phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
"[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubPOrder_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubPOrder_isSubLattice' '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", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]"	:= (SubChoice_isSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_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
"[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubChoice_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubChoice_isSubLattice' '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", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]"	:= (SubPOrder_isBSubLattice.Build _ _ _ _ U (opred0 _)) (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubPOrder_isBSubLattice' '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", "opred0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubPOrder_isBSubLattice.Build _ _ _ disp U (opred0 _)) (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubPOrder_isBSubLattice' '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", "opred0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]"	:= (SubChoice_isBSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred0 _)) (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isBSubLattice' '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", "opred0", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubChoice_isBSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred0 _)) (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubChoice_isBSubLattice' '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", "opred0", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]"	:= (SubPOrder_isTSubLattice.Build _ _ _ _ U (opred1 _)) (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubPOrder_isTSubLattice' '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", "opred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubPOrder_isTSubLattice.Build _ _ _ disp U (opred1 _)) (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubPOrder_isTSubLattice' '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", "opred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]"	:= (SubChoice_isTSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred1 _)) (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isTSubLattice' '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", "opred1", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubChoice_isTSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred1 _)) (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubChoice_isTSubLattice' '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", "opred1", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]"	:= (SubPOrder_isTBSubLattice.Build _ _ _ _ U (opred0 _) (opred1 _)) (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubPOrder_isTBSubLattice' '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", "opred0", "opred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubPOrder_isTBSubLattice.Build _ _ _ disp U (opred0 _) (opred1 _)) (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubPOrder_isTBSubLattice' '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", "opred0", "opred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]"	:= (SubChoice_isTBSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _)) (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isTBSubLattice' '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", "opred0", "opred1", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]"	:= (SubChoice_isTBSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _)) (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubChoice_isTBSubLattice' '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", "opred0", "opred1", "opredI", "opredU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]"	:= (SubLattice_isSubOrder.Build _ _ _ _ U) (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubLattice_isSubOrder' '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
"[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]"	:= (SubLattice_isSubOrder.Build _ _ _ disp U) (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubLattice_isSubOrder' '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_isSubOrder' 'of' U 'by' <: ]"	:= (SubPOrder_isSubOrder.Build _ _ _ _ U) (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubPOrder_isSubOrder' '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
"[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]"	:= (SubPOrder_isSubOrder.Build _ _ _ disp U) (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubPOrder_isSubOrder' '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
"[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]"	:= (SubChoice_isSubOrder.Build _ _ _ _ U) (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isSubOrder' '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_isSubOrder' 'of' U 'by' <: 'with' disp ]"	:= (SubChoice_isSubOrder.Build _ _ _ disp U) (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.	Notation	[ 'SubChoice_isSubOrder' '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
"[ 'POrder' 'of' T 'by' <: ]"	:= (POrder.copy T%type (sub_type T%type)) (format "[ 'POrder' 'of' T 'by' <: ]") : form_scope.	Notation	[ 'POrder' 'of' T '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",...	[ "copy", "sub_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Order' 'of' T 'by' <: ]"	:= (Total.copy T%type (sub_type T%type)) (only parsing) : form_scope.	Notation	[ 'Order' 'of' T '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",...	[ "copy", "sub_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Dconvex : {in D &, forall i j k, i < k < j -> k \in D}.		Hypothesis	Dconvex	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
incn_inP : {in D, forall i, i.+1 \in D -> f i < f i.+1} -> {in D &, {mono f : i j / i <= j}}.	Proof. by move=> f_inc; apply/le_mono_in/homo_ltn_lt_in. Qed.	Lemma	incn_inP	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", "homo_ltn_lt_in", "le_mono_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
decn_inP : {in D, forall i, i.+1 \in D -> f i > f i.+1} -> {in D &, {mono f : i j /~ i <= j}}.	Proof. by move=> f_dec; apply/le_nmono_in/nhomo_ltn_lt_in. Qed.	Lemma	decn_inP	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_nmono_in", "nhomo_ltn_lt_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
incnP : (forall i, f i < f i.+1) -> {mono f : i j / i <= j}.	Proof. by move=> f_inc; apply/le_mono/homo_ltn_lt. Qed.	Lemma	incnP	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", "homo_ltn_lt", "le_mono" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
decnP : (forall i, f i > f i.+1) -> {mono f : i j /~ i <= j}.	Proof. by move=> f_dec; apply/le_nmono/nhomo_ltn_lt. Qed.	Lemma	decnP	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_nmono", "nhomo_ltn_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcd	:= (@meet dvd_display _).	Notation	gcd	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",...	[ "dvd_display", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'gcd' T"	:= (@meet dvd_display T) (at level 10, T at level 8, only parsing) : function_scope.	Notation	@ 'gcd' 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",...	[ "dvd_display", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcm	:= (@join dvd_display _).	Notation	lcm	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",...	[ "dvd_display", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'lcm' T"	:= (@join dvd_display T) (at level 10, T at level 8, only parsing) : function_scope.	Notation	@ 'lcm' 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",...	[ "dvd_display", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i <- r \| P ) F"	:= (\big[gcd/nat0]_(i <- r \| P%B) F%O) : order_scope.	Notation	\gcd_ ( i <- r \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i <- r ) F"	:= (\big[gcd/nat0]_(i <- r) F%O) : order_scope.	Notation	\gcd_ ( i <- r ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i \| P ) F"	:= (\big[gcd/nat0]_(i \| P%B) F%O) : order_scope.	Notation	\gcd_ ( i \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ i F"	:= (\big[gcd/nat0]_i F%O) : order_scope.	Notation	\gcd_ i 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i : I \| P ) F"	:= (\big[gcd/nat0]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\gcd_ ( i : I \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i : I ) F"	:= (\big[gcd/nat0]_(i : I) F%O) (only parsing) : order_scope.	Notation	\gcd_ ( i : I ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( m <= i < n \| P ) F"	:= (\big[gcd/nat0]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\gcd_ ( m <= i < n \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( m <= i < n ) F"	:= (\big[gcd/nat0]_(m <= i < n) F%O) : order_scope.	Notation	\gcd_ ( m <= i < n ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i < n \| P ) F"	:= (\big[gcd/nat0]_(i < n \| P%B) F%O) : order_scope.	Notation	\gcd_ ( i < n \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i < n ) F"	:= (\big[gcd/nat0]_(i < n) F%O) : order_scope.	Notation	\gcd_ ( i < n ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i 'in' A \| P ) F"	:= (\big[gcd/nat0]_(i in A \| P%B) F%O) : order_scope.	Notation	\gcd_ ( i 'in' A \| P ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\gcd_ ( i 'in' A ) F"	:= (\big[gcd/nat0]_(i in A) F%O) : order_scope.	Notation	\gcd_ ( i 'in' A ) 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",...	[ "gcd", "nat0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i <- r \| P ) F"	:= (\big[lcm/nat1]_(i <- r \| P%B) F%O) : order_scope.	Notation	\lcm_ ( i <- r \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i <- r ) F"	:= (\big[lcm/nat1]_(i <- r) F%O) : order_scope.	Notation	\lcm_ ( i <- r ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i \| P ) F"	:= (\big[lcm/nat1]_(i \| P%B) F%O) : order_scope.	Notation	\lcm_ ( i \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ i F"	:= (\big[lcm/nat1]_i F%O) : order_scope.	Notation	\lcm_ i 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i : I \| P ) F"	:= (\big[lcm/nat1]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\lcm_ ( i : I \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i : I ) F"	:= (\big[lcm/nat1]_(i : I) F%O) (only parsing) : order_scope.	Notation	\lcm_ ( i : I ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( m <= i < n \| P ) F"	:= (\big[lcm/nat1]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\lcm_ ( m <= i < n \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( m <= i < n ) F"	:= (\big[lcm/nat1]_(m <= i < n) F%O) : order_scope.	Notation	\lcm_ ( m <= i < n ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i < n \| P ) F"	:= (\big[lcm/nat1]_(i < n \| P%B) F%O) : order_scope.	Notation	\lcm_ ( i < n \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i < n ) F"	:= (\big[lcm/nat1]_(i < n) F%O) : order_scope.	Notation	\lcm_ ( i < n ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i 'in' A \| P ) F"	:= (\big[lcm/nat1]_(i in A \| P%B) F%O) : order_scope.	Notation	\lcm_ ( i 'in' A \| P ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\lcm_ ( i 'in' A ) F"	:= (\big[lcm/nat1]_(i in A) F%O) : order_scope.	Notation	\lcm_ ( i 'in' A ) 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",...	[ "lcm", "nat1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcmnn n : lcmn n n = n.	Proof. by case: n => // n; rewrite /lcmn gcdnn mulnK. Qed.	Lemma	lcmnn	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",...	[ "gcdnn", "lcmn", "mulnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_def m n : (m %\| n) = (gcdn m n == m)%N.	Proof. by apply/gcdn_idPl/eqP. Qed.	Lemma	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",...	[ "apply", "gcdn", "gcdn_idPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinKI n m : gcdn m (lcmn m n) = m.	Proof. by rewrite (gcdn_idPl _)// dvdn_lcml. Qed.	Lemma	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",...	[ "dvdn_lcml", "gcdn", "gcdn_idPl", "lcmn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKU n m : lcmn m (gcdn m n) = m.	Proof. by rewrite (lcmn_idPl _)// dvdn_gcdl. Qed.	Lemma	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",...	[ "dvdn_gcdl", "gcdn", "lcmn", "lcmn_idPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : left_distributive gcdn lcmn.	Proof. move=> [\|m'] [\|n'] [\|p'] //=; rewrite ?lcmnn ?lcm0n ?lcmn0 ?gcd0n ?gcdn0//. - by rewrite gcdnC meetKU. - by rewrite lcmnC gcdnC meetKU. apply: eqn_from_log; rewrite ?(gcdn_gt0, lcmn_gt0)//= => p. by rewrite !(logn_gcd, logn_lcm) ?(gcdn_gt0, lcmn_gt0)// minn_maxl. Qed.	Lemma	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",...	[ "apply", "eqn_from_log", "gcd0n", "gcdn", "gcdn0", "gcdnC", "gcdn_gt0", "lcm0n", "lcmn", "lcmn0", "lcmnC", "lcmn_gt0", "lcmnn", "logn_gcd", "logn_lcm", "meetKU", "minn_maxl", "n'" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_anti : antisymmetric dvdn.	Proof. by move=> a b => /andP[] /gcdn_idPl + /gcdn_idPr => ->. Qed.	Fact	dvdn_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",...	[ "dvdn", "gcdn_idPl", "gcdn_idPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdvdE (m n : t) : (m %<\| n) = (n != m) && (m %\| n).	Proof. exact/lt_def. Qed.	Lemma	sdvdE	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_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcdE : gcd = gcdn :> (t -> t -> t).	Proof. by []. Qed.	Lemma	gcdE	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",...	[ "gcd", "gcdn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcmE : lcm = lcmn :> (t -> t -> t).	Proof. by []. Qed.	Lemma	lcmE	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",...	[ "lcm", "lcmn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdvdEnat	:= sdvdE.	Definition	sdvdEnat	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",...	[ "sdvdE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcdEnat	:= gcdE.	Definition	gcdEnat	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",...	[ "gcdE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcmEnat	:= lcmE.	Definition	lcmEnat	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",...	[ "lcmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
andbE x y : x && y = if (x < y)%N then x else y.	Proof. by case: x y => [] []. Qed.	Fact	andbE	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
orbE x y : x \|\| y = if (x < y)%N then y else x.	Proof. by case: x y => [] []. Qed.	Fact	orbE	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
anti : antisymmetric (leq : rel bool).	Proof. by move=> x y /anti_leq /(congr1 odd); rewrite !oddb. 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",...	[ "anti_leq", "leq", "odd", "oddb", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orKb b a : a && (a \|\| b) = a.	Proof. by rewrite orbC orKb. Qed.	Fact	orKb	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
andKb y x : x \|\| x && y = x.	Proof. by rewrite andbC andKb. Qed.	Fact	andKb	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
leEmeet x y : (x <= y) = (x && y == x).	Proof. by case: x; case: y. Qed.	Fact	leEmeet	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
andEbool : meet = andb.	Proof. by []. Qed.	Lemma	andEbool	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
orEbool : meet = andb.	Proof. by []. Qed.	Lemma	orEbool	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
subEbool x y : x `\` y = x && ~~ y.	Proof. by []. Qed.	Lemma	subEbool	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
complEbool : compl = negb.	Proof. by []. Qed.	Lemma	complEbool	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",...	[ "compl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEbool	:= leEbool.	Definition	leEbool	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
ltEbool	:= ltEbool.	Definition	ltEbool	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
andEbool	:= andEbool.	Definition	andEbool	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
orEbool	:= orEbool.	Definition	orEbool	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
subEbool	:= subEbool.	Definition	subEbool	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
complEbool	:= complEbool.	Definition	complEbool	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
"\bot"	:= (@bottom (prod_display _ _) _).	Notation	\bot	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", "prod_display" ]	notations later. Do not remove them.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\top"	:= (@top (prod_display _ _) _).	Notation	\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",...	[ "prod_display", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet	:= (@meet (prod_display _ _) _).	Notation	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",...	[ "prod_display" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join	:= (@join (prod_display _ _) _).	Notation	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",...	[ "prod_display" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
min	:= (@min (prod_display _ _) _).	Notation	min	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",...	[ "prod_display" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max	:= (@max (prod_display _ _) _).	Notation	max	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",...	[ "prod_display" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x `&^p` y"	:= (meet x y) : order_scope.	Notation	x `&^p` y	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
"x `\|^p` y"	:= (join x y) : order_scope.	Notation	x `\|^p` y	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
"\join^p_ ( i <- r \| P ) F"	:= (\big[join / \bot]_(i <- r \| P%B) F%O) : order_scope.	Notation	\join^p_ ( i <- r \| P ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( i <- r ) F"	:= (\big[join / \bot]_(i <- r) F%O) : order_scope.	Notation	\join^p_ ( i <- r ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( i \| P ) F"	:= (\big[join / \bot]_(i \| P%B) F%O) : order_scope.	Notation	\join^p_ ( i \| P ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ i F"	:= (\big[join / \bot]_i F%O) : order_scope.	Notation	\join^p_ i 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( i : I \| P ) F"	:= (\big[join / \bot]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\join^p_ ( i : I \| P ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( i : I ) F"	:= (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.	Notation	\join^p_ ( i : I ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( m <= i < n \| P ) F"	:= (\big[join / \bot]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\join^p_ ( m <= i < n \| P ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^p_ ( m <= i < n ) F"	:= (\big[join / \bot]_(m <= i < n) F%O) : order_scope.	Notation	\join^p_ ( m <= i < n ) 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",...	[ "bot", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubPOrder_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubPOrder_isSubLattice' '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", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]"

:= (SubChoice_isSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_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

"[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubChoice_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _)) (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubChoice_isSubLattice' '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", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]"

:= (SubPOrder_isBSubLattice.Build _ _ _ _ U (opred0 _)) (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubPOrder_isBSubLattice' '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", "opred0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubPOrder_isBSubLattice.Build _ _ _ disp U (opred0 _)) (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubPOrder_isBSubLattice' '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", "opred0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]"

:= (SubChoice_isBSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred0 _)) (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isBSubLattice' '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", "opred0", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubChoice_isBSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred0 _)) (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubChoice_isBSubLattice' '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", "opred0", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]"

:= (SubPOrder_isTSubLattice.Build _ _ _ _ U (opred1 _)) (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubPOrder_isTSubLattice' '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", "opred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubPOrder_isTSubLattice.Build _ _ _ disp U (opred1 _)) (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubPOrder_isTSubLattice' '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", "opred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]"

:= (SubChoice_isTSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred1 _)) (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isTSubLattice' '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", "opred1", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubChoice_isTSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred1 _)) (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubChoice_isTSubLattice' '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", "opred1", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]"

:= (SubPOrder_isTBSubLattice.Build _ _ _ _ U (opred0 _) (opred1 _)) (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubPOrder_isTBSubLattice' '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", "opred0", "opred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubPOrder_isTBSubLattice.Build _ _ _ disp U (opred0 _) (opred1 _)) (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubPOrder_isTBSubLattice' '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", "opred0", "opred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]"

:= (SubChoice_isTBSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _)) (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isTBSubLattice' '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", "opred0", "opred1", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]"

:= (SubChoice_isTBSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _)) (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubChoice_isTBSubLattice' '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", "opred0", "opred1", "opredI", "opredU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]"

:= (SubLattice_isSubOrder.Build _ _ _ _ U) (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubLattice_isSubOrder' '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

"[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]"

:= (SubLattice_isSubOrder.Build _ _ _ disp U) (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubLattice_isSubOrder' '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_isSubOrder' 'of' U 'by' <: ]"

:= (SubPOrder_isSubOrder.Build _ _ _ _ U) (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubPOrder_isSubOrder' '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

"[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]"

:= (SubPOrder_isSubOrder.Build _ _ _ disp U) (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubPOrder_isSubOrder' '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

"[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]"

:= (SubChoice_isSubOrder.Build _ _ _ _ U) (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isSubOrder' '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_isSubOrder' 'of' U 'by' <: 'with' disp ]"

:= (SubChoice_isSubOrder.Build _ _ _ disp U) (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: 'with' disp ]") : form_scope.

Notation

[ 'SubChoice_isSubOrder' '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

"[ 'POrder' 'of' T 'by' <: ]"

:= (POrder.copy T%type (sub_type T%type)) (format "[ 'POrder' 'of' T 'by' <: ]") : form_scope.

Notation

[ 'POrder' 'of' T '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",...

[ "copy", "sub_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Order' 'of' T 'by' <: ]"

:= (Total.copy T%type (sub_type T%type)) (only parsing) : form_scope.

Notation

[ 'Order' 'of' T '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",...

[ "copy", "sub_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Dconvex : {in D &, forall i j k, i < k < j -> k \in D}.

Hypothesis

Dconvex

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

incn_inP : {in D, forall i, i.+1 \in D -> f i < f i.+1} -> {in D &, {mono f : i j / i <= j}}.

Proof. by move=> f_inc; apply/le_mono_in/homo_ltn_lt_in. Qed.

Lemma

incn_inP

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", "homo_ltn_lt_in", "le_mono_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

decn_inP : {in D, forall i, i.+1 \in D -> f i > f i.+1} -> {in D &, {mono f : i j /~ i <= j}}.

Proof. by move=> f_dec; apply/le_nmono_in/nhomo_ltn_lt_in. Qed.

Lemma

decn_inP

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_nmono_in", "nhomo_ltn_lt_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

incnP : (forall i, f i < f i.+1) -> {mono f : i j / i <= j}.

Proof. by move=> f_inc; apply/le_mono/homo_ltn_lt. Qed.

Lemma

incnP

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", "homo_ltn_lt", "le_mono" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

decnP : (forall i, f i > f i.+1) -> {mono f : i j /~ i <= j}.

Proof. by move=> f_dec; apply/le_nmono/nhomo_ltn_lt. Qed.

Lemma

decnP

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_nmono", "nhomo_ltn_lt" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcd

:= (@meet dvd_display _).

Notation

gcd

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",...

[ "dvd_display", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"@ 'gcd' T"

:= (@meet dvd_display T) (at level 10, T at level 8, only parsing) : function_scope.

Notation

@ 'gcd' 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",...

[ "dvd_display", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcm

:= (@join dvd_display _).

Notation

lcm

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",...

[ "dvd_display", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"@ 'lcm' T"

:= (@join dvd_display T) (at level 10, T at level 8, only parsing) : function_scope.

Notation

@ 'lcm' 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",...

[ "dvd_display", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i <- r | P ) F"

:= (\big[gcd/nat0]_(i <- r | P%B) F%O) : order_scope.

Notation

\gcd_ ( i <- r | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i <- r ) F"

:= (\big[gcd/nat0]_(i <- r) F%O) : order_scope.

Notation

\gcd_ ( i <- r ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i | P ) F"

:= (\big[gcd/nat0]_(i | P%B) F%O) : order_scope.

Notation

\gcd_ ( i | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ i F"

:= (\big[gcd/nat0]_i F%O) : order_scope.

Notation

\gcd_ i 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i : I | P ) F"

:= (\big[gcd/nat0]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\gcd_ ( i : I | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i : I ) F"

:= (\big[gcd/nat0]_(i : I) F%O) (only parsing) : order_scope.

Notation

\gcd_ ( i : I ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( m <= i < n | P ) F"

:= (\big[gcd/nat0]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\gcd_ ( m <= i < n | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( m <= i < n ) F"

:= (\big[gcd/nat0]_(m <= i < n) F%O) : order_scope.

Notation

\gcd_ ( m <= i < n ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i < n | P ) F"

:= (\big[gcd/nat0]_(i < n | P%B) F%O) : order_scope.

Notation

\gcd_ ( i < n | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i < n ) F"

:= (\big[gcd/nat0]_(i < n) F%O) : order_scope.

Notation

\gcd_ ( i < n ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i 'in' A | P ) F"

:= (\big[gcd/nat0]_(i in A | P%B) F%O) : order_scope.

Notation

\gcd_ ( i 'in' A | P ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\gcd_ ( i 'in' A ) F"

:= (\big[gcd/nat0]_(i in A) F%O) : order_scope.

Notation

\gcd_ ( i 'in' A ) 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",...

[ "gcd", "nat0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i <- r | P ) F"

:= (\big[lcm/nat1]_(i <- r | P%B) F%O) : order_scope.

Notation

\lcm_ ( i <- r | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i <- r ) F"

:= (\big[lcm/nat1]_(i <- r) F%O) : order_scope.

Notation

\lcm_ ( i <- r ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i | P ) F"

:= (\big[lcm/nat1]_(i | P%B) F%O) : order_scope.

Notation

\lcm_ ( i | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ i F"

:= (\big[lcm/nat1]_i F%O) : order_scope.

Notation

\lcm_ i 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i : I | P ) F"

:= (\big[lcm/nat1]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\lcm_ ( i : I | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i : I ) F"

:= (\big[lcm/nat1]_(i : I) F%O) (only parsing) : order_scope.

Notation

\lcm_ ( i : I ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( m <= i < n | P ) F"

:= (\big[lcm/nat1]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\lcm_ ( m <= i < n | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( m <= i < n ) F"

:= (\big[lcm/nat1]_(m <= i < n) F%O) : order_scope.

Notation

\lcm_ ( m <= i < n ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i < n | P ) F"

:= (\big[lcm/nat1]_(i < n | P%B) F%O) : order_scope.

Notation

\lcm_ ( i < n | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i < n ) F"

:= (\big[lcm/nat1]_(i < n) F%O) : order_scope.

Notation

\lcm_ ( i < n ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i 'in' A | P ) F"

:= (\big[lcm/nat1]_(i in A | P%B) F%O) : order_scope.

Notation

\lcm_ ( i 'in' A | P ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\lcm_ ( i 'in' A ) F"

:= (\big[lcm/nat1]_(i in A) F%O) : order_scope.

Notation

\lcm_ ( i 'in' A ) 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",...

[ "lcm", "nat1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcmnn n : lcmn n n = n.

Proof. by case: n => // n; rewrite /lcmn gcdnn mulnK. Qed.

Lemma

lcmnn

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",...

[ "gcdnn", "lcmn", "mulnK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_def m n : (m %| n) = (gcdn m n == m)%N.

Proof. by apply/gcdn_idPl/eqP. Qed.

Lemma

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",...

[ "apply", "gcdn", "gcdn_idPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinKI n m : gcdn m (lcmn m n) = m.

Proof. by rewrite (gcdn_idPl _)// dvdn_lcml. Qed.

Lemma

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",...

[ "dvdn_lcml", "gcdn", "gcdn_idPl", "lcmn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetKU n m : lcmn m (gcdn m n) = m.

Proof. by rewrite (lcmn_idPl _)// dvdn_gcdl. Qed.

Lemma

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",...

[ "dvdn_gcdl", "gcdn", "lcmn", "lcmn_idPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetUl : left_distributive gcdn lcmn.

Proof. move=> [|m'] [|n'] [|p'] //=; rewrite ?lcmnn ?lcm0n ?lcmn0 ?gcd0n ?gcdn0//. - by rewrite gcdnC meetKU. - by rewrite lcmnC gcdnC meetKU. apply: eqn_from_log; rewrite ?(gcdn_gt0, lcmn_gt0)//= => p. by rewrite !(logn_gcd, logn_lcm) ?(gcdn_gt0, lcmn_gt0)// minn_maxl. Qed.

Lemma

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",...

[ "apply", "eqn_from_log", "gcd0n", "gcdn", "gcdn0", "gcdnC", "gcdn_gt0", "lcm0n", "lcmn", "lcmn0", "lcmnC", "lcmn_gt0", "lcmnn", "logn_gcd", "logn_lcm", "meetKU", "minn_maxl", "n'" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_anti : antisymmetric dvdn.

Proof. by move=> a b => /andP[] /gcdn_idPl + /gcdn_idPr => ->. Qed.

Fact

dvdn_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",...

[ "dvdn", "gcdn_idPl", "gcdn_idPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdvdE (m n : t) : (m %<| n) = (n != m) && (m %| n).

Proof. exact/lt_def. Qed.

Lemma

sdvdE

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_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcdE : gcd = gcdn :> (t -> t -> t).

Proof. by []. Qed.

Lemma

gcdE

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",...

[ "gcd", "gcdn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcmE : lcm = lcmn :> (t -> t -> t).

Proof. by []. Qed.

Lemma

lcmE

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",...

[ "lcm", "lcmn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdvdEnat

:= sdvdE.

Definition

sdvdEnat

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",...

[ "sdvdE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcdEnat

:= gcdE.

Definition

gcdEnat

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",...

[ "gcdE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcmEnat

:= lcmE.

Definition

lcmEnat

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",...

[ "lcmE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

andbE x y : x && y = if (x < y)%N then x else y.

Proof. by case: x y => [] []. Qed.

Fact

andbE

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

orbE x y : x || y = if (x < y)%N then y else x.

Proof. by case: x y => [] []. Qed.

Fact

orbE

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

anti : antisymmetric (leq : rel bool).

Proof. by move=> x y /anti_leq /(congr1 odd); rewrite !oddb. 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",...

[ "anti_leq", "leq", "odd", "oddb", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orKb b a : a && (a || b) = a.

Proof. by rewrite orbC orKb. Qed.

Fact

orKb

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

andKb y x : x || x && y = x.

Proof. by rewrite andbC andKb. Qed.

Fact

andKb

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

leEmeet x y : (x <= y) = (x && y == x).

Proof. by case: x; case: y. Qed.

Fact

leEmeet

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

andEbool : meet = andb.

Proof. by []. Qed.

Lemma

andEbool

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

orEbool : meet = andb.

Proof. by []. Qed.

Lemma

orEbool

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

subEbool x y : x `\` y = x && ~~ y.

Proof. by []. Qed.

Lemma

subEbool

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

complEbool : compl = negb.

Proof. by []. Qed.

Lemma

complEbool

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",...

[ "compl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEbool

:= leEbool.

Definition

leEbool

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

ltEbool

:= ltEbool.

Definition

ltEbool

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

andEbool

:= andEbool.

Definition

andEbool

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

orEbool

:= orEbool.

Definition

orEbool

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

subEbool

:= subEbool.

Definition

subEbool

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

complEbool

:= complEbool.

Definition

complEbool

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

"\bot"

:= (@bottom (prod_display _ _) _).

Notation

\bot

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", "prod_display" ]

notations later. Do not remove them.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\top"

:= (@top (prod_display _ _) _).

Notation

\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",...

[ "prod_display", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet

:= (@meet (prod_display _ _) _).

Notation

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",...

[ "prod_display" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join

:= (@join (prod_display _ _) _).

Notation

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",...

[ "prod_display" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

min

:= (@min (prod_display _ _) _).

Notation

min

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",...

[ "prod_display" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max

:= (@max (prod_display _ _) _).

Notation

max

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",...

[ "prod_display" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x `&^p` y"

:= (meet x y) : order_scope.

Notation

x `&^p` y

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

"x `|^p` y"

:= (join x y) : order_scope.

Notation

x `|^p` y

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

"\join^p_ ( i <- r | P ) F"

:= (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.

Notation

\join^p_ ( i <- r | P ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( i <- r ) F"

:= (\big[join / \bot]_(i <- r) F%O) : order_scope.

Notation

\join^p_ ( i <- r ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( i | P ) F"

:= (\big[join / \bot]_(i | P%B) F%O) : order_scope.

Notation

\join^p_ ( i | P ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ i F"

:= (\big[join / \bot]_i F%O) : order_scope.

Notation

\join^p_ i 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( i : I | P ) F"

:= (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\join^p_ ( i : I | P ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( i : I ) F"

:= (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.

Notation

\join^p_ ( i : I ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( m <= i < n | P ) F"

:= (\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\join^p_ ( m <= i < n | P ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^p_ ( m <= i < n ) F"

:= (\big[join / \bot]_(m <= i < n) F%O) : order_scope.

Notation

\join^p_ ( m <= i < n ) 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",...

[ "bot", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d