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Coq-MathComp

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complEsubset
:= @complEsubset.
Definition
complEsubset
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mono_unique d (T T' : finPOrderType d) (f g : T -> T') : total (<=%O : rel T) -> (#|T'| <= #|T|)%N -> {mono f : x y / x <= y} -> {mono g : x y / x <= y} -> f =1 g.
Proof. move=> le_total leT'T lef leg x0; move: {+}x0. suff: finfun f = finfun g by move=> /ffunP + x => /(_ x); rewrite !ffunE. apply: (can_inj fgraphK); apply/val_inj => /=; rewrite !codomE. under eq_map do rewrite ffunE; under [RHS]eq_map do rewrite ffunE. have [finj ginj] := (inc_inj lef, inc_inj leg). have [f' fK f...
Lemma
mono_unique
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "T'", "all_map", "apply", "codomE", "enum", "enum_uniq", "eq_map", "eq_map_all", "fK", "ffunE", "ffunP", "fgraphK", "gK", "inc_inj", "inj_card_bij", "le_anti", "le_total", "le_trans", "mapP", "map_inj_uniq", "mem_enum", "mono_sorted_enum", "permPl", "perm_all", "perm_...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
(leT_total : total (<=%O : rel T)).
Hypothesis
leT_total
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "rel", "total" ]
which requires the type to be nonempty and we do not want to rule this out
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_val A : {mono @enum_val _ _ A : i j / i <= j}.
Proof. apply: le_mono => i j le_ij. rewrite /enum_val (set_nth_default (enum_default j)) -?cardE//. apply: (sorted_ltn_nth lt_trans); rewrite -?topredE/= -?cardE//. by rewrite lt_sorted_uniq_le enum_uniq/= sort_sorted. Qed.
Lemma
le_enum_val
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "cardE", "enum_default", "enum_uniq", "enum_val", "le_mono", "lt_sorted_uniq_le", "lt_trans", "set_nth_default", "sort_sorted", "sorted_ltn_nth" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_rank_in x0 A (Ax0 : x0 \in A) : {in A &, {mono enum_rank_in Ax0 : x y / x <= y}}.
Proof. apply: can_mono_in (@in2W _ _ predT predT _ (@le_enum_val A)) => //. exact/onW_can_in/enum_rankK_in. Qed.
Lemma
le_enum_rank_in
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "enum_rankK_in", "enum_rank_in", "le_enum_val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_rank : {mono @enum_rank d T : i j / i <= j}.
Proof. exact: (can_mono (@enum_rankK _ _) (@le_enum_val predT)). Qed.
Lemma
le_enum_rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "enum_rank", "enum_rankK", "le_enum_val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_val
:= le_enum_val.
Notation
le_enum_val
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_rank_in
:= le_enum_rank_in.
Notation
le_enum_rank_in
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_enum_rank
:= le_enum_rank.
Notation
le_enum_rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordsum
:= 'I_(\sum_i p_ i)%N.
Notation
ordsum
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
T
:= {i & 'I_(p_ i)}.
Notation
T
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card : #|{: T}| = \sum_i p_ i.
Proof. rewrite card_tagged sumnE/= big_map big_enum. by apply: eq_bigr => i _; rewrite card_ord. Qed.
Lemma
card
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "big_enum", "big_map", "card_ord", "card_tagged", "eq_bigr", "sumnE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig : ordsum -> T
:= enum_val \o cast_ord (esym card).
Definition
sig
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "card", "cast_ord", "enum_val", "ordsum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank : T -> ordsum
:= cast_ord card \o enum_rank.
Definition
rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "card", "cast_ord", "enum_rank", "ordsum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sigK : cancel sig rank.
Proof. by move=> s; rewrite /sig/rank/= enum_valK cast_ord_comp cast_ord_id. Qed.
Lemma
sigK
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "cast_ord_comp", "cast_ord_id", "enum_valK", "rank", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig_inj : injective sig.
Proof. exact: can_inj sigK. Qed.
Lemma
sig_inj
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "sig", "sigK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rankK : cancel rank sig.
Proof. by move=> p; rewrite /sig/rank/= cast_ord_comp cast_ord_id enum_rankK. Qed.
Lemma
rankK
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "cast_ord_comp", "cast_ord_id", "enum_rankK", "rank", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_inj : injective rank.
Proof. exact: can_inj rankK. Qed.
Lemma
rank_inj
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "rank", "rankK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig1 s : 'I_n
:= tag (sig s).
Definition
sig1
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig2 s : 'I_(p_ (sig1 s))
:= tagged (sig s).
Definition
sig2
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "sig", "sig1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rank i (j : 'I_(p_ i))
:= rank (Tagged _ j).
Definition
Rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "rank" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sigE12 s : sig s = @Tagged _ (sig1 s) _ (sig2 s).
Proof. by rewrite /sig1 /sig2; case: sig. Qed.
Lemma
sigE12
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "sig", "sig1", "sig2" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rankE p : rank p = @Rank (tag p) (tagged p).
Proof. by case: p. Qed.
Lemma
rankE
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "rank" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig2K s : Rank (sig2 s) = s.
Proof. by rewrite -rankE sigK. Qed.
Lemma
sig2K
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "rankE", "sig2", "sigK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rank1K i0 (k : 'I_(p_ i0)) : sig1 (Rank k) = i0.
Proof. by rewrite /sig1 /Rank/= rankK/=. Qed.
Lemma
Rank1K
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "i0", "rankK", "sig1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rank2K i0 (k : 'I_(p_ i0)) : sig2 (Rank k) = cast_ord (congr1 p_ (esym (Rank1K k))) k.
Proof. by apply: val_inj; rewrite /sig2/sig1/Rank/= rankK. Qed.
Lemma
Rank2K
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "Rank1K", "apply", "cast_ord", "i0", "rankK", "sig1", "sig2", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_bij : bijective rank.
Proof. by exists sig. Qed.
Lemma
rank_bij
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "rank", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig_bij : bijective sig.
Proof. by exists rank. Qed.
Lemma
sig_bij
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "rank", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_bij_on : {on [pred _ | true], bijective rank}.
Proof. exact/onW_bij/rank_bij. Qed.
Lemma
rank_bij_on
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "on", "rank", "rank_bij" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sig_bij_on : {on [pred _ | true], bijective sig}.
Proof. exact/onW_bij/sig_bij. Qed.
Lemma
sig_bij_on
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "on", "sig", "sig_bij" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_sig : {mono sig : i j / i <= j}.
Proof. by move=> i j; rewrite /sig/= le_enum_val//; apply: le_total. Qed.
Lemma
le_sig
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "le_enum_val", "le_total", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_sig1 : {homo sig1 : i j / i <= j}.
Proof. by move=> i j; rewrite /sig1/= -le_sig leEsig/=; case: leP. Qed.
Lemma
le_sig1
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leEsig", "leP", "le_sig", "sig1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_rank : {mono rank : p q / p <= q}.
Proof. exact: can_mono le_sig. Qed.
Lemma
le_rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_sig", "rank" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_Rank i : {mono @Rank i : j k / j <= k}.
Proof. by move=> j k; rewrite /Rank le_rank/= leEsig/= tagged_asE lexx. Qed.
Lemma
le_Rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "leEsig", "le_rank", "lexx", "tagged_asE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_sig : {mono sig : i j / i < j}.
Proof. by move=> i j; rewrite !ltNge le_sig. Qed.
Lemma
lt_sig
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_sig", "ltNge", "sig" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_rank : {mono rank : p q / p < q}.
Proof. by move=> p q; rewrite !ltNge le_rank. Qed.
Lemma
lt_rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_rank", "ltNge", "rank" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_Rank i : {mono @Rank i : j k / j < k}.
Proof. by move=> j k; rewrite !ltNge le_Rank. Qed.
Lemma
lt_Rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "le_Rank", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_Rank i i' (j : 'I_(p_ i)) (j': 'I_(p_ i')) : (Rank j == Rank j' :> nat) = (i == i') && (j == j' :> nat).
Proof. rewrite val_eqE /Rank -(can_eq sigK) !rankK. case: (i =P i') => ii' /=; last by case: eqVneq => // -[]. by case: _ / ii' in j' *; rewrite eq_Tagged. Qed.
Lemma
eq_Rank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "can_eq", "eqVneq", "eq_Tagged", "last", "nat", "rankK", "sigK", "val_eqE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rankEsum p : rank p = \sum_(i < n | (i < tag p)%N) p_ i + tagged p :> nat.
Proof. pose sum p := \sum_(i < n | (i < tag p)%N) p_ i + tagged p. rewrite -/(sum _); have sumlt : forall p, (sum p < \sum_i p_ i)%N. rewrite /sum => -[/= i j]. rewrite [ltnRHS](bigID [pred i' : 'I__ | (i' < i)%N])/= ltn_add2l. by rewrite (bigD1 i) ?ltnn//= ltn_addr. suff: rank =1 (fun p => Ordinal (sumlt p)) by ...
Lemma
rankEsum
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "bigD1", "bigID", "card", "card_ord", "eq_bigl", "i0", "last", "leEord", "le_mono", "le_rank", "le_total", "leqRHS", "leq_add", "leq_addr", "leq_trans", "ltEord", "ltEsig", "ltnP", "ltnRHS", "ltn_add2l", "ltn_addr", "ltn_trans", "ltngtP", "ltnn", "mono_uniq...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
RankEsum i j : @Rank i j = \sum_(k < n | (k < i)%N) p_ k + j :> nat.
Proof. by rewrite /Rank rankEsum/=. Qed.
Lemma
RankEsum
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "nat", "rankEsum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rect s : s = \sum_(i < n | (i < sig1 s)%N) p_ i + sig2 s :> nat.
Proof. by rewrite -[s]sigK rankEsum /= sigK. Qed.
Lemma
rect
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "nat", "rankEsum", "sig1", "sig2", "sigK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqRank (i0 j : nat) (li0 : (i0 < n)%N) (lj : (j < p_ (Ordinal li0))%N) : (\sum_(i < n | (i < i0)%N) p_ i) + j = Rank (Ordinal lj) :> nat.
Proof. by rewrite RankEsum. Qed.
Lemma
eqRank
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "Rank", "RankEsum", "i0", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
disp_t
:= Disp {d1 : unit; d2 : unit}.
Record
disp_t
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y"
:= (le x y) : order_scope.
Notation
x <= y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y"
:= (lt x y) : order_scope.
Notation
x < y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable : rel T
:= fun (x y : T) => (x <= y) || (y <= x).
Definition
comparable
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "rel" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >=< y"
:= (comparable x y) : order_scope.
Notation
x >=< y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >< y"
:= (~~ (x >=< y)) : order_scope.
Notation
x >< y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge : simpl_rel T
:= [rel x y | y <= x].
Definition
ge
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "rel" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt : simpl_rel T
:= [rel x y | y < x].
Definition
gt
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "rel" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leif (x y : T) C : Prop
:= ((x <= y) * ((x == y) = C))%type.
Definition
leif
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_of_leif x y C (le_xy : @leif x y C)
:= le_xy.1 : le x y.
Definition
le_of_leif
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "le", "leif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif (x y : T) C
:= if C then x <= y else x < y.
Definition
lteif
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_xor_gt (x y : T) : T -> T -> T -> T -> bool -> bool -> Set
:= | LeNotGt of x <= y : le_xor_gt x y x x y y true false | GtNotLe of y < x : le_xor_gt x y y y x x false true.
Variant
le_xor_gt
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_xor_ge (x y : T) : T -> T -> T -> T -> bool -> bool -> Set
:= | LtNotGe of x < y : lt_xor_ge x y x x y y false true | GeNotLt of y <= x : lt_xor_ge x y y y x x true false.
Variant
lt_xor_ge
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min (x y : T)
:= if x < y then x else y.
Definition
min
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max (x y : T)
:= if x < y then y else x.
Definition
max
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
compare (x y : T) : T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> Set
:= | CompareLt of x < y : compare x y x x y y false false false true false true | CompareGt of y < x : compare x y y y x x false false true false true false | CompareEq of x = y : compare x y x x x x true true true true false false.
Variant
compare
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
incompare (x y : T) : T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> Set
:= | InCompareLt of x < y : incompare x y x x y y false false false true false true true true | InCompareGt of y < x : incompare x y y y x x false false true false true false true true | InCompare of x >< y : incompare x y x y y x false false false false false false false false | InCompareEq of x =...
Variant
incompare
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_min {I : finType}
:= @extremum T I le.
Definition
arg_min
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "extremum", "le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_max {I : finType}
:= @extremum T I ge.
Definition
arg_max
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "extremum", "ge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_fun f g x
:= min (f x) (g x).
Definition
min_fun
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_fun f g x
:= max (f x) (g x).
Definition
max_fun
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nondecreasing disp' (T' : preorderType disp') (f : T -> T') : Prop
:= {homo f : x y / x <= y}.
Definition
nondecreasing
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "T'" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nondecreasing
:= nondecreasing.
Notation
nondecreasing
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min
:= min.
Notation
min
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max
:= max.
Notation
max
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<=%O"
:= le : function_scope.
Notation
<=%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=%O"
:= ge : function_scope.
Notation
>=%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "ge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<%O"
:= lt : function_scope.
Notation
<%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">%O"
:= gt : function_scope.
Notation
>%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "gt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<?=%O"
:= leif : function_scope.
Notation
<?=%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "leif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<?<=%O"
:= lteif : function_scope.
Notation
<?<=%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lteif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=<%O"
:= comparable : function_scope.
Notation
>=<%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"><%O"
:= (fun x y => ~~ (comparable x y)) : function_scope.
Notation
><%O
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<= y"
:= (ge y) : order_scope.
Notation
<= y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "ge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<= y :> T"
:= (<= (y : T)) (only parsing) : order_scope.
Notation
<= y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">= y"
:= (le y) : order_scope.
Notation
>= y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">= y :> T"
:= (>= (y : T)) (only parsing) : order_scope.
Notation
>= y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"< y"
:= (gt y) : order_scope.
Notation
< y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "gt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"< y :> T"
:= (< (y : T)) (only parsing) : order_scope.
Notation
< y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"> y"
:= (lt y) : order_scope.
Notation
> y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"> y :> T"
:= (> (y : T)) (only parsing) : order_scope.
Notation
> y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y :> T"
:= ((x : T) <= (y : T)) (only parsing) : order_scope.
Notation
x <= y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >= y"
:= (y <= x) (only parsing) : order_scope.
Notation
x >= y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >= y :> T"
:= ((x : T) >= (y : T)) (only parsing) : order_scope.
Notation
x >= y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y"
:= (lt x y) : order_scope.
Notation
x < y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y :> T"
:= ((x : T) < (y : T)) (only parsing) : order_scope.
Notation
x < y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x > y"
:= (y < x) (only parsing) : order_scope.
Notation
x > y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x > y :> T"
:= ((x : T) > (y : T)) (only parsing) : order_scope.
Notation
x > y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y <= z"
:= ((x <= y) && (y <= z)) : order_scope.
Notation
x <= y <= z
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y <= z"
:= ((x < y) && (y <= z)) : order_scope.
Notation
x < y <= z
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y < z"
:= ((x <= y) && (y < z)) : order_scope.
Notation
x <= y < z
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y < z"
:= ((x < y) && (y < z)) : order_scope.
Notation
x < y < z
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y ?= 'iff' C"
:= (leif x y C) : order_scope.
Notation
x <= y ?= 'iff' C
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "leif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y ?= 'iff' C :> T"
:= ((x : T) <= (y : T) ?= iff C) (only parsing) : order_scope.
Notation
x <= y ?= 'iff' C :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y ?<= 'if' C"
:= (lteif x y C) : order_scope.
Notation
x < y ?<= 'if' C
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "lteif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y ?<= 'if' C :> T"
:= ((x : T) < (y : T) ?<= if C) (only parsing) : order_scope.
Notation
x < y ?<= 'if' C :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=< y"
:= [pred x | comparable x y] : order_scope.
Notation
>=< y
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=< y :> T"
:= (>=< (y : T)) (only parsing) : order_scope.
Notation
>=< y :> T
order
order/preorder.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d