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Coq-MathComp

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le_refl : reflexive le.
Proof. by move=> x; case: (le x x) (le_total x x). Qed.
Fact
le_refl
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", "le_total" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
T'
:= T.
Definition
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
meetE x y : meet x y = x `&` y.
Proof. by rewrite meet_def. Qed.
Fact
meetE
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",...
[ "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinE x y : join x y = x `|` y.
Proof. by rewrite join_def. Qed.
Fact
joinE
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",...
[ "join" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetC : commutative meet.
Proof. by move=> *; rewrite !meetE meetC. Qed.
Fact
meetC
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",...
[ "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinC : commutative join.
Proof. by move=> *; rewrite !joinE joinC. Qed.
Fact
joinC
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",...
[ "join", "joinE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetA : associative meet.
Proof. by move=> *; rewrite !meetE meetA. Qed.
Fact
meetA
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",...
[ "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinA : associative join.
Proof. by move=> *; rewrite !joinE joinA. Qed.
Fact
joinA
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",...
[ "join", "joinE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinKI y x : meet x (join x y) = x.
Proof. by rewrite meetE joinE joinKI. Qed.
Fact
joinKI
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",...
[ "join", "joinE", "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKU y x : join x (meet x y) = x.
Proof. by rewrite meetE joinE meetKU. Qed.
Fact
meetKU
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",...
[ "join", "joinE", "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : left_distributive meet join.
Proof. by move=> *; rewrite !meetE !joinE meetUl. Qed.
Fact
meetUl
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",...
[ "join", "joinE", "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetxx : idempotent_op meet.
Proof. by move=> *; rewrite meetE meetxx. Qed.
Fact
meetxx
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",...
[ "idempotent_op", "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_def x y : (x <= y) = (meet x y == x).
Proof. by rewrite meetE (eq_meetl x y). Qed.
Fact
le_def
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",...
[ "eq_meetl", "meet", "meetE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_def x y : lt x y = (y != x) && le x y.
Proof. by rewrite le_def; case: eqVneq => //= ->; rewrite lt_irr. Qed.
Fact
lt_def
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",...
[ "eqVneq", "le", "le_def", "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_def_le x y : meet x y = if lt x y then x else y.
Proof. by rewrite meet_def lt_def; case: eqP. Qed.
Fact
meet_def_le
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",...
[ "lt", "lt_def", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
join_def_le x y : join x y = if lt x y then y else x.
Proof. by rewrite join_def lt_def; case: eqP. Qed.
Fact
join_def_le
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",...
[ "join", "lt", "lt_def" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_anti : antisymmetric le.
Proof. move=> x y; rewrite !le_def; case: eqVneq => //= _ /andP [] hxy. by move/(lt_trans hxy); rewrite lt_irr. Qed.
Fact
le_anti
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",...
[ "eqVneq", "le", "le_def", "lt_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_trans : transitive le.
Proof. move=> y x z; rewrite !le_def; case: eqVneq => [->|_] //=. by case: eqVneq => [-> ->|_ hxy /(lt_trans hxy) ->]; rewrite orbT. Qed.
Fact
le_trans
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",...
[ "eqVneq", "le", "le_def", "lt_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_total : total le.
Proof. by move=> x y; rewrite !le_def; case: eqVneq => //; exact: lt_total. Qed.
Fact
le_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",...
[ "eqVneq", "le", "le_def", "lt_total", "total" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
totalT : total (<=%O : rel T).
Proof. by move=> x y; rewrite -!f_mono le_total. Qed.
Fact
totalT
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",...
[ "f_mono", "le_total", "rel", "total" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
anti : antisymmetric (le f).
Proof. by move=> ? ? /le_anti; apply: pcan_inj. Qed.
Fact
anti
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", "le_anti" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_def x y : lt f x y = (y != x) && le f x y.
Proof. by rewrite /lt lt_def (inj_eq (pcan_inj f_can)). Qed.
Fact
lt_def
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",...
[ "inj_eq", "le", "lt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Pcan
:= isPOrder.Build disp (Choice.Pack (Choice.class T)) lt_def (@refl T disp' T' f) anti (@trans T disp' T' f).
Definition
Pcan
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",...
[ "Build", "T'", "anti", "class", "lt_def", "refl", "trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Can f' (f_can : cancel f f')
:= Pcan (can_pcan f_can).
Definition
Can
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",...
[ "Pcan" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCanIsPartial
:= CancelPartial.Pcan.
Notation
PCanIsPartial
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",...
[ "Pcan" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
CanIsPartial
:= CancelPartial.Can.
Notation
CanIsPartial
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",...
[ "Can" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCanIsTotal : DistrLattice_isTotal _ (pcan_type f_can)
:= Total.on (pcan_type f_can).
Definition
PCanIsTotal
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", "pcan_type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
CanIsTotal : DistrLattice_isTotal _ (can_type f_can)
:= Total.on (can_type f_can).
Definition
CanIsTotal
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",...
[ "can_type", "on" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bottom
:= f' \bot%O.
Definition
bottom
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",...
[ "bot" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isobottom x : (bottom <= x)%O.
Proof. by rewrite /bottom -f_mono f'_can Order.le0x. Qed.
Lemma
isobottom
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",...
[ "bottom", "f_mono", "le0x" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
top
:= f' \top%O.
Definition
top
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
isotop x : (x <= top)%O.
Proof. by rewrite /top -f_mono f'_can Order.lex1. Qed.
Lemma
isotop
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",...
[ "f_mono", "lex1", "top" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet (x y : T)
:= f' (meet (f x) (f y)).
Definition
meet
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
join (x y : T)
:= f' (join (f x) (f y)).
Definition
join
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
meetC : commutative meet.
Proof. by move=> x y; rewrite /meet meetC. Qed.
Fact
meetC
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",...
[ "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinC : commutative join.
Proof. by move=> x y; rewrite /join joinC. Qed.
Fact
joinC
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",...
[ "join" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetA : associative meet.
Proof. by move=> y x z; rewrite /meet !f'_can meetA. Qed.
Fact
meetA
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",...
[ "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinA : associative join.
Proof. by move=> y x z; rewrite /join !f'_can joinA. Qed.
Fact
joinA
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",...
[ "join" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinKI y x : meet x (join x y) = x.
Proof. by rewrite /meet /join f'_can joinKI f_can. Qed.
Fact
joinKI
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",...
[ "join", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKI y x : join x (meet x y) = x.
Proof. by rewrite /join /meet f'_can meetKU f_can. Qed.
Fact
meetKI
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",...
[ "join", "meet", "meetKU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_eql x y : (x <= y) = (meet x y == x).
Proof. by rewrite /meet -(can_eq f_can) f'_can eq_meetl f_mono. Qed.
Fact
meet_eql
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",...
[ "can_eq", "eq_meetl", "f_mono", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : left_distributive (meet : T -> T -> T) join.
Proof. by move=> x y z; rewrite /meet /join /= !f'_can meetUl. Qed.
Fact
meetUl
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",...
[ "join", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
omorph_lt (d : disp_t) (T : porderType d) (d' : disp_t) (T' : porderType d') (f : {omorphism T -> T'}) : injective f -> {homo f : x y / x < y}.
Proof. by move/inj_homo_lt; apply; apply: omorph_le. Qed.
Lemma
omorph_lt
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'", "apply", "disp_t", "inj_homo_lt", "omorph_le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_morphism d (T : latticeType d) d' (T' : latticeType d') (f : T -> T') : Prop
:= {morph f : x y / x `&` y}.
Definition
meet_morphism
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'" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
join_morphism d (T : latticeType d) d' (T' : latticeType d') (f : T -> T') : Prop
:= {morph f : x y / x `|` y}.
Definition
join_morphism
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'" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'mlmorphism' T -> T' }"
:= (@MeetLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'mlmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'jlmorphism' T -> T' }"
:= (@JoinLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'jlmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'lmorphism' T -> T' }"
:= (@LatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'lmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'mlmorphism' 'of' f 'as' g ]"
:= (MeetLatticeMorphism.clone _ _ _ _ f%function g) (format "[ 'mlmorphism' 'of' f 'as' g ]") : form_scope.
Notation
[ 'mlmorphism' 'of' f 'as' g ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'mlmorphism' 'of' f ]"
:= (MeetLatticeMorphism.clone _ _ _ _ f%function _) (format "[ 'mlmorphism' 'of' f ]") : form_scope.
Notation
[ 'mlmorphism' 'of' f ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'jlmorphism' 'of' f 'as' g ]"
:= (JoinLatticeMorphism.clone _ _ _ _ f%function g) (format "[ 'jlmorphism' 'of' f 'as' g ]") : form_scope.
Notation
[ 'jlmorphism' 'of' f 'as' g ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'jlmorphism' 'of' f ]"
:= (JoinLatticeMorphism.clone _ _ _ _ f%function _) (format "[ 'jlmorphism' 'of' f ]") : form_scope.
Notation
[ 'jlmorphism' 'of' f ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'lmorphism' 'of' f 'as' g ]"
:= (LatticeMorphism.clone _ _ _ _ f%function g) (format "[ 'lmorphism' 'of' f 'as' g ]") : form_scope.
Notation
[ 'lmorphism' 'of' f 'as' g ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'lmorphism' 'of' f ]"
:= (LatticeMorphism.clone _ _ _ _ f%function _) (format "[ 'lmorphism' 'of' f ]") : form_scope.
Notation
[ 'lmorphism' 'of' f ]
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",...
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
omorphI (f : {mlmorphism T -> T'}) : {morph f : x y / x `&` y}.
Proof. exact: omorphI_subproof. Qed.
Lemma
omorphI
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'" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
omorphU (f : {jlmorphism T -> T'}) : {morph f : x y / x `|` y}.
Proof. exact: omorphU_subproof. Qed.
Lemma
omorphU
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'" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_meet_morphism : meet_morphism (@idfun T).
Proof. by []. Qed.
Fact
idfun_is_meet_morphism
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",...
[ "meet_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_meet_morphism : meet_morphism (f \o g).
Proof. by move=> x y; rewrite /= !omorphI. Qed.
Fact
comp_is_meet_morphism
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",...
[ "meet_morphism", "omorphI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_join_morphism : join_morphism (@idfun T).
Proof. by []. Qed.
Fact
idfun_is_join_morphism
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",...
[ "join_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_join_morphism : join_morphism (f \o g).
Proof. by move=> x y; rewrite /= !omorphU. Qed.
Fact
comp_is_join_morphism
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",...
[ "join_morphism", "omorphU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'blmorphism' T -> T' }"
:= (@BLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'blmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'tlmorphism' T -> T' }"
:= (@TLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'tlmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'tblmorphism' T -> T' }"
:= (@TBLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation
{ 'tblmorphism' T -> 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",...
[ "T'", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
omorph0 : f \bot = \bot.
Proof. exact: omorph0_subproof. Qed.
Lemma
omorph0
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",...
[ "bot" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_bottom_morphism : (@idfun T) \bot = \bot.
Proof. by []. Qed.
Fact
idfun_is_bottom_morphism
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",...
[ "bot" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_bottom_morphism : (f \o g) \bot = \bot.
Proof. by rewrite /= !omorph0. Qed.
Fact
comp_is_bottom_morphism
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",...
[ "bot", "omorph0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
omorph1 : f \top = \top.
Proof. exact: omorph1_subproof. Qed.
Lemma
omorph1
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",...
[ "top" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_top_morphism : (@idfun T) \top = \top.
Proof. by []. Qed.
Fact
idfun_is_top_morphism
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",...
[ "top" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_top_morphism : (f \o g) \top = \top.
Proof. by rewrite /= !omorph1. Qed.
Fact
comp_is_top_morphism
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",...
[ "omorph1", "top" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_closed
:= {in S &, forall u v, u `&` v \in S}.
Definition
meet_closed
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
join_closed
:= {in S &, forall u v, u `|` v \in S}.
Definition
join_closed
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
opredI (S : meetLatticeClosed T) : {in S &, forall u v, u `&` v \in S}.
Proof. exact: opredI. Qed.
Lemma
opredI
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
opredU (S : joinLatticeClosed T) : {in S &, forall u v, u `|` v \in S}.
Proof. exact: opredU. Qed.
Lemma
opredU
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
opred0 (S : bLatticeClosed T) : \bot \in S.
Proof. exact: opred0. Qed.
Lemma
opred0
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",...
[ "bot" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opred_joins (S : bJoinLatticeClosed T) I r (P : pred I) F : (forall i, P i -> F i \in S) -> \join_(i <- r | P i) F i \in S.
Proof. by move=> FS; elim/big_ind: _; [exact: opred0 | exact: opredU |]. Qed.
Lemma
opred_joins
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",...
[ "big_ind", "opred0", "opredU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opred1 (S : tLatticeClosed T) : \top \in S.
Proof. exact: opred1. Qed.
Lemma
opred1
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",...
[ "top" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opred_meets (S : tMeetLatticeClosed T) I r (P : pred I) F : (forall i, P i -> F i \in S) -> \meet_(i <- r | P i) F i \in S.
Proof. by move=> FS; elim/big_ind: _; [exact: opred1 | exact: opredI |]. Qed.
Lemma
opred_meets
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",...
[ "big_ind", "opred1", "opredI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetU (u1 u2 : U) : U
:= inU (opredI (valP u1) (valP u2)).
Let
meetU
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",...
[ "inU", "opredI", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinU (u1 u2 : U) : U
:= inU (opredU (valP u1) (valP u2)).
Let
joinU
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",...
[ "inU", "opredU", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUC : commutative meetU.
Proof. by move=> x y; apply: val_inj; rewrite !SubK meetC. Qed.
Let
meetUC
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",...
[ "SubK", "apply", "meetC", "meetU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinUC : commutative joinU.
Proof. by move=> x y; apply: val_inj; rewrite !SubK joinC. Qed.
Let
joinUC
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",...
[ "SubK", "apply", "joinC", "joinU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUA : associative meetU.
Proof. by move=> x y z; apply: val_inj; rewrite !SubK meetA. Qed.
Let
meetUA
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",...
[ "SubK", "apply", "meetA", "meetU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinUA : associative joinU.
Proof. by move=> x y z; apply: val_inj; rewrite !SubK joinA. Qed.
Let
joinUA
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",...
[ "SubK", "apply", "joinA", "joinU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinUKI y x : meetU x (joinU x y) = x.
Proof. by apply: val_inj; rewrite !SubK joinKI. Qed.
Lemma
joinUKI
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",...
[ "SubK", "apply", "joinKI", "joinU", "meetU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUKU y x : joinU x (meetU x y) = x.
Proof. by apply: val_inj; rewrite !SubK meetKU. Qed.
Let
meetUKU
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",...
[ "SubK", "apply", "joinU", "meetKU", "meetU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_meetU x y : (x <= y) = (meetU x y == x).
Proof. by rewrite -le_val -(inj_eq val_inj) SubK leEmeet. Qed.
Let
le_meetU
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",...
[ "SubK", "inj_eq", "leEmeet", "meetU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valI : meet_morphism (val : U -> T).
Proof. by move=> x y; rewrite !SubK. Qed.
Fact
valI
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",...
[ "SubK", "meet_morphism", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valU : join_morphism (val : U -> T).
Proof. by move=> x y; rewrite !SubK. Qed.
Fact
valU
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",...
[ "SubK", "join_morphism", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zeroU : U
:= inU opred0_subproof.
Let
zeroU
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",...
[ "inU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0x x : zeroU <= x.
Proof. by rewrite -le_val /= SubK le0x. Qed.
Fact
le0x
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",...
[ "SubK", "zeroU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val0 : (val : U -> T) \bot = \bot.
Proof. by rewrite SubK. Qed.
Fact
val0
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",...
[ "SubK", "bot", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oneU : U
:= inU opred1_subproof.
Let
oneU
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",...
[ "inU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lex1 x : x <= oneU.
Proof. by rewrite -le_val /= SubK lex1. Qed.
Fact
lex1
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",...
[ "SubK", "oneU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val1 : (val : U -> T) \top = \top.
Proof. by rewrite SubK. Qed.
Fact
val1
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",...
[ "SubK", "top", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
totalU : total (<=%O : rel U).
Proof. by move=> x y; rewrite -!le_val le_total. Qed.
Lemma
totalU
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_total", "rel", "total" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opredI : meet_closed S.
Proof. by move=> x y Sx Sy; rewrite meetEtotal; case: leP. Qed.
Fact
opredI
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",...
[ "leP", "meetEtotal", "meet_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opredU : join_closed S.
Proof. by move=> x y Sx Sy; rewrite joinEtotal; case: leP. Qed.
Fact
opredU
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",...
[ "joinEtotal", "join_closed", "leP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubPOrder' 'of' U 'by' <: ]"
:= (SubChoice_isSubPOrder.Build _ _ _ _ U) (format "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: ]") : form_scope.
Notation
[ 'SubChoice_isSubPOrder' 'of' U 'by' <: ]
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",...
[ "Build" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubPOrder' 'of' U 'by' <: 'with' disp ]"
:= (SubChoice_isSubPOrder.Build _ _ _ disp U) (format "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.
Notation
[ 'SubChoice_isSubPOrder' 'of' U 'by' <: 'with' disp ]
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",...
[ "Build" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isSubLattice' 'of' U 'by' <: ]"
:= (SubPOrder_isSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: ]") : form_scope.
Notation
[ 'SubPOrder_isSubLattice' 'of' U 'by' <: ]
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",...
[ "Build", "opredI", "opredU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d