan fK). | Definition | CanHasChoice | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"PCanHasChoice",
"fK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_hasChoice : hasChoice (seq T). | Proof.
pose r f := [fun xs => fun x : T => f (x :: xs) : option (seq T)].
pose fix f sP ns xs {struct ns} :=
if ns is n :: ns1 then let fr := r (f sP ns1) xs in obind fr (find fr n)
else if sP xs then Some xs else None.
exists (fun sP nn => f sP (dc nn) nil) => [sP n ys | sP [ys] | sP sQ eqPQ n].
- elim: {n}(dc n) ... | Fact | seq_hasChoice | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"cat_rcons",
"cats0",
"code",
"codeK",
"complete",
"correct",
"dc",
"extensional",
"find",
"last_ind",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_hasChoice : hasChoice {i : I & T_ i}. | Proof.
pose mkT i (x : T_ i) := Tagged T_ x.
pose ft tP n i := omap (mkT i) (find (tP \o mkT i) n).
pose fi tP ni nt := obind (ft tP nt) (find (ft tP nt) ni).
pose f tP n := if dc n is [:: ni; nt] then fi tP ni nt else None.
exists f => [tP n u | tP [[i x] tPxi] | sP sQ eqPQ n].
- rewrite /f /fi; case: (dc n) => [|ni [... | Fact | tagged_hasChoice | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"apply",
"code",
"codeK",
"complete",
"correct",
"dc",
"extensional",
"find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_hasChoice : hasChoice nat. | Proof.
pose f := [fun (P : pred nat) n => if P n then Some n else None].
exists f => [P n m | P [n Pn] | P Q eqPQ n] /=; last by rewrite eqPQ.
by case: ifP => // Pn [<-].
by exists n; rewrite Pn.
Qed. | Fact | nat_hasChoice | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"last",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'Choice' 'of' T 'by' <: ]" | := (Choice.copy T%type (sub_type T%type))
(format "[ 'Choice' 'of' T 'by' <: ]") : form_scope. | Notation | [ 'Choice' 'of' T 'by' <: ] | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"copy",
"sub_type",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pickle_inv n | :=
obind (fun x : T => if pickle x == n then Some x else None) (unpickle n). | Definition | pickle_inv | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pickle_invK : ocancel pickle_inv pickle. | Proof.
by rewrite /pickle_inv => n; case def_x: (unpickle n) => //= [x]; case: eqP.
Qed. | Lemma | pickle_invK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"pickle_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pickleK_inv : pcancel pickle pickle_inv. | Proof. by rewrite /pickle_inv => x; rewrite pickleK /= eqxx. Qed. | Lemma | pickleK_inv | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"eqxx",
"pickle_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcan_pickleK sT f f' :
@pcancel T sT f f' -> pcancel (pickle \o f) (pcomp f' unpickle). | Proof. by move=> fK x; rewrite /pcomp pickleK /= fK. Qed. | Lemma | pcan_pickleK | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"fK",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCanIsCountable sT (f : sT -> T) f' (fK : pcancel f f') | :=
isCountable.Build sT (pcan_pickleK fK). | Definition | PCanIsCountable | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"Build",
"fK",
"isCountable",
"pcan_pickleK",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CanIsCountable sT f f' (fK : cancel f f') | :=
@PCanIsCountable sT _ _ (can_pcan fK). | Definition | CanIsCountable | boot | boot/choice.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"Choice.InternalTheory",
"CodeSeq"
] | [
"PCanIsCountable",
"fK",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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