phanerozoic/Coq-MathComp · Datasets at Hugging Face

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
all_allpairsP (S : eqType) (T : S -> eqType) (R : Type) (p : pred R) (f : forall x : S, T x -> R) (s : seq S) (t : forall x : S, seq (T x)) : reflect (forall (x : S) (y : T x), x \in s -> y \in t x -> p (f x y)) (all p [seq f x y \| x <- s, y <- t x]).	Proof. elim: s => [\|x s IHs]; first by constructor. rewrite /= all_cat all_map /preim. apply/(iffP andP)=> [[/allP /= ? ? x' y x'_in_xs]\|p_xs_t]. by move: x'_in_xs y => /[1!inE] /predU1P [-> //\|? ?]; exact: IHs. split; first by apply/allP => ?; exact/p_xs_t/mem_head. by apply/IHs => x' y x'_in_s; apply: p_xs_t; rewri...	Lemma	all_allpairsP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allP", "all_cat", "all_map", "apply", "inE", "mem_head", "predU1P", "seq", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allpairsP f s t (z : R) : reflect (exists p, [/\ p.1 \in s, p.2 \in t & z = f p.1 p.2]) (z \in [seq f x y \| x <- s, y <- t]).	Proof. by apply: (iffP allpairsPdep) => [[x[y]]\|[[x y]]]; [exists (x, y)\|exists x, y]. Qed.	Lemma	allpairsP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allpairsPdep", "apply", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allpairs_f f s t x y : x \in s -> y \in t -> f x y \in [seq f x y \| x <- s, y <- t].	Proof. exact: allpairs_f_dep. Qed.	Lemma	allpairs_f	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allpairs_f_dep", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_allpairs f1 f2 s t : {in s & t, f1 =2 f2} <-> [seq f1 x y : R \| x <- s, y <- t] = [seq f2 x y \| x <- s, y <- t].	Proof. split=> [eq_f \| /eq_in_allpairs_dep-eq_f x y /eq_f/(_ y)//]. by apply/eq_in_allpairs_dep=> x /eq_f. Qed.	Lemma	eq_in_allpairs	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "eq_f", "eq_in_allpairs_dep", "f1", "f2", "seq", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_allpairs f s1 t1 s2 t2 : perm_eq s1 s2 -> perm_eq t1 t2 -> perm_eq [seq f x y \| x <- s1, y <- t1] [seq f x y \| x <- s2, y <- t2].	Proof. by move=> perm_s perm_t; apply: perm_allpairs_dep. Qed.	Lemma	perm_allpairs	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "perm_allpairs_dep", "perm_eq", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_allpairs f s1 t1 s2 t2 : s1 =i s2 -> t1 =i t2 -> [seq f x y \| x <- s1, y <- t1] =i [seq f x y \| x <- s2, y <- t2].	Proof. by move=> eq_s eq_t; apply: mem_allpairs_dep. Qed.	Lemma	mem_allpairs	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "mem_allpairs_dep", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allpairs_uniq f s t (st := [seq (x, y) \| x <- s, y <- t]) : uniq s -> uniq t -> {in st &, injective (uncurry f)} -> uniq [seq f x y \| x <- s, y <- t].	Proof. move=> Us Ut inj_f; rewrite -(map_allpairs (uncurry f) (@pair S T)) -/st. rewrite map_inj_in_uniq // allpairs_uniq_dep {Us Ut st inj_f}//. by apply: in2W => -[x1 y1] [x2 y2] /= [-> ->]. Qed.	Lemma	allpairs_uniq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allpairs_uniq_dep", "apply", "inj_f", "map_allpairs", "map_inj_in_uniq", "seq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel xs ys	:= all [pred x \| all (r x) ys] xs.	Definition	allrel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel0l ys : allrel [::] ys.	Proof. by []. Qed.	Lemma	allrel0l	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel0r xs : allrel xs [::].	Proof. by elim: xs. Qed.	Lemma	allrel0r	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_consl x xs ys : allrel (x :: xs) ys = all (r x) ys && allrel xs ys.	Proof. by []. Qed.	Lemma	allrel_consl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_consr xs y ys : allrel xs (y :: ys) = all (r^~ y) xs && allrel xs ys.	Proof. exact: all_predI. Qed.	Lemma	allrel_consr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "all_predI", "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_cons2 x y xs ys : allrel (x :: xs) (y :: ys) = [&& r x y, all (r x) ys, all (r^~ y) xs & allrel xs ys].	Proof. by rewrite /= allrel_consr -andbA. Qed.	Lemma	allrel_cons2	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel", "allrel_consr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel1l x ys : allrel [:: x] ys = all (r x) ys.	Proof. exact: andbT. Qed.	Lemma	allrel1l	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel1r xs y : allrel xs [:: y] = all (r^~ y) xs.	Proof. by rewrite allrel_consr allrel0r andbT. Qed.	Lemma	allrel1r	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel", "allrel0r", "allrel_consr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_catl xs xs' ys : allrel (xs ++ xs') ys = allrel xs ys && allrel xs' ys.	Proof. exact: all_cat. Qed.	Lemma	allrel_catl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_cat", "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_catr xs ys ys' : allrel xs (ys ++ ys') = allrel xs ys && allrel xs ys'.	Proof. elim: ys => /= [\|y ys ihys]; first by rewrite allrel0r. by rewrite !allrel_consr ihys andbA. Qed.	Lemma	allrel_catr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrel0r", "allrel_consr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_maskl m xs ys : allrel xs ys -> allrel (mask m xs) ys.	Proof. by elim: m xs => [\|[] m IHm] [\|x xs] //= /andP [xys /IHm->]; rewrite ?xys. Qed.	Lemma	allrel_maskl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "mask" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_maskr m xs ys : allrel xs ys -> allrel xs (mask m ys).	Proof. by elim: xs => //= x xs IHxs /andP [/all_mask->]. Qed.	Lemma	allrel_maskr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_mask", "allrel", "mask" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_filterl a xs ys : allrel xs ys -> allrel (filter a xs) ys.	Proof. by rewrite filter_mask; apply: allrel_maskl. Qed.	Lemma	allrel_filterl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrel_maskl", "apply", "filter", "filter_mask" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_filterr a xs ys : allrel xs ys -> allrel xs (filter a ys).	Proof. by rewrite filter_mask; apply: allrel_maskr. Qed.	Lemma	allrel_filterr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrel_maskr", "apply", "filter", "filter_mask" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_allpairsE xs ys : allrel xs ys = all id [seq r x y \| x <- xs, y <- ys].	Proof. by elim: xs => //= x xs ->; rewrite all_cat all_map. Qed.	Lemma	allrel_allpairsE	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "all_cat", "all_map", "allrel", "id", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
all2rel r xs	:= (allrel r xs xs).	Notation	all2rel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_allrel {T S : Type} (P : {pred T}) (Q : {pred S}) (r r' : T -> S -> bool) : {in P & Q, forall x y, r x y -> r' x y} -> forall xs ys, all P xs -> all Q ys -> allrel r xs ys -> allrel r' xs ys.	Proof. move=> rr' + ys; elim=> //= x xs IHxs /andP [Px Pxs] Qys. rewrite !allrel_consl => /andP [+ {}/IHxs-> //]; rewrite andbT. by elim: ys Qys => //= y ys IHys /andP [Qy Qys] /andP [/rr'-> // /IHys->]. Qed.	Lemma	sub_in_allrel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "Px", "all", "allrel", "allrel_consl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_allrel {T S : Type} (r r' : T -> S -> bool) : (forall x y, r x y -> r' x y) -> forall xs ys, allrel r xs ys -> allrel r' xs ys.	Proof. by move=> rr' xs ys; apply/sub_in_allrel/all_predT/all_predT; apply: in2W. Qed.	Lemma	sub_allrel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predT", "allrel", "apply", "sub_in_allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_allrel {T S : Type} (P : {pred T}) (Q : {pred S}) r r' : {in P & Q, r =2 r'} -> forall xs ys, all P xs -> all Q ys -> allrel r xs ys = allrel r' xs ys.	Proof. move=> rr' xs ys Pxs Qys. by apply/idP/idP; apply/sub_in_allrel/Qys/Pxs => ? ? ? ?; rewrite rr'. Qed.	Lemma	eq_in_allrel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel", "apply", "sub_in_allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_allrel {T S : Type} (r r' : T -> S -> bool) : r =2 r' -> allrel r =2 allrel r'.	Proof. by move=> rr' xs ys; apply/eq_in_allrel/all_predT/all_predT. Qed.	Lemma	eq_allrel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predT", "allrel", "apply", "eq_in_allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrelC {T S : Type} (r : T -> S -> bool) xs ys : allrel r xs ys = allrel (fun y => r^~ y) ys xs.	Proof. by elim: xs => [\|x xs ih]; [elim: ys \| rewrite allrel_consr -ih]. Qed.	Lemma	allrelC	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrel_consr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_mapl {T T' S : Type} (f : T' -> T) (r : T -> S -> bool) xs ys : allrel r (map f xs) ys = allrel (fun x => r (f x)) xs ys.	Proof. exact: all_map. Qed.	Lemma	allrel_mapl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "T'", "all_map", "allrel", "map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_mapr {T S S' : Type} (f : S' -> S) (r : T -> S -> bool) xs ys : allrel r xs (map f ys) = allrel (fun x y => r x (f y)) xs ys.	Proof. by rewrite allrelC allrel_mapl allrelC. Qed.	Lemma	allrel_mapr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrelC", "allrel_mapl", "map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrelP {T S : eqType} {r : T -> S -> bool} {xs ys} : reflect {in xs & ys, forall x y, r x y} (allrel r xs ys).	Proof. by rewrite allrel_allpairsE; exact: all_allpairsP. Qed.	Lemma	allrelP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_allpairsP", "allrel", "allrel_allpairsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrelT {T S : Type} (xs : seq T) (ys : seq S) : allrel (fun _ _ => true) xs ys = true.	Proof. by elim: xs => //= ? ?; rewrite allrel_consl all_predT. Qed.	Lemma	allrelT	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predT", "allrel", "allrel_consl", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_relI {T S : Type} (r r' : T -> S -> bool) xs ys : allrel (fun x y => r x y && r' x y) xs ys = allrel r xs ys && allrel r' xs ys.	Proof. by rewrite -all_predI; apply: eq_all => ?; rewrite /= -all_predI. Qed.	Lemma	allrel_relI	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predI", "allrel", "apply", "eq_all" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_revl {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r (rev s1) s2 = allrel r s1 s2.	Proof. exact: all_rev. Qed.	Lemma	allrel_revl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_rev", "allrel", "rev", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_revr {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r s1 (rev s2) = allrel r s1 s2.	Proof. by rewrite allrelC allrel_revl allrelC. Qed.	Lemma	allrel_revr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrelC", "allrel_revl", "rev", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
allrel_rev2 {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r (rev s1) (rev s2) = allrel r s1 s2.	Proof. by rewrite allrel_revr allrel_revl. Qed.	Lemma	allrel_rev2	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrel_revl", "allrel_revr", "rev", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_allrel_meml {T : eqType} {S} (r : T -> S -> bool) (s1 s1' : seq T) s2 : s1 =i s1' -> allrel r s1 s2 = allrel r s1' s2.	Proof. by move=> eqs1; apply: eq_all_r. Qed.	Lemma	eq_allrel_meml	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "apply", "eq_all_r", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_allrel_memr {T} {S : eqType} (r : T -> S -> bool) s1 (s2 s2' : seq S) : s2 =i s2' -> allrel r s1 s2 = allrel r s1 s2'.	Proof. by rewrite ![allrel _ s1 _]allrelC; apply: eq_allrel_meml. Qed.	Lemma	eq_allrel_memr	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "allrelC", "apply", "eq_allrel_meml", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_allrel_mem2 {T S : eqType} (r : T -> S -> bool) (s1 s1' : seq T) (s2 s2' : seq S) : s1 =i s1' -> s2 =i s2' -> allrel r s1 s2 = allrel r s1' s2'.	Proof. by move=> /eq_allrel_meml -> /eq_allrel_memr ->. Qed.	Lemma	eq_allrel_mem2	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allrel", "eq_allrel_meml", "eq_allrel_memr", "s1", "s2", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(rsym : symmetric r).		Hypothesis	rsym	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
all2rel1 x : all2rel r [:: x] = r x x.	Proof. by rewrite /allrel /= !andbT. Qed.	Lemma	all2rel1	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all2rel", "allrel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
all2rel2 x y : all2rel r [:: x; y] = r x x && r y y && r x y.	Proof. by rewrite /allrel /= rsym; do 3 case: r. Qed.	Lemma	all2rel2	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all2rel", "allrel", "rsym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
all2rel_cons x xs : all2rel r (x :: xs) = [&& r x x, all (r x) xs & all2rel r xs].	Proof. rewrite allrel_cons2; congr andb; rewrite andbA -all_predI; congr andb. by elim: xs => //= y xs ->; rewrite rsym andbb. Qed.	Lemma	all2rel_cons	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "all2rel", "all_predI", "allrel_cons2", "rsym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise xs : bool	:= if xs is x :: xs then all (r x) xs && pairwise xs else true.	Fixpoint	pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_cons x xs : pairwise (x :: xs) = all (r x) xs && pairwise xs.	Proof. by []. Qed.	Lemma	pairwise_cons	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_cat xs ys : pairwise (xs ++ ys) = [&& allrel r xs ys, pairwise xs & pairwise ys].	Proof. by elim: xs => //= x xs ->; rewrite all_cat -!andbA; bool_congr. Qed.	Lemma	pairwise_cat	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_cat", "allrel", "pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_rcons xs x : pairwise (rcons xs x) = all (r^~ x) xs && pairwise xs.	Proof. by rewrite -cats1 pairwise_cat allrel1r andbT. Qed.	Lemma	pairwise_rcons	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allrel1r", "cats1", "pairwise", "pairwise_cat", "rcons" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise2 x y : pairwise [:: x; y] = r x y.	Proof. by rewrite /= !andbT. Qed.	Lemma	pairwise2	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_mask m xs : pairwise xs -> pairwise (mask m xs).	Proof. by elim: m xs => [\|[] m IHm] [\|x xs] //= /andP [? ?]; rewrite ?IHm // all_mask. Qed.	Lemma	pairwise_mask	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_mask", "mask", "pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_filter a xs : pairwise xs -> pairwise (filter a xs).	Proof. by rewrite filter_mask; apply: pairwise_mask. Qed.	Lemma	pairwise_filter	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "filter", "filter_mask", "pairwise", "pairwise_mask" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwiseP x0 xs : reflect {in gtn (size xs) &, {homo nth x0 xs : i j / i < j >-> r i j}} (pairwise xs).	Proof. elim: xs => /= [\|x xs IHxs]; first exact: (iffP idP). apply: (iffP andP) => [[r_x_xs pxs] i j\|Hnth]; rewrite -?topredE /= ?ltnS. by case: i j => [\|i] [\|j] //= gti gtj ij; [exact/all_nthP \| exact/IHxs]. split; last by apply/IHxs => // i j; apply/(Hnth i.+1 j.+1). by apply/(all_nthP x0) => i gti; apply/(Hnth 0 i...	Lemma	pairwiseP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_nthP", "apply", "gtn", "last", "ltnS", "nth", "pairwise", "size", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_all2rel : reflexive r -> symmetric r -> forall xs, pairwise xs = all2rel r xs.	Proof. by move=> r_refl r_sym; elim => //= x xs ->; rewrite all2rel_cons // r_refl. Qed.	Lemma	pairwise_all2rel	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all2rel", "all2rel_cons", "pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_pairwise {T : Type} (P : {pred T}) (r r' : rel T) : {in P &, subrel r r'} -> forall xs, all P xs -> pairwise r xs -> pairwise r' xs.	Proof. move=> rr'; elim=> //= x xs IHxs /andP [Px Pxs] /andP [+ {}/IHxs->] //. rewrite andbT; elim: xs Pxs => //= x' xs IHxs /andP [? ?] /andP [+ /IHxs->] //. by rewrite andbT; apply: rr'. Qed.	Lemma	sub_in_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "Px", "all", "apply", "pairwise", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_pairwise {T : Type} (r r' : rel T) xs : subrel r r' -> pairwise r xs -> pairwise r' xs.	Proof. by move=> rr'; apply/sub_in_pairwise/all_predT; apply: in2W. Qed.	Lemma	sub_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predT", "apply", "pairwise", "rel", "sub_in_pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_pairwise {T : Type} (P : {pred T}) (r r' : rel T) : {in P &, r =2 r'} -> forall xs, all P xs -> pairwise r xs = pairwise r' xs.	Proof. move=> rr' xs Pxs. by apply/idP/idP; apply/sub_in_pairwise/Pxs => ? ? ? ?; rewrite rr'. Qed.	Lemma	eq_in_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "apply", "pairwise", "rel", "sub_in_pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_pairwise {T : Type} (r r' : rel T) : r =2 r' -> pairwise r =i pairwise r'.	Proof. by move=> rr' xs; apply/eq_in_pairwise/all_predT. Qed.	Lemma	eq_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predT", "apply", "eq_in_pairwise", "pairwise", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_map {T T' : Type} (f : T' -> T) (r : rel T) xs : pairwise r (map f xs) = pairwise (relpre f r) xs.	Proof. by elim: xs => //= x xs ->; rewrite all_map. Qed.	Lemma	pairwise_map	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "T'", "all_map", "map", "pairwise", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_relI {T : Type} (r r' : rel T) (s : seq T) : pairwise [rel x y \| r x y && r' x y] s = pairwise r s && pairwise r' s.	Proof. by elim: s => //= x s ->; rewrite andbACA all_predI. Qed.	Lemma	pairwise_relI	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predI", "pairwise", "rel", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subseq_pairwise xs ys : subseq xs ys -> pairwise r ys -> pairwise r xs.	Proof. by case/subseqP => m _ ->; apply: pairwise_mask. Qed.	Lemma	subseq_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "pairwise", "pairwise_mask", "subseq", "subseqP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
uniq_pairwise xs : uniq xs = pairwise [rel x y \| x != y] xs.	Proof. elim: xs => //= x xs ->; congr andb; rewrite -has_pred1 -all_predC. by elim: xs => //= x' xs ->; case: eqVneq. Qed.	Lemma	uniq_pairwise	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all_predC", "eqVneq", "has_pred1", "pairwise", "rel", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_uniq xs : irreflexive r -> pairwise r xs -> uniq xs.	Proof. move=> r_irr; rewrite uniq_pairwise; apply/sub_pairwise => x y. by apply: contraTneq => ->; rewrite r_irr. Qed.	Lemma	pairwise_uniq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "contraTneq", "pairwise", "sub_pairwise", "uniq", "uniq_pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_eq : antisymmetric r -> forall xs ys, pairwise r xs -> pairwise r ys -> perm_eq xs ys -> xs = ys.	Proof. move=> r_asym; elim=> [\|x xs IHxs] [\|y ys] //=; try by move=> ? ? /perm_size. move=> /andP [r_x_xs pxs] /andP [r_y_ys pys] eq_xs_ys. move: (mem_head y ys) (mem_head x xs). rewrite -(perm_mem eq_xs_ys) [x \in _](perm_mem eq_xs_ys) !inE. case: eqVneq eq_xs_ys => /= [->\|ne_xy] eq_xs_ys ys_x xs_y. by rewrite (IHxs...	Lemma	pairwise_eq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allP", "apply", "eqVneq", "inE", "mem_head", "pairwise", "perm_cons", "perm_eq", "perm_mem", "perm_size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_trans s : antisymmetric r -> pairwise r s -> {in s & &, transitive r}.	Proof. move=> /(_ _ _ _)/eqP r_anti + y x z => /pairwiseP-/(_ y) ltP ys xs zs. have [-> //\|neqxy] := eqVneq x y; have [-> //\|neqzy] := eqVneq z y. move=> lxy lyz; move: ys xs zs lxy neqxy lyz neqzy. move=> /(nthP y)[j jlt <-] /(nthP y)[i ilt <-] /(nthP y)[k klt <-]. have [ltij\|ltji\|->] := ltngtP i j; last 2 first. - by...	Lemma	pairwise_trans	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "eqVneq", "last", "ltP", "ltn_trans", "ltngtP", "nthP", "pairwise", "pairwiseP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
incr_tally bs x	:= if bs isn't b :: bs then [:: (x, 1)] else if x == b.1 then (x, b.2.+1) :: bs else b :: incr_tally bs x.	Fixpoint	incr_tally	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tally s	:= foldl incr_tally [::] s.	Definition	tally	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "foldl", "incr_tally" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
wf_tally	:= [qualify a bs : seq (T * nat) \| uniq (unzip1 bs) && (0 \notin unzip2 bs)].	Definition	wf_tally	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "nat", "seq", "uniq", "unzip1", "unzip2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tally_seq bs	:= flatten [seq nseq b.2 b.1 \| b <- bs].	Definition	tally_seq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "flatten", "nseq", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tseq	:= tally_seq.	Notation	tseq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "tally_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_tally_seq bs : size (tally_seq bs) = sumn (unzip2 bs).	Proof. by rewrite size_flatten /shape -map_comp; under eq_map do rewrite /= size_nseq. Qed.	Lemma	size_tally_seq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "eq_map", "map_comp", "shape", "size", "size_flatten", "size_nseq", "sumn", "tally_seq", "unzip2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tally_seqK : {in wf_tally, cancel tally_seq tally}.	Proof. move=> bs /andP[]; elim: bs => [\|[x [\|n]] bs IHbs] //= /andP[bs'x Ubs] bs'0. rewrite inE /tseq /tally /= -[n.+1]addn1 in bs'0 *. elim: n 1 => /= [\|n IHn] m; last by rewrite eqxx IHn addnS. rewrite -{}[in RHS]IHbs {Ubs bs'0}// /tally /tally_seq add0n. elim: bs bs'x [::] => [\|[y n] bs IHbs] //= /[1!inE] /norP[y'x ...	Lemma	tally_seqK	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "add0n", "addn1", "addnS", "eq_sym", "eqxx", "inE", "last", "tally", "tally_seq", "tseq", "wf_tally" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
incr_tallyP x : {homo incr_tally^~ x : bs / bs \in wf_tally}.	Proof. move=> bs /andP[]; rewrite unfold_in. elim: bs => [\|[y [\|n]] bs IHbs] //= /andP[bs'y Ubs] /[1!inE] /= bs'0. have [<- \| y'x] /= := eqVneq y; first by rewrite bs'y Ubs. rewrite -andbA {}IHbs {Ubs bs'0}// andbT. elim: bs bs'y => [\|b bs IHbs] /=; rewrite inE ?y'x // => /norP[b'y bs'y]. by case: ifP => _; rewrite /= ...	Lemma	incr_tallyP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "eqVneq", "inE", "incr_tally", "wf_tally" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tallyP s : tally s \is a wf_tally.	Proof. rewrite /tally; set bs := [::]; have: bs \in wf_tally by []. by elim: s bs => //= x s IHs bs /(incr_tallyP x)/IHs. Qed.	Lemma	tallyP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "incr_tallyP", "tally", "wf_tally" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tallyK s : perm_eq (tally_seq (tally s)) s.	Proof. rewrite -[s in perm_eq _ s]cats0 -[nil]/(tseq [::]) /tally. elim: s [::] => //= x s IHs bs; rewrite {IHs}(permPl (IHs _)). rewrite perm_sym -cat1s perm_catCA {s}perm_cat2l. elim: bs => //= b bs IHbs; case: eqP => [-> \| _] //=. by rewrite -cat1s perm_catCA perm_cat2l. Qed.	Lemma	tallyK	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "cat1s", "cats0", "permPl", "perm_cat2l", "perm_catCA", "perm_eq", "perm_sym", "tally", "tally_seq", "tseq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tallyEl s : perm_eq (unzip1 (tally s)) (undup s).	Proof. have /andP[Ubs bs'0] := tallyP s; set bs := tally s in Ubs bs'0 . rewrite uniq_perm ?undup_uniq {Ubs}// => x. rewrite mem_undup -(perm_mem (tallyK s)) -/bs. elim: bs => [\|[y [\|m]] bs IHbs] //= in bs'0 . by rewrite inE IHbs // mem_cat mem_nseq. Qed.	Lemma	tallyEl	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "inE", "mem_cat", "mem_nseq", "mem_undup", "perm_eq", "perm_mem", "tally", "tallyK", "tallyP", "undup", "undup_uniq", "uniq_perm", "unzip1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tallyE s : perm_eq (tally s) [seq (x, count_mem x s) \| x <- undup s].	Proof. have /andP[Ubs _] := tallyP s; pose b := [fun s x => (x, count_mem x (tseq s))]. suffices /permPl->: perm_eq (tally s) (map (b (tally s)) (unzip1 (tally s))). congr perm_eq: (perm_map (b (tally s)) (tallyEl s)). by under eq_map do rewrite /= (permP (tallyK s)). elim: (tally s) Ubs => [\|[x m] bs IH] //= /andP...	Lemma	tallyE	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "addn0", "apply", "count_cat", "count_mem", "count_memPn", "count_nseq", "eq_in_map", "eq_map", "eqxx", "inE", "map", "memPnC", "mem_cat", "mem_nseq", "mul1n", "permP", "permPl", "perm_cons", "perm_eq", "perm_map", "seq", "tally", "tallyEl", "tallyK", "tallyP", "tse...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_tally s1 s2 : perm_eq s1 s2 -> perm_eq (tally s1) (tally s2).	Proof. move=> eq_s12; apply: (@perm_trans _ [seq (x, count_mem x s2) \| x <- undup s1]). by congr perm_eq: (tallyE s1); under eq_map do rewrite (permP eq_s12). by rewrite (permPr (tallyE s2)); apply/perm_map/perm_undup/(perm_mem eq_s12). Qed.	Lemma	perm_tally	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "count_mem", "eq_map", "permP", "permPr", "perm_eq", "perm_map", "perm_mem", "perm_trans", "perm_undup", "s1", "s2", "seq", "tally", "tallyE", "undup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_tally_seq bs1 bs2 : perm_eq bs1 bs2 -> perm_eq (tally_seq bs1) (tally_seq bs2).	Proof. by move=> Ebs12; rewrite perm_flatten ?perm_map. Qed.	Lemma	perm_tally_seq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "perm_eq", "perm_flatten", "perm_map", "tally_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_tseq	:= perm_tally_seq.	Notation	perm_tseq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "perm_tally_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_count_undup s : perm_eq (flatten [seq nseq (count_mem x s) x \| x <- undup s]) s.	Proof. by rewrite -(permPr (tallyK s)) (permPr (perm_tseq (tallyE s))) /tseq -map_comp. Qed.	Lemma	perm_count_undup	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "count_mem", "flatten", "map_comp", "nseq", "permPr", "perm_eq", "perm_tseq", "seq", "tallyE", "tallyK", "tseq", "undup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cons_perms_ perms_rec (s : seq T) bs bs2 acc	:= if bs isn't b :: bs1 then acc else if b isn't (x, m.+1) then cons_perms_ perms_rec s bs1 bs2 acc else let acc_xs := perms_rec (x :: s) ((x, m) :: bs1 ++ bs2) acc in cons_perms_ perms_rec s bs1 (b :: bs2) acc_xs.	Fixpoint	cons_perms_	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "perms_rec", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perms_rec n s bs acc	:= if n isn't n.+1 then s :: acc else cons_perms_ (perms_rec n) s bs [::] acc.	Fixpoint	perms_rec	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "cons_perms_" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cons_perms n	:= (cons_perms_ (perms_rec n) [::]).	Notation	cons_perms	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "cons_perms_", "perms_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permutations s	:= perms_rec (size s) [::] (tally s) [::].	Definition	permutations	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "perms_rec", "size", "tally" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permsP s : exists n bs, [/\ permutations s = perms_rec n [::] bs [::], size (tseq bs) == n, perm_eq (tseq bs) s & uniq (unzip1 bs)].	Proof. have /andP[Ubs _] := tallyP s; exists (size s), (tally s). by rewrite (perm_size (tallyK s)) tallyK. Qed.	Let	permsP	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "perm_eq", "perm_size", "perms_rec", "permutations", "size", "tally", "tallyK", "tallyP", "tseq", "uniq", "unzip1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bsCA	:= (permEl (perm_catCA _ [:: _] _)).	Notation	bsCA	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "permEl", "perm_catCA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cons_permsE : forall n x bs bs1 bs2, let cp := cons_perms n bs bs2 in let perms s := perms_rec n s bs1 [::] in cp (perms [:: x]) = cp [::] ++ [seq rcons t x \| t <- perms [::]].	Proof. pose is_acc f := forall acc, f acc = f [::] ++ acc. (* f is accumulating. ) have cpE: forall f & forall s bs, is_acc (f s bs), is_acc (cons_perms_ f _ _ _). move=> s bs bs2 f fE acc; elim: bs => [\|[x [\|m]] bs IHbs] //= in s bs2 acc . by rewrite fE IHbs catA -IHbs. have prE: is_acc (perms_rec _ _ _) by elim...	Let	cons_permsE	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "catA", "cats1", "cons_perms", "cons_perms_", "cp", "eq_map", "fE", "map_cat", "map_comp", "perms", "perms_rec", "rcons", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_permutations s t : (t \in permutations s) = perm_eq t s.	Proof. have{s} [n [bs [-> Dn /permPr<- _]]] := permsP s. elim: n => [\|n IHn] /= in t bs Dn . by rewrite inE (nilP Dn); apply/eqP/perm_nilP. rewrite -[bs in tseq bs]cats0 in Dn ; have x0 : T by case: (tseq _) Dn. rewrite -[RHS](@andb_idl (last x0 t \in tseq bs)). case/lastP: t {IHn} => [\|t x] Dt; first by rewrite ...	Lemma	mem_permutations	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "bsCA", "cat_cons", "cats0", "cons_permsE", "inE", "last", "lastP", "last_rcons", "mapP", "mem_cat", "mem_head", "mem_nseq", "mem_rcons", "nilP", "permPr", "perm_cons", "perm_eq", "perm_mem", "perm_nilP", "perm_rcons", "perm_size", "perm_tseq", "permsP", "per...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permutations_uniq s : uniq (permutations s).	Proof. have{s} [n [bs [-> Dn _ Ubs]]] := permsP s. elim: n => //= n IHn in bs Dn Ubs ; rewrite -[bs]cats0 /unzip1 in Dn Ubs. elim: bs [::] => [\|[x [\|m]] bs IHbs] //= bs2 in Dn Ubs . by case/andP: Ubs => _ /IHbs->. rewrite /= cons_permsE cat_uniq has_sym andbCA andbC. rewrite {}IHbs; first 1 last; last by rewrite (p...	Lemma	permutations_uniq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "bsCA", "cat_uniq", "cats0", "cons_permsE", "hasPn", "has_cat", "has_map", "has_sym", "inE", "last", "mapP", "map_cat", "map_inj_uniq", "mem_cat", "perm_map", "perm_size", "perm_tseq", "perm_uniq", "permsP", "perms_rec", "permutations", "rcons_inj", "rcons_injl...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perms	:= permutations.	Notation	perms	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "permutations" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permutationsE s : 0 < size s -> perm_eq (perms s) [seq x :: t \| x <- undup s, t <- perms (rem x s)].	Proof. move=> nt_s; apply/uniq_perm=> [\|\|t]; first exact: permutations_uniq. apply/allpairs_uniq_dep=> [\|x _\|]; rewrite ?undup_uniq ?permutations_uniq //. by case=> [_ _] [x t] _ _ [-> ->]. rewrite mem_permutations; apply/idP/allpairsPdep=> [Dt \| [x [t1 []]]]. rewrite -(perm_size Dt) in nt_s; case: t nt_s => // ...	Lemma	permutationsE	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "allpairsPdep", "allpairs_uniq_dep", "apply", "mem_head", "mem_permutations", "mem_undup", "permPl", "perm_cons", "perm_eq", "perm_mem", "perm_size", "perm_sym", "perm_to_rem", "perms", "permutations_uniq", "rem", "seq", "size", "split", "undup", "undup_uniq", "uniq_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permutationsErot x s (le_x := fun t => iota 0 (index x t + 1)) : perm_eq (perms (x :: s)) [seq rot i (x :: t) \| t <- perms s, i <- le_x t].	Proof. have take'x t i: i <= index x t -> i <= size t /\ x \notin take i t. move=> le_i_x; have le_i_t: i <= size t := leq_trans le_i_x (index_size x t). case: (nthP x) => // -[j lt_j_i /eqP]; rewrite size_takel // in lt_j_i. by rewrite nth_take // [_ == _](before_find x (leq_trans lt_j_i le_i_x)). pose xrot t i ...	Lemma	permutationsErot	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "addn0", "addn1", "allpairsPdep", "allpairs_uniq_dep", "apply", "before_find", "cat_cons", "cat_rcons", "drop", "drop_nth", "eqxx", "index", "index_cat", "index_mem", "index_size", "iota", "iota_uniq", "leq_addr", "leq_trans", "leqnn", "ltnS", "mem_head", "mem_iota", "m...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_permutations s : uniq s -> size (permutations s) = (size s)`!.	Proof. move Dn: (size s) => n Us; elim: n s => [[]\|n IHn s] //= in Dn Us *. rewrite (perm_size (permutationsE _)) ?Dn // undup_id // factS -Dn. rewrite -(size_iota 0 n`!) -(size_allpairs (fun=>id)) !size_allpairs_dep. by apply/congr1/eq_in_map=> x sx; rewrite size_iota IHn ?size_rem ?Dn ?rem_uniq. Qed.	Lemma	size_permutations	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "eq_in_map", "factS", "id", "perm_size", "permutations", "permutationsE", "rem_uniq", "size", "size_allpairs", "size_allpairs_dep", "size_iota", "size_rem", "undup_id", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permutations_all_uniq s : uniq s -> all uniq (permutations s).	Proof. by move=> Us; apply/allP=> t; rewrite mem_permutations => /perm_uniq->. Qed.	Lemma	permutations_all_uniq	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "all", "allP", "apply", "mem_permutations", "perm_uniq", "permutations", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_permutations s t : perm_eq s t -> perm_eq (permutations s) (permutations t).	Proof. move=> Est; apply/uniq_perm; try exact: permutations_uniq. by move=> u; rewrite !mem_permutations (permPr Est). Qed.	Lemma	perm_permutations	boot	boot/seq.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]	[ "apply", "mem_permutations", "permPr", "perm_eq", "permutations", "permutations_uniq", "uniq_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
change_type ty ty' (x : ty) (strategy : ty = ty') : ty'	:= ecast ty ty strategy x.	Definition	change_type	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
simplrefl	:= (ltac: (simpl; reflexivity)) (only parsing).	Notation	simplrefl	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cbvrefl	:= (ltac: (cbv; reflexivity)) (only parsing).	Notation	cbvrefl	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vmrefl	:= (ltac: (vm_compute; reflexivity)) (only parsing).	Notation	vmrefl	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ten	:= (xO (xI (xO xH))).	Notation	ten	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[]	** Conversion with a decimal representation for printing/parsing	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
of_uint_acc (d:Decimal.uint) (acc:positive)	:= match d with \| Decimal.Nil => acc \| Decimal.D0 l => of_uint_acc l (mul ten acc) \| Decimal.D1 l => of_uint_acc l (add 1 (mul ten acc)) \| Decimal.D2 l => of_uint_acc l (add 1~0 (mul ten acc)) \| Decimal.D3 l => of_uint_acc l (add 1~1 (mul ten acc)) \| Decimal.D4 l => of_uint_acc l (add 1~0~0 (mul ten acc))...	Fixpoint	of_uint_acc	boot	boot/ssrAC.v	[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]	[ "Nil", "add", "mul", "ten" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

all_allpairsP (S : eqType) (T : S -> eqType) (R : Type) (p : pred R) (f : forall x : S, T x -> R) (s : seq S) (t : forall x : S, seq (T x)) : reflect (forall (x : S) (y : T x), x \in s -> y \in t x -> p (f x y)) (all p [seq f x y | x <- s, y <- t x]).

Proof. elim: s => [|x s IHs]; first by constructor. rewrite /= all_cat all_map /preim. apply/(iffP andP)=> [[/allP /= ? ? x' y x'_in_xs]|p_xs_t]. by move: x'_in_xs y => /[1!inE] /predU1P [-> //|? ?]; exact: IHs. split; first by apply/allP => ?; exact/p_xs_t/mem_head. by apply/IHs => x' y x'_in_s; apply: p_xs_t; rewri...

Lemma

all_allpairsP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allP", "all_cat", "all_map", "apply", "inE", "mem_head", "predU1P", "seq", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allpairsP f s t (z : R) : reflect (exists p, [/\ p.1 \in s, p.2 \in t & z = f p.1 p.2]) (z \in [seq f x y | x <- s, y <- t]).

Proof. by apply: (iffP allpairsPdep) => [[x[y]]|[[x y]]]; [exists (x, y)|exists x, y]. Qed.

Lemma

allpairsP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allpairsPdep", "apply", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allpairs_f f s t x y : x \in s -> y \in t -> f x y \in [seq f x y | x <- s, y <- t].

Proof. exact: allpairs_f_dep. Qed.

Lemma

allpairs_f

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allpairs_f_dep", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_allpairs f1 f2 s t : {in s & t, f1 =2 f2} <-> [seq f1 x y : R | x <- s, y <- t] = [seq f2 x y | x <- s, y <- t].

Proof. split=> [eq_f | /eq_in_allpairs_dep-eq_f x y /eq_f/(_ y)//]. by apply/eq_in_allpairs_dep=> x /eq_f. Qed.

Lemma

eq_in_allpairs

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "eq_f", "eq_in_allpairs_dep", "f1", "f2", "seq", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_allpairs f s1 t1 s2 t2 : perm_eq s1 s2 -> perm_eq t1 t2 -> perm_eq [seq f x y | x <- s1, y <- t1] [seq f x y | x <- s2, y <- t2].

Proof. by move=> perm_s perm_t; apply: perm_allpairs_dep. Qed.

Lemma

perm_allpairs

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "perm_allpairs_dep", "perm_eq", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_allpairs f s1 t1 s2 t2 : s1 =i s2 -> t1 =i t2 -> [seq f x y | x <- s1, y <- t1] =i [seq f x y | x <- s2, y <- t2].

Proof. by move=> eq_s eq_t; apply: mem_allpairs_dep. Qed.

Lemma

mem_allpairs

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "mem_allpairs_dep", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allpairs_uniq f s t (st := [seq (x, y) | x <- s, y <- t]) : uniq s -> uniq t -> {in st &, injective (uncurry f)} -> uniq [seq f x y | x <- s, y <- t].

Proof. move=> Us Ut inj_f; rewrite -(map_allpairs (uncurry f) (@pair S T)) -/st. rewrite map_inj_in_uniq // allpairs_uniq_dep {Us Ut st inj_f}//. by apply: in2W => -[x1 y1] [x2 y2] /= [-> ->]. Qed.

Lemma

allpairs_uniq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allpairs_uniq_dep", "apply", "inj_f", "map_allpairs", "map_inj_in_uniq", "seq", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel xs ys

:= all [pred x | all (r x) ys] xs.

Definition

allrel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel0l ys : allrel [::] ys.

Proof. by []. Qed.

Lemma

allrel0l

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel0r xs : allrel xs [::].

Proof. by elim: xs. Qed.

Lemma

allrel0r

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_consl x xs ys : allrel (x :: xs) ys = all (r x) ys && allrel xs ys.

Proof. by []. Qed.

Lemma

allrel_consl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_consr xs y ys : allrel xs (y :: ys) = all (r^~ y) xs && allrel xs ys.

Proof. exact: all_predI. Qed.

Lemma

allrel_consr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "all_predI", "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_cons2 x y xs ys : allrel (x :: xs) (y :: ys) = [&& r x y, all (r x) ys, all (r^~ y) xs & allrel xs ys].

Proof. by rewrite /= allrel_consr -andbA. Qed.

Lemma

allrel_cons2

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel", "allrel_consr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel1l x ys : allrel [:: x] ys = all (r x) ys.

Proof. exact: andbT. Qed.

Lemma

allrel1l

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel1r xs y : allrel xs [:: y] = all (r^~ y) xs.

Proof. by rewrite allrel_consr allrel0r andbT. Qed.

Lemma

allrel1r

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel", "allrel0r", "allrel_consr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_catl xs xs' ys : allrel (xs ++ xs') ys = allrel xs ys && allrel xs' ys.

Proof. exact: all_cat. Qed.

Lemma

allrel_catl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_cat", "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_catr xs ys ys' : allrel xs (ys ++ ys') = allrel xs ys && allrel xs ys'.

Proof. elim: ys => /= [|y ys ihys]; first by rewrite allrel0r. by rewrite !allrel_consr ihys andbA. Qed.

Lemma

allrel_catr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrel0r", "allrel_consr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_maskl m xs ys : allrel xs ys -> allrel (mask m xs) ys.

Proof. by elim: m xs => [|[] m IHm] [|x xs] //= /andP [xys /IHm->]; rewrite ?xys. Qed.

Lemma

allrel_maskl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "mask" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_maskr m xs ys : allrel xs ys -> allrel xs (mask m ys).

Proof. by elim: xs => //= x xs IHxs /andP [/all_mask->]. Qed.

Lemma

allrel_maskr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_mask", "allrel", "mask" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_filterl a xs ys : allrel xs ys -> allrel (filter a xs) ys.

Proof. by rewrite filter_mask; apply: allrel_maskl. Qed.

Lemma

allrel_filterl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrel_maskl", "apply", "filter", "filter_mask" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_filterr a xs ys : allrel xs ys -> allrel xs (filter a ys).

Proof. by rewrite filter_mask; apply: allrel_maskr. Qed.

Lemma

allrel_filterr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrel_maskr", "apply", "filter", "filter_mask" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_allpairsE xs ys : allrel xs ys = all id [seq r x y | x <- xs, y <- ys].

Proof. by elim: xs => //= x xs ->; rewrite all_cat all_map. Qed.

Lemma

allrel_allpairsE

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "all_cat", "all_map", "allrel", "id", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

all2rel r xs

:= (allrel r xs xs).

Notation

all2rel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_in_allrel {T S : Type} (P : {pred T}) (Q : {pred S}) (r r' : T -> S -> bool) : {in P & Q, forall x y, r x y -> r' x y} -> forall xs ys, all P xs -> all Q ys -> allrel r xs ys -> allrel r' xs ys.

Proof. move=> rr' + ys; elim=> //= x xs IHxs /andP [Px Pxs] Qys. rewrite !allrel_consl => /andP [+ {}/IHxs-> //]; rewrite andbT. by elim: ys Qys => //= y ys IHys /andP [Qy Qys] /andP [/rr'-> // /IHys->]. Qed.

Lemma

sub_in_allrel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "Px", "all", "allrel", "allrel_consl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_allrel {T S : Type} (r r' : T -> S -> bool) : (forall x y, r x y -> r' x y) -> forall xs ys, allrel r xs ys -> allrel r' xs ys.

Proof. by move=> rr' xs ys; apply/sub_in_allrel/all_predT/all_predT; apply: in2W. Qed.

Lemma

sub_allrel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predT", "allrel", "apply", "sub_in_allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_allrel {T S : Type} (P : {pred T}) (Q : {pred S}) r r' : {in P & Q, r =2 r'} -> forall xs ys, all P xs -> all Q ys -> allrel r xs ys = allrel r' xs ys.

Proof. move=> rr' xs ys Pxs Qys. by apply/idP/idP; apply/sub_in_allrel/Qys/Pxs => ? ? ? ?; rewrite rr'. Qed.

Lemma

eq_in_allrel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel", "apply", "sub_in_allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_allrel {T S : Type} (r r' : T -> S -> bool) : r =2 r' -> allrel r =2 allrel r'.

Proof. by move=> rr' xs ys; apply/eq_in_allrel/all_predT/all_predT. Qed.

Lemma

eq_allrel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predT", "allrel", "apply", "eq_in_allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrelC {T S : Type} (r : T -> S -> bool) xs ys : allrel r xs ys = allrel (fun y => r^~ y) ys xs.

Proof. by elim: xs => [|x xs ih]; [elim: ys | rewrite allrel_consr -ih]. Qed.

Lemma

allrelC

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrel_consr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_mapl {T T' S : Type} (f : T' -> T) (r : T -> S -> bool) xs ys : allrel r (map f xs) ys = allrel (fun x => r (f x)) xs ys.

Proof. exact: all_map. Qed.

Lemma

allrel_mapl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "T'", "all_map", "allrel", "map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_mapr {T S S' : Type} (f : S' -> S) (r : T -> S -> bool) xs ys : allrel r xs (map f ys) = allrel (fun x y => r x (f y)) xs ys.

Proof. by rewrite allrelC allrel_mapl allrelC. Qed.

Lemma

allrel_mapr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrelC", "allrel_mapl", "map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrelP {T S : eqType} {r : T -> S -> bool} {xs ys} : reflect {in xs & ys, forall x y, r x y} (allrel r xs ys).

Proof. by rewrite allrel_allpairsE; exact: all_allpairsP. Qed.

Lemma

allrelP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_allpairsP", "allrel", "allrel_allpairsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrelT {T S : Type} (xs : seq T) (ys : seq S) : allrel (fun _ _ => true) xs ys = true.

Proof. by elim: xs => //= ? ?; rewrite allrel_consl all_predT. Qed.

Lemma

allrelT

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predT", "allrel", "allrel_consl", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_relI {T S : Type} (r r' : T -> S -> bool) xs ys : allrel (fun x y => r x y && r' x y) xs ys = allrel r xs ys && allrel r' xs ys.

Proof. by rewrite -all_predI; apply: eq_all => ?; rewrite /= -all_predI. Qed.

Lemma

allrel_relI

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predI", "allrel", "apply", "eq_all" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_revl {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r (rev s1) s2 = allrel r s1 s2.

Proof. exact: all_rev. Qed.

Lemma

allrel_revl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_rev", "allrel", "rev", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_revr {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r s1 (rev s2) = allrel r s1 s2.

Proof. by rewrite allrelC allrel_revl allrelC. Qed.

Lemma

allrel_revr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrelC", "allrel_revl", "rev", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

allrel_rev2 {T S : Type} (r : T -> S -> bool) (s1 : seq T) (s2 : seq S) : allrel r (rev s1) (rev s2) = allrel r s1 s2.

Proof. by rewrite allrel_revr allrel_revl. Qed.

Lemma

allrel_rev2

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrel_revl", "allrel_revr", "rev", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_allrel_meml {T : eqType} {S} (r : T -> S -> bool) (s1 s1' : seq T) s2 : s1 =i s1' -> allrel r s1 s2 = allrel r s1' s2.

Proof. by move=> eqs1; apply: eq_all_r. Qed.

Lemma

eq_allrel_meml

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "apply", "eq_all_r", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_allrel_memr {T} {S : eqType} (r : T -> S -> bool) s1 (s2 s2' : seq S) : s2 =i s2' -> allrel r s1 s2 = allrel r s1 s2'.

Proof. by rewrite ![allrel _ s1 _]allrelC; apply: eq_allrel_meml. Qed.

Lemma

eq_allrel_memr

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "allrelC", "apply", "eq_allrel_meml", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_allrel_mem2 {T S : eqType} (r : T -> S -> bool) (s1 s1' : seq T) (s2 s2' : seq S) : s1 =i s1' -> s2 =i s2' -> allrel r s1 s2 = allrel r s1' s2'.

Proof. by move=> /eq_allrel_meml -> /eq_allrel_memr ->. Qed.

Lemma

eq_allrel_mem2

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allrel", "eq_allrel_meml", "eq_allrel_memr", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(rsym : symmetric r).

Hypothesis

rsym

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

all2rel1 x : all2rel r [:: x] = r x x.

Proof. by rewrite /allrel /= !andbT. Qed.

Lemma

all2rel1

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all2rel", "allrel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

all2rel2 x y : all2rel r [:: x; y] = r x x && r y y && r x y.

Proof. by rewrite /allrel /= rsym; do 3 case: r. Qed.

Lemma

all2rel2

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all2rel", "allrel", "rsym" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

all2rel_cons x xs : all2rel r (x :: xs) = [&& r x x, all (r x) xs & all2rel r xs].

Proof. rewrite allrel_cons2; congr andb; rewrite andbA -all_predI; congr andb. by elim: xs => //= y xs ->; rewrite rsym andbb. Qed.

Lemma

all2rel_cons

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "all2rel", "all_predI", "allrel_cons2", "rsym" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise xs : bool

:= if xs is x :: xs then all (r x) xs && pairwise xs else true.

Fixpoint

pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_cons x xs : pairwise (x :: xs) = all (r x) xs && pairwise xs.

Proof. by []. Qed.

Lemma

pairwise_cons

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_cat xs ys : pairwise (xs ++ ys) = [&& allrel r xs ys, pairwise xs & pairwise ys].

Proof. by elim: xs => //= x xs ->; rewrite all_cat -!andbA; bool_congr. Qed.

Lemma

pairwise_cat

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_cat", "allrel", "pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_rcons xs x : pairwise (rcons xs x) = all (r^~ x) xs && pairwise xs.

Proof. by rewrite -cats1 pairwise_cat allrel1r andbT. Qed.

Lemma

pairwise_rcons

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allrel1r", "cats1", "pairwise", "pairwise_cat", "rcons" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise2 x y : pairwise [:: x; y] = r x y.

Proof. by rewrite /= !andbT. Qed.

Lemma

pairwise2

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_mask m xs : pairwise xs -> pairwise (mask m xs).

Proof. by elim: m xs => [|[] m IHm] [|x xs] //= /andP [? ?]; rewrite ?IHm // all_mask. Qed.

Lemma

pairwise_mask

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_mask", "mask", "pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_filter a xs : pairwise xs -> pairwise (filter a xs).

Proof. by rewrite filter_mask; apply: pairwise_mask. Qed.

Lemma

pairwise_filter

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "filter", "filter_mask", "pairwise", "pairwise_mask" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwiseP x0 xs : reflect {in gtn (size xs) &, {homo nth x0 xs : i j / i < j >-> r i j}} (pairwise xs).

Proof. elim: xs => /= [|x xs IHxs]; first exact: (iffP idP). apply: (iffP andP) => [[r_x_xs pxs] i j|Hnth]; rewrite -?topredE /= ?ltnS. by case: i j => [|i] [|j] //= gti gtj ij; [exact/all_nthP | exact/IHxs]. split; last by apply/IHxs => // i j; apply/(Hnth i.+1 j.+1). by apply/(all_nthP x0) => i gti; apply/(Hnth 0 i...

Lemma

pairwiseP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_nthP", "apply", "gtn", "last", "ltnS", "nth", "pairwise", "size", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_all2rel : reflexive r -> symmetric r -> forall xs, pairwise xs = all2rel r xs.

Proof. by move=> r_refl r_sym; elim => //= x xs ->; rewrite all2rel_cons // r_refl. Qed.

Lemma

pairwise_all2rel

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all2rel", "all2rel_cons", "pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_in_pairwise {T : Type} (P : {pred T}) (r r' : rel T) : {in P &, subrel r r'} -> forall xs, all P xs -> pairwise r xs -> pairwise r' xs.

Proof. move=> rr'; elim=> //= x xs IHxs /andP [Px Pxs] /andP [+ {}/IHxs->] //. rewrite andbT; elim: xs Pxs => //= x' xs IHxs /andP [? ?] /andP [+ /IHxs->] //. by rewrite andbT; apply: rr'. Qed.

Lemma

sub_in_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "Px", "all", "apply", "pairwise", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_pairwise {T : Type} (r r' : rel T) xs : subrel r r' -> pairwise r xs -> pairwise r' xs.

Proof. by move=> rr'; apply/sub_in_pairwise/all_predT; apply: in2W. Qed.

Lemma

sub_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predT", "apply", "pairwise", "rel", "sub_in_pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_pairwise {T : Type} (P : {pred T}) (r r' : rel T) : {in P &, r =2 r'} -> forall xs, all P xs -> pairwise r xs = pairwise r' xs.

Proof. move=> rr' xs Pxs. by apply/idP/idP; apply/sub_in_pairwise/Pxs => ? ? ? ?; rewrite rr'. Qed.

Lemma

eq_in_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "apply", "pairwise", "rel", "sub_in_pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_pairwise {T : Type} (r r' : rel T) : r =2 r' -> pairwise r =i pairwise r'.

Proof. by move=> rr' xs; apply/eq_in_pairwise/all_predT. Qed.

Lemma

eq_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predT", "apply", "eq_in_pairwise", "pairwise", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_map {T T' : Type} (f : T' -> T) (r : rel T) xs : pairwise r (map f xs) = pairwise (relpre f r) xs.

Proof. by elim: xs => //= x xs ->; rewrite all_map. Qed.

Lemma

pairwise_map

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "T'", "all_map", "map", "pairwise", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_relI {T : Type} (r r' : rel T) (s : seq T) : pairwise [rel x y | r x y && r' x y] s = pairwise r s && pairwise r' s.

Proof. by elim: s => //= x s ->; rewrite andbACA all_predI. Qed.

Lemma

pairwise_relI

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predI", "pairwise", "rel", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subseq_pairwise xs ys : subseq xs ys -> pairwise r ys -> pairwise r xs.

Proof. by case/subseqP => m _ ->; apply: pairwise_mask. Qed.

Lemma

subseq_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "pairwise", "pairwise_mask", "subseq", "subseqP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

uniq_pairwise xs : uniq xs = pairwise [rel x y | x != y] xs.

Proof. elim: xs => //= x xs ->; congr andb; rewrite -has_pred1 -all_predC. by elim: xs => //= x' xs ->; case: eqVneq. Qed.

Lemma

uniq_pairwise

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all_predC", "eqVneq", "has_pred1", "pairwise", "rel", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_uniq xs : irreflexive r -> pairwise r xs -> uniq xs.

Proof. move=> r_irr; rewrite uniq_pairwise; apply/sub_pairwise => x y. by apply: contraTneq => ->; rewrite r_irr. Qed.

Lemma

pairwise_uniq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "contraTneq", "pairwise", "sub_pairwise", "uniq", "uniq_pairwise" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_eq : antisymmetric r -> forall xs ys, pairwise r xs -> pairwise r ys -> perm_eq xs ys -> xs = ys.

Proof. move=> r_asym; elim=> [|x xs IHxs] [|y ys] //=; try by move=> ? ? /perm_size. move=> /andP [r_x_xs pxs] /andP [r_y_ys pys] eq_xs_ys. move: (mem_head y ys) (mem_head x xs). rewrite -(perm_mem eq_xs_ys) [x \in _](perm_mem eq_xs_ys) !inE. case: eqVneq eq_xs_ys => /= [->|ne_xy] eq_xs_ys ys_x xs_y. by rewrite (IHxs...

Lemma

pairwise_eq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allP", "apply", "eqVneq", "inE", "mem_head", "pairwise", "perm_cons", "perm_eq", "perm_mem", "perm_size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairwise_trans s : antisymmetric r -> pairwise r s -> {in s & &, transitive r}.

Proof. move=> /(_ _ _ _)/eqP r_anti + y x z => /pairwiseP-/(_ y) ltP ys xs zs. have [-> //|neqxy] := eqVneq x y; have [-> //|neqzy] := eqVneq z y. move=> lxy lyz; move: ys xs zs lxy neqxy lyz neqzy. move=> /(nthP y)[j jlt <-] /(nthP y)[i ilt <-] /(nthP y)[k klt <-]. have [ltij|ltji|->] := ltngtP i j; last 2 first. - by...

Lemma

pairwise_trans

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "eqVneq", "last", "ltP", "ltn_trans", "ltngtP", "nthP", "pairwise", "pairwiseP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

incr_tally bs x

:= if bs isn't b :: bs then [:: (x, 1)] else if x == b.1 then (x, b.2.+1) :: bs else b :: incr_tally bs x.

Fixpoint

incr_tally

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tally s

:= foldl incr_tally [::] s.

Definition

tally

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "foldl", "incr_tally" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

wf_tally

:= [qualify a bs : seq (T * nat) | uniq (unzip1 bs) && (0 \notin unzip2 bs)].

Definition

wf_tally

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "nat", "seq", "uniq", "unzip1", "unzip2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tally_seq bs

:= flatten [seq nseq b.2 b.1 | b <- bs].

Definition

tally_seq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "flatten", "nseq", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tseq

:= tally_seq.

Notation

tseq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "tally_seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_tally_seq bs : size (tally_seq bs) = sumn (unzip2 bs).

Proof. by rewrite size_flatten /shape -map_comp; under eq_map do rewrite /= size_nseq. Qed.

Lemma

size_tally_seq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "eq_map", "map_comp", "shape", "size", "size_flatten", "size_nseq", "sumn", "tally_seq", "unzip2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tally_seqK : {in wf_tally, cancel tally_seq tally}.

Proof. move=> bs /andP[]; elim: bs => [|[x [|n]] bs IHbs] //= /andP[bs'x Ubs] bs'0. rewrite inE /tseq /tally /= -[n.+1]addn1 in bs'0 *. elim: n 1 => /= [|n IHn] m; last by rewrite eqxx IHn addnS. rewrite -{}[in RHS]IHbs {Ubs bs'0}// /tally /tally_seq add0n. elim: bs bs'x [::] => [|[y n] bs IHbs] //= /[1!inE] /norP[y'x ...

Lemma

tally_seqK

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "add0n", "addn1", "addnS", "eq_sym", "eqxx", "inE", "last", "tally", "tally_seq", "tseq", "wf_tally" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

incr_tallyP x : {homo incr_tally^~ x : bs / bs \in wf_tally}.

Proof. move=> bs /andP[]; rewrite unfold_in. elim: bs => [|[y [|n]] bs IHbs] //= /andP[bs'y Ubs] /[1!inE] /= bs'0. have [<- | y'x] /= := eqVneq y; first by rewrite bs'y Ubs. rewrite -andbA {}IHbs {Ubs bs'0}// andbT. elim: bs bs'y => [|b bs IHbs] /=; rewrite inE ?y'x // => /norP[b'y bs'y]. by case: ifP => _; rewrite /= ...

Lemma

incr_tallyP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "eqVneq", "inE", "incr_tally", "wf_tally" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tallyP s : tally s \is a wf_tally.

Proof. rewrite /tally; set bs := [::]; have: bs \in wf_tally by []. by elim: s bs => //= x s IHs bs /(incr_tallyP x)/IHs. Qed.

Lemma

tallyP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "incr_tallyP", "tally", "wf_tally" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tallyK s : perm_eq (tally_seq (tally s)) s.

Proof. rewrite -[s in perm_eq _ s]cats0 -[nil]/(tseq [::]) /tally. elim: s [::] => //= x s IHs bs; rewrite {IHs}(permPl (IHs _)). rewrite perm_sym -cat1s perm_catCA {s}perm_cat2l. elim: bs => //= b bs IHbs; case: eqP => [-> | _] //=. by rewrite -cat1s perm_catCA perm_cat2l. Qed.

Lemma

tallyK

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "cat1s", "cats0", "permPl", "perm_cat2l", "perm_catCA", "perm_eq", "perm_sym", "tally", "tally_seq", "tseq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tallyEl s : perm_eq (unzip1 (tally s)) (undup s).

Proof. have /andP[Ubs bs'0] := tallyP s; set bs := tally s in Ubs bs'0 *. rewrite uniq_perm ?undup_uniq {Ubs}// => x. rewrite mem_undup -(perm_mem (tallyK s)) -/bs. elim: bs => [|[y [|m]] bs IHbs] //= in bs'0 *. by rewrite inE IHbs // mem_cat mem_nseq. Qed.

Lemma

tallyEl

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "inE", "mem_cat", "mem_nseq", "mem_undup", "perm_eq", "perm_mem", "tally", "tallyK", "tallyP", "undup", "undup_uniq", "uniq_perm", "unzip1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tallyE s : perm_eq (tally s) [seq (x, count_mem x s) | x <- undup s].

Proof. have /andP[Ubs _] := tallyP s; pose b := [fun s x => (x, count_mem x (tseq s))]. suffices /permPl->: perm_eq (tally s) (map (b (tally s)) (unzip1 (tally s))). congr perm_eq: (perm_map (b (tally s)) (tallyEl s)). by under eq_map do rewrite /= (permP (tallyK s)). elim: (tally s) Ubs => [|[x m] bs IH] //= /andP...

Lemma

tallyE

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "addn0", "apply", "count_cat", "count_mem", "count_memPn", "count_nseq", "eq_in_map", "eq_map", "eqxx", "inE", "map", "memPnC", "mem_cat", "mem_nseq", "mul1n", "permP", "permPl", "perm_cons", "perm_eq", "perm_map", "seq", "tally", "tallyEl", "tallyK", "tallyP", "tse...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_tally s1 s2 : perm_eq s1 s2 -> perm_eq (tally s1) (tally s2).

Proof. move=> eq_s12; apply: (@perm_trans _ [seq (x, count_mem x s2) | x <- undup s1]). by congr perm_eq: (tallyE s1); under eq_map do rewrite (permP eq_s12). by rewrite (permPr (tallyE s2)); apply/perm_map/perm_undup/(perm_mem eq_s12). Qed.

Lemma

perm_tally

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "count_mem", "eq_map", "permP", "permPr", "perm_eq", "perm_map", "perm_mem", "perm_trans", "perm_undup", "s1", "s2", "seq", "tally", "tallyE", "undup" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_tally_seq bs1 bs2 : perm_eq bs1 bs2 -> perm_eq (tally_seq bs1) (tally_seq bs2).

Proof. by move=> Ebs12; rewrite perm_flatten ?perm_map. Qed.

Lemma

perm_tally_seq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "perm_eq", "perm_flatten", "perm_map", "tally_seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_tseq

:= perm_tally_seq.

Notation

perm_tseq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "perm_tally_seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_count_undup s : perm_eq (flatten [seq nseq (count_mem x s) x | x <- undup s]) s.

Proof. by rewrite -(permPr (tallyK s)) (permPr (perm_tseq (tallyE s))) /tseq -map_comp. Qed.

Lemma

perm_count_undup

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "count_mem", "flatten", "map_comp", "nseq", "permPr", "perm_eq", "perm_tseq", "seq", "tallyE", "tallyK", "tseq", "undup" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cons_perms_ perms_rec (s : seq T) bs bs2 acc

:= if bs isn't b :: bs1 then acc else if b isn't (x, m.+1) then cons_perms_ perms_rec s bs1 bs2 acc else let acc_xs := perms_rec (x :: s) ((x, m) :: bs1 ++ bs2) acc in cons_perms_ perms_rec s bs1 (b :: bs2) acc_xs.

Fixpoint

cons_perms_

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "perms_rec", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perms_rec n s bs acc

:= if n isn't n.+1 then s :: acc else cons_perms_ (perms_rec n) s bs [::] acc.

Fixpoint

perms_rec

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "cons_perms_" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cons_perms n

:= (cons_perms_ (perms_rec n) [::]).

Notation

cons_perms

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "cons_perms_", "perms_rec" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permutations s

:= perms_rec (size s) [::] (tally s) [::].

Definition

permutations

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "perms_rec", "size", "tally" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permsP s : exists n bs, [/\ permutations s = perms_rec n [::] bs [::], size (tseq bs) == n, perm_eq (tseq bs) s & uniq (unzip1 bs)].

Proof. have /andP[Ubs _] := tallyP s; exists (size s), (tally s). by rewrite (perm_size (tallyK s)) tallyK. Qed.

Let

permsP

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "perm_eq", "perm_size", "perms_rec", "permutations", "size", "tally", "tallyK", "tallyP", "tseq", "uniq", "unzip1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bsCA

:= (permEl (perm_catCA _ [:: _] _)).

Notation

bsCA

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "permEl", "perm_catCA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cons_permsE : forall n x bs bs1 bs2, let cp := cons_perms n bs bs2 in let perms s := perms_rec n s bs1 [::] in cp (perms [:: x]) = cp [::] ++ [seq rcons t x | t <- perms [::]].

Proof. pose is_acc f := forall acc, f acc = f [::] ++ acc. (* f is accumulating. *) have cpE: forall f & forall s bs, is_acc (f s bs), is_acc (cons_perms_ f _ _ _). move=> s bs bs2 f fE acc; elim: bs => [|[x [|m]] bs IHbs] //= in s bs2 acc *. by rewrite fE IHbs catA -IHbs. have prE: is_acc (perms_rec _ _ _) by elim...

Let

cons_permsE

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "catA", "cats1", "cons_perms", "cons_perms_", "cp", "eq_map", "fE", "map_cat", "map_comp", "perms", "perms_rec", "rcons", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_permutations s t : (t \in permutations s) = perm_eq t s.

Proof. have{s} [n [bs [-> Dn /permPr<- _]]] := permsP s. elim: n => [|n IHn] /= in t bs Dn *. by rewrite inE (nilP Dn); apply/eqP/perm_nilP. rewrite -[bs in tseq bs]cats0 in Dn *; have x0 : T by case: (tseq _) Dn. rewrite -[RHS](@andb_idl (last x0 t \in tseq bs)). case/lastP: t {IHn} => [|t x] Dt; first by rewrite ...

Lemma

mem_permutations

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "bsCA", "cat_cons", "cats0", "cons_permsE", "inE", "last", "lastP", "last_rcons", "mapP", "mem_cat", "mem_head", "mem_nseq", "mem_rcons", "nilP", "permPr", "perm_cons", "perm_eq", "perm_mem", "perm_nilP", "perm_rcons", "perm_size", "perm_tseq", "permsP", "per...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permutations_uniq s : uniq (permutations s).

Proof. have{s} [n [bs [-> Dn _ Ubs]]] := permsP s. elim: n => //= n IHn in bs Dn Ubs *; rewrite -[bs]cats0 /unzip1 in Dn Ubs. elim: bs [::] => [|[x [|m]] bs IHbs] //= bs2 in Dn Ubs *. by case/andP: Ubs => _ /IHbs->. rewrite /= cons_permsE cat_uniq has_sym andbCA andbC. rewrite {}IHbs; first 1 last; last by rewrite (p...

Lemma

permutations_uniq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "bsCA", "cat_uniq", "cats0", "cons_permsE", "hasPn", "has_cat", "has_map", "has_sym", "inE", "last", "mapP", "map_cat", "map_inj_uniq", "mem_cat", "perm_map", "perm_size", "perm_tseq", "perm_uniq", "permsP", "perms_rec", "permutations", "rcons_inj", "rcons_injl...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perms

:= permutations.

Notation

perms

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "permutations" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permutationsE s : 0 < size s -> perm_eq (perms s) [seq x :: t | x <- undup s, t <- perms (rem x s)].

Proof. move=> nt_s; apply/uniq_perm=> [||t]; first exact: permutations_uniq. apply/allpairs_uniq_dep=> [|x _|]; rewrite ?undup_uniq ?permutations_uniq //. by case=> [_ _] [x t] _ _ [-> ->]. rewrite mem_permutations; apply/idP/allpairsPdep=> [Dt | [x [t1 []]]]. rewrite -(perm_size Dt) in nt_s; case: t nt_s => // ...

Lemma

permutationsE

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "allpairsPdep", "allpairs_uniq_dep", "apply", "mem_head", "mem_permutations", "mem_undup", "permPl", "perm_cons", "perm_eq", "perm_mem", "perm_size", "perm_sym", "perm_to_rem", "perms", "permutations_uniq", "rem", "seq", "size", "split", "undup", "undup_uniq", "uniq_perm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permutationsErot x s (le_x := fun t => iota 0 (index x t + 1)) : perm_eq (perms (x :: s)) [seq rot i (x :: t) | t <- perms s, i <- le_x t].

Proof. have take'x t i: i <= index x t -> i <= size t /\ x \notin take i t. move=> le_i_x; have le_i_t: i <= size t := leq_trans le_i_x (index_size x t). case: (nthP x) => // -[j lt_j_i /eqP]; rewrite size_takel // in lt_j_i. by rewrite nth_take // [_ == _](before_find x (leq_trans lt_j_i le_i_x)). pose xrot t i ...

Lemma

permutationsErot

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "addn0", "addn1", "allpairsPdep", "allpairs_uniq_dep", "apply", "before_find", "cat_cons", "cat_rcons", "drop", "drop_nth", "eqxx", "index", "index_cat", "index_mem", "index_size", "iota", "iota_uniq", "leq_addr", "leq_trans", "leqnn", "ltnS", "mem_head", "mem_iota", "m...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_permutations s : uniq s -> size (permutations s) = (size s)`!.

Proof. move Dn: (size s) => n Us; elim: n s => [[]|n IHn s] //= in Dn Us *. rewrite (perm_size (permutationsE _)) ?Dn // undup_id // factS -Dn. rewrite -(size_iota 0 n`!) -(size_allpairs (fun=>id)) !size_allpairs_dep. by apply/congr1/eq_in_map=> x sx; rewrite size_iota IHn ?size_rem ?Dn ?rem_uniq. Qed.

Lemma

size_permutations

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "eq_in_map", "factS", "id", "perm_size", "permutations", "permutationsE", "rem_uniq", "size", "size_allpairs", "size_allpairs_dep", "size_iota", "size_rem", "undup_id", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permutations_all_uniq s : uniq s -> all uniq (permutations s).

Proof. by move=> Us; apply/allP=> t; rewrite mem_permutations => /perm_uniq->. Qed.

Lemma

permutations_all_uniq

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "all", "allP", "apply", "mem_permutations", "perm_uniq", "permutations", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_permutations s t : perm_eq s t -> perm_eq (permutations s) (permutations t).

Proof. move=> Est; apply/uniq_perm; try exact: permutations_uniq. by move=> u; rewrite !mem_permutations (permPr Est). Qed.

Lemma

perm_permutations

boot

boot/seq.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat" ]

[ "apply", "mem_permutations", "permPr", "perm_eq", "permutations", "permutations_uniq", "uniq_perm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

change_type ty ty' (x : ty) (strategy : ty = ty') : ty'

:= ecast ty ty strategy x.

Definition

change_type

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

simplrefl

:= (ltac: (simpl; reflexivity)) (only parsing).

Notation

simplrefl

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cbvrefl

:= (ltac: (cbv; reflexivity)) (only parsing).

Notation

cbvrefl

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vmrefl

:= (ltac: (vm_compute; reflexivity)) (only parsing).

Notation

vmrefl

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ten

:= (xO (xI (xO xH))).

Notation

ten

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[]

** Conversion with a decimal representation for printing/parsing

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

of_uint_acc (d:Decimal.uint) (acc:positive)

:= match d with | Decimal.Nil => acc | Decimal.D0 l => of_uint_acc l (mul ten acc) | Decimal.D1 l => of_uint_acc l (add 1 (mul ten acc)) | Decimal.D2 l => of_uint_acc l (add 1~0 (mul ten acc)) | Decimal.D3 l => of_uint_acc l (add 1~1 (mul ten acc)) | Decimal.D4 l => of_uint_acc l (add 1~0~0 (mul ten acc))...

Fixpoint

of_uint_acc

boot

boot/ssrAC.v

[ "HB", "structures", "Corelib", "PosDef", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnat", "eqtype", "seq", "bigop", "Pos", "AC.Syntax", "AC.Exports" ]

[ "Nil", "add", "mul", "ten" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d