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
"{ 'itv' 'nat' & i }"	:= (def nat_sem (Itv.Real i%Z)) : type_scope.	Notation	{ 'itv' 'nat' & i }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "def", "nat_sem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'i01' R }"	:= {itv R & `[0, 1]} : type_scope.	Notation	{ 'i01' R }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'i01' 'nat' }"	:= {itv nat & `[0, 1]} : type_scope.	Notation	{ 'i01' 'nat' }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'posnum' R }"	:= (@posnum _ (Phant R)) : ring_scope.	Notation	{ 'posnum' R }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "posnum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'nonneg' R }"	:= (@nonneg _ (Phant R)) : ring_scope.	Notation	{ 'nonneg' R }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "nonneg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'posnum' 'nat' }"	:= (def nat_sem (Itv.Real `]0%Z, +oo[ )) : type_scope.	Notation	{ 'posnum' 'nat' }	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "def", "nat_sem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:itv"	:= (from (Phantom _ x)) : ring_scope.	Notation	x %:itv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'itv' 'of' x ]"	:= (fromP (Phantom _ x)) : ring_scope.	Notation	[ 'itv' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "fromP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num	:= r.	Notation	num	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:inum"	:= (r x) (only parsing) : ring_scope.	Notation	x %:inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:num"	:= (r x) : ring_scope.	Notation	x %:num	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:posnum"	:= (@r _ _ (Real `]0%Z, +oo[) x) : ring_scope.	Notation	x %:posnum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:nngnum"	:= (@r _ _ (Real `[0%Z, +oo[) x) : ring_scope.	Notation	x %:nngnum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec	:= (Itv.spec (@Itv.num_sem _)).	Notation	num_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "num_sem", "spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_def R	:= (Itv.def (@Itv.num_sem R)).	Notation	num_def	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "def", "num_sem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_itv_bound R	:= (@map_itv_bound _ R intr).	Notation	num_itv_bound	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "intr", "map_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_spec	:= (Itv.spec Itv.nat_sem).	Notation	nat_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "nat_sem", "spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_def	:= (Itv.def Itv.nat_sem).	Notation	nat_def	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "def", "nat_sem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
itv	:= (Itv.def f i).	Notation	itv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nR	:= (num_def R (Itv.Real i)).	Notation	nR	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "num_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
itv_le_total_subproof : total (<=%O : rel nR).	Proof. move=> x y; apply: real_comparable. - by case: x => [x /=/andP[]]. - by case: y => [y /=/andP[]]. Qed.	Lemma	itv_le_total_subproof	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "nR", "real_comparable", "rel", "total" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
top_typ_spec T f (x : T) : Itv.spec f Itv.Top x.	Proof. by []. Qed.	Lemma	top_typ_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
top_typ T f	:= Itv.Typ (@top_typ_spec T f).	Canonical	top_typ	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "top_typ_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_domain_typ_spec (R : realDomainType) (x : R) : num_spec (Itv.Real `]-oo, +oo[) x.	Proof. by rewrite /Itv.num_sem/= num_real. Qed.	Lemma	real_domain_typ_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "num_real", "num_sem", "num_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_domain_typ (R : realDomainType)	:= Itv.Typ (@real_domain_typ_spec R).	Canonical	real_domain_typ	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "real_domain_typ_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_field_typ_spec (R : realFieldType) (x : R) : num_spec (Itv.Real `]-oo, +oo[) x.	Proof. exact: real_domain_typ_spec. Qed.	Lemma	real_field_typ_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "num_spec", "real_domain_typ_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_field_typ (R : realFieldType)	:= Itv.Typ (@real_field_typ_spec R).	Canonical	real_field_typ	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "real_field_typ_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_typ_spec (x : nat) : nat_spec (Itv.Real `[0, +oo[) x.	Proof. by []. Qed.	Lemma	nat_typ_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "nat", "nat_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_typ	:= Itv.Typ nat_typ_spec.	Canonical	nat_typ	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "nat_typ_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
typ_inum_spec (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) : Itv.spec (@Itv.sort_sem _ xt) i x.	Proof. by move: xt x => []. Qed.	Lemma	typ_inum_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "sort", "spec", "typ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
typ_inum (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt)	:= Itv.mk (typ_inum_spec x).	Canonical	typ_inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "sort", "typ", "typ_inum_spec" ]	This adds _ <- Itv.r ( typ_inum ) to canonical projections (c.f., Print Canonical Projections Itv.r) meaning that if no other canonical instance (with a registered head symbol) is found, a canonical instance of Itv.typ, like the ones above, will be looked for.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unify {T} f (x y : T)	:= Unify : f x y = true.	Class	unify	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unify' {T} f (x y : T)	:= Unify' : f x y = true.	Class	unify'	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unify'P {T} f (x y : T) : unify' f x y -> unify f x y	:= id.	Instance	unify'P	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "id", "unify", "unify'" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unify_itv ix iy	:= (unify Itv.sub ix iy).	Notation	unify_itv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "sub", "unify" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
top_wider_anything i : unify_itv i Itv.Top.	Proof. by case: i. Qed.	Instance	top_wider_anything	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_wider_anyreal i : unify_itv (Itv.Real i) (Itv.Real `]-oo, +oo[).	Proof. by case: i => [l u]; apply/andP; rewrite !bnd_simp. Qed.	Instance	real_wider_anyreal	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "bnd_simp", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_num_itv_bound (x y : itv_bound int) : (num_itv_bound R x <= num_itv_bound R y)%O = (x <= y)%O.	Proof. by case: x y => [[] x \| x] [[] y \| y]//=; rewrite !bnd_simp ?ler_int ?ltr_int. Qed.	Lemma	le_num_itv_bound	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "bnd_simp", "int", "itv_bound", "ler_int", "ltr_int", "num_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_itv_bound_le_BLeft (x : itv_bound int) (y : int) : (num_itv_bound R x <= BLeft (y%:~R : R))%O = (x <= BLeft y)%O.	Proof. rewrite -[BLeft y%:~R]/(map_itv_bound intr (BLeft y)). by rewrite le_num_itv_bound. Qed.	Lemma	num_itv_bound_le_BLeft	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "int", "intr", "itv_bound", "le_num_itv_bound", "map_itv_bound", "num_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
BRight_le_num_itv_bound (x : int) (y : itv_bound int) : (BRight (x%:~R : R) <= num_itv_bound R y)%O = (BRight x <= y)%O.	Proof. rewrite -[BRight x%:~R]/(map_itv_bound intr (BRight x)). by rewrite le_num_itv_bound. Qed.	Lemma	BRight_le_num_itv_bound	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BRight", "int", "intr", "itv_bound", "le_num_itv_bound", "map_itv_bound", "num_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec_sub (x y : Itv.t) : Itv.sub x y -> forall z : R, num_spec x z -> num_spec y z.	Proof. case: x y => [\| x] [\| y] //= x_sub_y z /andP[rz]; rewrite /Itv.num_sem rz/=. move: x y x_sub_y => [lx ux] [ly uy] /andP[lel leu] /=. move=> /andP[lxz zux]; apply/andP; split. - by apply: le_trans lxz; rewrite le_num_itv_bound. - by apply: le_trans zux _; rewrite le_num_itv_bound. Qed.	Lemma	num_spec_sub	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "le_num_itv_bound", "le_trans", "num_sem", "num_spec", "split", "sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_spec_sub (n m : Itv.t) : Itv.sub n m -> forall l : nat, nat_spec n l -> nat_spec m l.	Proof. case: n m => [\| n] [\| m] //= n_sub_m l. move: n m n_sub_m => [ln un] [lm um] /andP[lel leu] /=. move=> /andP[lxz zux]; apply/andP; split. - by apply: le_trans lxz. - by apply: le_trans zux _. Qed.	Lemma	nat_spec_sub	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "le_trans", "nat", "nat_spec", "split", "sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
empty_itv	:= Itv.Real `[1, 0]%Z.	Definition	empty_itv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bottom x : ~ unify_itv i empty_itv.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /andP[] /le_trans /[apply]; rewrite ler10. Qed.	Lemma	bottom	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "empty_itv", "in_itv", "le_trans", "ler10", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt0 x : unify_itv i (Itv.Real `]0%Z, +oo[) -> 0 < x%:num :> R.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_]. by rewrite /= in_itv/= andbT. Qed.	Lemma	gt0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtn0 (x : nat_def i) : unify_itv i (Itv.Real `]0, +oo[) -> (0 < x%:num)%N.	Proof. by case: x => x /= /[swap] /nat_spec_sub /[apply] /andP[_] //. Qed.	Lemma	gtn0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "nat_def", "nat_spec_sub", "num", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0F x : unify_itv i (Itv.Real `]0%Z, +oo[) -> (x%:num <= 0 :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT => /lt_geF. Qed.	Lemma	le0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0 x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> x%:num < 0 :> R.	Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.	Lemma	lt0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge0F x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> (0 <= x%:num :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /lt_geF. Qed.	Lemma	ge0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge0 x : unify_itv i (Itv.Real `[0%Z, +oo[) -> 0 <= x%:num :> R.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT. Qed.	Lemma	ge0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0F x : unify_itv i (Itv.Real `[0%Z, +oo[) -> (x%:num < 0 :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT => /le_gtF. Qed.	Lemma	lt0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0 x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> x%:num <= 0 :> R.	Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.	Lemma	le0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt0F x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> (0 < x%:num :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /le_gtF. Qed.	Lemma	gt0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cmp0 x : unify_itv i (Itv.Real `]-oo, +oo[) -> 0 >=< x%:num.	Proof. by case: i x => [//\| i' [x /=/andP[]]]. Qed.	Lemma	cmp0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "num", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neq0 x : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> x%:num != 0 :> R.	Proof. case: i x => [//\| [l u] [x /= Px]]; apply: contra => /eqP x0 /=. move: Px; rewrite x0 => /and3P[_ /= l0 u0]; apply/andP; split. - by case: l l0 => [[] l /= \|//]; rewrite !bnd_simp ?lerz0 ?ltrz0. - by case: u u0 => [[] u /= \|//]; rewrite !bnd_simp ?ler0z ?ltr0z. Qed.	Lemma	neq0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "Px", "apply", "bnd_simp", "l0", "ler0z", "lerz0", "ltr0z", "ltrz0", "num", "split", "sub", "unify" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq0F x : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (x%:num == 0 :> R) = false.	Proof. by move=> u; apply/negbTE/neq0. Qed.	Lemma	eq0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "neq0", "num", "sub", "unify" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neqn0 (n : nat_def i) : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (n%:num != 0)%N.	Proof. case: i n => [//\| [l u] [n /= Pn]]; apply: contra => /eqP n0 /=. move: Pn; rewrite n0 => /andP[l0 u0]; apply/andP; split. - by case: l l0 => [[] l /= \|//]; rewrite !bnd_simp ?lerz0 ?ltrz0. - by case: u u0 => [[] u /= \|//]; rewrite !bnd_simp ?ler0z ?ltr0z. Qed.	Lemma	neqn0	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "bnd_simp", "l0", "ler0z", "lerz0", "ltr0z", "ltrz0", "nat_def", "num", "split", "sub", "unify" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqn0F (n : nat_def i) : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (n%:num == 0)%N = false.	Proof. by move=> u; apply/negbTE/neqn0. Qed.	Lemma	eqn0F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "nat_def", "neqn0", "num", "sub", "unify" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt1 x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> x%:num < 1 :> R.	Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.	Lemma	lt1	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge1F x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> (1 <= x%:num :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /lt_geF. Qed.	Lemma	ge1F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le1 x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> x%:num <= 1 :> R.	Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.	Lemma	le1	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt1F x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> (1 < x%:num :> R) = false.	Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /le_gtF. Qed.	Lemma	gt1F	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_itv_subproof x i' : Itv.sub i i' -> num_spec i' x%:num.	Proof. by case: x => x /= /[swap] /num_spec_sub; apply. Qed.	Lemma	widen_itv_subproof	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "num", "num_spec", "num_spec_sub", "sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_itv x i' (uni : unify_itv i i')	:= Itv.mk (widen_itv_subproof x uni).	Definition	widen_itv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "unify_itv", "widen_itv_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_itvE x (uni : unify_itv i i) : @widen_itv x i uni = x.	Proof. exact/val_inj. Qed.	Lemma	widen_itvE	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "unify_itv", "val_inj", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
posE x (uni : unify_itv i (Itv.Real `]0%Z, +oo[)) : (widen_itv x%:num%:itv uni)%:num = x%:num.	Proof. by []. Qed.	Lemma	posE	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "num", "unify_itv", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nngE x (uni : unify_itv i (Itv.Real `[0%Z, +oo[)) : (widen_itv x%:num%:itv uni)%:num = x%:num.	Proof. by []. Qed.	Lemma	nngE	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "num", "unify_itv", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_natitv_subproof n i' : Itv.sub i i' -> nat_spec i' n%:num.	Proof. by case: n => n /= /[swap] /nat_spec_sub; apply. Qed.	Lemma	widen_natitv_subproof	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "nat_spec", "nat_spec_sub", "num", "sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_natitv n i' (uni : unify_itv i i')	:= Itv.mk (widen_natitv_subproof n uni).	Definition	widen_natitv	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "unify_itv", "widen_natitv_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_natitvE n (uni : unify_itv i i) : @widen_natitv n i uni = n.	Proof. exact/val_inj. Qed.	Lemma	widen_natitvE	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "unify_itv", "val_inj", "widen_natitv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'gt0' 'of' x ]"	:= (ltac:(refine (gt0 x%:itv))) (only parsing).	Notation	[ 'gt0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "gt0", "itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'lt0' 'of' x ]"	:= (ltac:(refine (lt0 x%:itv))) (only parsing).	Notation	[ 'lt0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "lt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'ge0' 'of' x ]"	:= (ltac:(refine (ge0 x%:itv))) (only parsing).	Notation	[ 'ge0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "ge0", "itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'le0' 'of' x ]"	:= (ltac:(refine (le0 x%:itv))) (only parsing).	Notation	[ 'le0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "le0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'cmp0' 'of' x ]"	:= (ltac:(refine (cmp0 x%:itv))) (only parsing).	Notation	[ 'cmp0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "cmp0", "itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'neq0' 'of' x ]"	:= (ltac:(refine (neq0 x%:itv))) (only parsing).	Notation	[ 'neq0' 'of' x ]	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:i01"	:= (widen_itv x%:itv : {i01 _}) (only parsing) : ring_scope.	Notation	x %:i01	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:i01"	:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0, 1]%Z) _) (only printing) : ring_scope.	Notation	x %:i01	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:n01"	:= (widen_natitv x%N%:itv : {i01 nat}) (only parsing) : ring_scope.	Notation	x %:n01	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "nat", "widen_natitv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:n01"	:= (@widen_natitv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0, 1]%Z) _) (only printing) : ring_scope.	Notation	x %:n01	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from", "widen_natitv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:pos"	:= (widen_itv x%:itv : {posnum _}) (only parsing) : ring_scope.	Notation	x %:pos	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "posnum", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:pos"	:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `]0%Z, +oo[) _) (only printing) : ring_scope.	Notation	x %:pos	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:nng"	:= (widen_itv x%:itv : {nonneg _}) (only parsing) : ring_scope.	Notation	x %:nng	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "nonneg", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:nng"	:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0%Z, +oo[) _) (only printing) : ring_scope.	Notation	x %:nng	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from", "widen_itv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:posnat"	:= (widen_natitv x%N%:itv : {posnum nat}) (only parsing) : ring_scope.	Notation	x %:posnat	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "itv", "nat", "posnum", "widen_natitv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:posnat"	:= (@widen_natitv (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `]0, +oo[) _) (only printing) : type_scope.	Notation	x %:posnat	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "from", "widen_natitv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec_zero : num_spec (Itv.Real `[0, 0]) (0 : R).	Proof. by apply/andP; split; [exact: real0 \| rewrite /= in_itv/= lexx]. Qed.	Lemma	num_spec_zero	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "lexx", "num_spec", "real0", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
zero_inum	:= Itv.mk num_spec_zero.	Canonical	zero_inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "num_spec_zero" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec_one : num_spec (Itv.Real `[1, 1]) (1 : R).	Proof. by apply/andP; split; [exact: real1 \| rewrite /= in_itv/= lexx]. Qed.	Lemma	num_spec_one	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "in_itv", "lexx", "num_spec", "real1", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
one_inum	:= Itv.mk num_spec_one.	Canonical	one_inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "num_spec_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_boundr (x : R) b : (BRight (- x)%R <= num_itv_bound R (opp_bound b))%O = (num_itv_bound R b <= BLeft x)%O.	Proof. by case: b => [[] b \| []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2. Qed.	Lemma	opp_boundr	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "BRight", "bnd_simp", "lerN2", "ltrN2", "mulrNz", "num_itv_bound", "opp_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_boundl (x : R) b : (num_itv_bound R (opp_bound b) <= BLeft (- x)%R)%O = (BRight x <= num_itv_bound R b)%O.	Proof. by case: b => [[] b \| []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2. Qed.	Lemma	opp_boundl	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "BRight", "bnd_simp", "lerN2", "ltrN2", "mulrNz", "num_itv_bound", "opp_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec_opp (i : Itv.t) (x : num_def R i) (r := Itv.real1 opp i) : num_spec r (- x%:num).	Proof. apply: Itv.spec_real1 (Itv.P x). case: x => x /= _ [l u] /and3P[xr lx xu]. rewrite /Itv.num_sem/= realN xr/=; apply/andP. by rewrite opp_boundl opp_boundr. Qed.	Lemma	num_spec_opp	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "apply", "num", "num_def", "num_sem", "num_spec", "opp", "opp_boundl", "opp_boundr", "real1", "realN", "spec_real1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_inum (i : Itv.t) (x : num_def R i)	:= Itv.mk (num_spec_opp x).	Canonical	opp_inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "num_def", "num_spec_opp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_itv_add_boundl (x1 x2 : R) b1 b2 : (num_itv_bound R b1 <= BLeft x1)%O -> (num_itv_bound R b2 <= BLeft x2)%O -> (num_itv_bound R (add_boundl b1 b2) <= BLeft (x1 + x2)%R)%O.	Proof. case: b1 b2 => [bb1 b1 \|//] [bb2 b2 \|//]. case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr. - exact: lerD. - exact: ler_ltD. - exact: ltr_leD. - exact: ltrD. Qed.	Lemma	num_itv_add_boundl	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "add_boundl", "bnd_simp", "lerD", "ler_ltD", "ltrD", "ltr_leD", "mulrzDr", "num_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_itv_add_boundr (x1 x2 : R) b1 b2 : (BRight x1 <= num_itv_bound R b1)%O -> (BRight x2 <= num_itv_bound R b2)%O -> (BRight (x1 + x2)%R <= num_itv_bound R (add_boundr b1 b2))%O.	Proof. case: b1 b2 => [bb1 b1 \|//] [bb2 b2 \|//]. case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr. - exact: ltrD. - exact: ltr_leD. - exact: ler_ltD. - exact: lerD. Qed.	Lemma	num_itv_add_boundr	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BRight", "add_boundr", "bnd_simp", "lerD", "ler_ltD", "ltrD", "ltr_leD", "mulrzDr", "num_itv_bound" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
num_spec_add (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi) (r := Itv.real2 add xi yi) : num_spec r (x%:num + y%:num).	Proof. apply: Itv.spec_real2 (Itv.P x) (Itv.P y). case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=. move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]]. rewrite /Itv.num_sem realD//=; apply/andP. by rewrite num_itv_add_boundl ?num_itv_add_boundr. Qed.	Lemma	num_spec_add	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "add", "apply", "num", "num_def", "num_itv_add_boundl", "num_itv_add_boundr", "num_sem", "num_spec", "real2", "realD", "spec_real2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_inum (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)	:= Itv.mk (num_spec_add x y).	Canonical	add_inum	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "mk", "num_def", "num_spec_add" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sign_spec (l u : itv_bound int) (x : R) : signi -> Set	:= \| ISignEqZero : l = BLeft 0 -> u = BRight 0 -> x = 0 -> sign_spec l u x (Known EqZero) \| ISignNonNeg : (BLeft 0%:Z <= l)%O -> (BRight 0%:Z < u)%O -> 0 <= x -> sign_spec l u x (Known NonNeg) \| ISignNonPos : (l < BLeft 0%:Z)%O -> (u <= BRight 0%:Z)%O -> x <= 0 -> ...	Variant	sign_spec	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "BRight", "int", "itv_bound", "real", "signi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signP (l u : itv_bound int) (x : R) : (num_itv_bound R l <= BLeft x)%O -> (BRight x <= num_itv_bound R u)%O -> x \in Num.real -> sign_spec l u x (sign (Interval l u)).	Proof. move=> + + xr; rewrite /sign/sign_boundl/sign_boundr. have [lneg\|lpos\|->] := ltgtP l; have [uneg\|upos\|->] := ltgtP u => lx xu. - apply: ISignNonPos => //; first exact: ltW. have:= le_trans xu (eqbRL (le_num_itv_bound _ _) (ltW uneg)). by rewrite bnd_simp. - exact: ISignBoth. - exact: ISignNonPos. - have:= @l...	Lemma	signP	algebra	algebra/interval_inference.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...	[ "BLeft", "BRight", "apply", "bnd_simp", "int", "itv_bound", "leBRight_ltBLeft", "le_anti", "le_lt_trans", "le_num_itv_bound", "le_trans", "ltBRight_leBLeft", "ltW", "ltgtP", "ltxx", "num_itv_bound", "real", "sign", "sign_boundl", "sign_boundr", "sign_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'itv' 'nat' & i }"

:= (def nat_sem (Itv.Real i%Z)) : type_scope.

Notation

{ 'itv' 'nat' & i }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "def", "nat_sem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'i01' R }"

:= {itv R & `[0, 1]} : type_scope.

Notation

{ 'i01' R }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'i01' 'nat' }"

:= {itv nat & `[0, 1]} : type_scope.

Notation

{ 'i01' 'nat' }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'posnum' R }"

:= (@posnum _ (Phant R)) : ring_scope.

Notation

{ 'posnum' R }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "posnum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'nonneg' R }"

:= (@nonneg _ (Phant R)) : ring_scope.

Notation

{ 'nonneg' R }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "nonneg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'posnum' 'nat' }"

:= (def nat_sem (Itv.Real `]0%Z, +oo[ )) : type_scope.

Notation

{ 'posnum' 'nat' }

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "def", "nat_sem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:itv"

:= (from (Phantom _ x)) : ring_scope.

Notation

x %:itv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'itv' 'of' x ]"

:= (fromP (Phantom _ x)) : ring_scope.

Notation

[ 'itv' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "fromP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num

:= r.

Notation

num

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:inum"

:= (r x) (only parsing) : ring_scope.

Notation

x %:inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:num"

:= (r x) : ring_scope.

Notation

x %:num

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:posnum"

:= (@r _ _ (Real `]0%Z, +oo[) x) : ring_scope.

Notation

x %:posnum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:nngnum"

:= (@r _ _ (Real `[0%Z, +oo[) x) : ring_scope.

Notation

x %:nngnum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec

:= (Itv.spec (@Itv.num_sem _)).

Notation

num_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "num_sem", "spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_def R

:= (Itv.def (@Itv.num_sem R)).

Notation

num_def

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "def", "num_sem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_itv_bound R

:= (@map_itv_bound _ R intr).

Notation

num_itv_bound

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "intr", "map_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_spec

:= (Itv.spec Itv.nat_sem).

Notation

nat_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "nat_sem", "spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_def

:= (Itv.def Itv.nat_sem).

Notation

nat_def

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "def", "nat_sem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

itv

:= (Itv.def f i).

Notation

itv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nR

:= (num_def R (Itv.Real i)).

Notation

nR

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "num_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

itv_le_total_subproof : total (<=%O : rel nR).

Proof. move=> x y; apply: real_comparable. - by case: x => [x /=/andP[]]. - by case: y => [y /=/andP[]]. Qed.

Lemma

itv_le_total_subproof

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "nR", "real_comparable", "rel", "total" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

top_typ_spec T f (x : T) : Itv.spec f Itv.Top x.

Proof. by []. Qed.

Lemma

top_typ_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

top_typ T f

:= Itv.Typ (@top_typ_spec T f).

Canonical

top_typ

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "top_typ_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

real_domain_typ_spec (R : realDomainType) (x : R) : num_spec (Itv.Real `]-oo, +oo[) x.

Proof. by rewrite /Itv.num_sem/= num_real. Qed.

Lemma

real_domain_typ_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "num_real", "num_sem", "num_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

real_domain_typ (R : realDomainType)

:= Itv.Typ (@real_domain_typ_spec R).

Canonical

real_domain_typ

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "real_domain_typ_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

real_field_typ_spec (R : realFieldType) (x : R) : num_spec (Itv.Real `]-oo, +oo[) x.

Proof. exact: real_domain_typ_spec. Qed.

Lemma

real_field_typ_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "num_spec", "real_domain_typ_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

real_field_typ (R : realFieldType)

:= Itv.Typ (@real_field_typ_spec R).

Canonical

real_field_typ

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "real_field_typ_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_typ_spec (x : nat) : nat_spec (Itv.Real `[0, +oo[) x.

Proof. by []. Qed.

Lemma

nat_typ_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "nat", "nat_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_typ

:= Itv.Typ nat_typ_spec.

Canonical

nat_typ

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "nat_typ_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

typ_inum_spec (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) : Itv.spec (@Itv.sort_sem _ xt) i x.

Proof. by move: xt x => []. Qed.

Lemma

typ_inum_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "sort", "spec", "typ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

typ_inum (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt)

:= Itv.mk (typ_inum_spec x).

Canonical

typ_inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "sort", "typ", "typ_inum_spec" ]

This adds _ <- Itv.r ( typ_inum ) to canonical projections (c.f., Print Canonical Projections Itv.r) meaning that if no other canonical instance (with a registered head symbol) is found, a canonical instance of Itv.typ, like the ones above, will be looked for.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unify {T} f (x y : T)

:= Unify : f x y = true.

Class

unify

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unify' {T} f (x y : T)

:= Unify' : f x y = true.

Class

unify'

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unify'P {T} f (x y : T) : unify' f x y -> unify f x y

:= id.

Instance

unify'P

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "id", "unify", "unify'" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unify_itv ix iy

:= (unify Itv.sub ix iy).

Notation

unify_itv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "sub", "unify" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

top_wider_anything i : unify_itv i Itv.Top.

Proof. by case: i. Qed.

Instance

top_wider_anything

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

real_wider_anyreal i : unify_itv (Itv.Real i) (Itv.Real `]-oo, +oo[).

Proof. by case: i => [l u]; apply/andP; rewrite !bnd_simp. Qed.

Instance

real_wider_anyreal

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "bnd_simp", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_num_itv_bound (x y : itv_bound int) : (num_itv_bound R x <= num_itv_bound R y)%O = (x <= y)%O.

Proof. by case: x y => [[] x | x] [[] y | y]//=; rewrite !bnd_simp ?ler_int ?ltr_int. Qed.

Lemma

le_num_itv_bound

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "bnd_simp", "int", "itv_bound", "ler_int", "ltr_int", "num_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_itv_bound_le_BLeft (x : itv_bound int) (y : int) : (num_itv_bound R x <= BLeft (y%:~R : R))%O = (x <= BLeft y)%O.

Proof. rewrite -[BLeft y%:~R]/(map_itv_bound intr (BLeft y)). by rewrite le_num_itv_bound. Qed.

Lemma

num_itv_bound_le_BLeft

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "int", "intr", "itv_bound", "le_num_itv_bound", "map_itv_bound", "num_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

BRight_le_num_itv_bound (x : int) (y : itv_bound int) : (BRight (x%:~R : R) <= num_itv_bound R y)%O = (BRight x <= y)%O.

Proof. rewrite -[BRight x%:~R]/(map_itv_bound intr (BRight x)). by rewrite le_num_itv_bound. Qed.

Lemma

BRight_le_num_itv_bound

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BRight", "int", "intr", "itv_bound", "le_num_itv_bound", "map_itv_bound", "num_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec_sub (x y : Itv.t) : Itv.sub x y -> forall z : R, num_spec x z -> num_spec y z.

Proof. case: x y => [| x] [| y] //= x_sub_y z /andP[rz]; rewrite /Itv.num_sem rz/=. move: x y x_sub_y => [lx ux] [ly uy] /andP[lel leu] /=. move=> /andP[lxz zux]; apply/andP; split. - by apply: le_trans lxz; rewrite le_num_itv_bound. - by apply: le_trans zux _; rewrite le_num_itv_bound. Qed.

Lemma

num_spec_sub

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "le_num_itv_bound", "le_trans", "num_sem", "num_spec", "split", "sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_spec_sub (n m : Itv.t) : Itv.sub n m -> forall l : nat, nat_spec n l -> nat_spec m l.

Proof. case: n m => [| n] [| m] //= n_sub_m l. move: n m n_sub_m => [ln un] [lm um] /andP[lel leu] /=. move=> /andP[lxz zux]; apply/andP; split. - by apply: le_trans lxz. - by apply: le_trans zux _. Qed.

Lemma

nat_spec_sub

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "le_trans", "nat", "nat_spec", "split", "sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

empty_itv

:= Itv.Real `[1, 0]%Z.

Definition

empty_itv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bottom x : ~ unify_itv i empty_itv.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /andP[] /le_trans /[apply]; rewrite ler10. Qed.

Lemma

bottom

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "empty_itv", "in_itv", "le_trans", "ler10", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt0 x : unify_itv i (Itv.Real `]0%Z, +oo[) -> 0 < x%:num :> R.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_]. by rewrite /= in_itv/= andbT. Qed.

Lemma

gt0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gtn0 (x : nat_def i) : unify_itv i (Itv.Real `]0, +oo[) -> (0 < x%:num)%N.

Proof. by case: x => x /= /[swap] /nat_spec_sub /[apply] /andP[_] //. Qed.

Lemma

gtn0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "nat_def", "nat_spec_sub", "num", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le0F x : unify_itv i (Itv.Real `]0%Z, +oo[) -> (x%:num <= 0 :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT => /lt_geF. Qed.

Lemma

le0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0 x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> x%:num < 0 :> R.

Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.

Lemma

lt0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge0F x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> (0 <= x%:num :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /lt_geF. Qed.

Lemma

ge0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge0 x : unify_itv i (Itv.Real `[0%Z, +oo[) -> 0 <= x%:num :> R.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT. Qed.

Lemma

ge0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0F x : unify_itv i (Itv.Real `[0%Z, +oo[) -> (x%:num < 0 :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= andbT => /le_gtF. Qed.

Lemma

lt0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le0 x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> x%:num <= 0 :> R.

Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.

Lemma

le0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt0F x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> (0 < x%:num :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /le_gtF. Qed.

Lemma

gt0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cmp0 x : unify_itv i (Itv.Real `]-oo, +oo[) -> 0 >=< x%:num.

Proof. by case: i x => [//| i' [x /=/andP[]]]. Qed.

Lemma

cmp0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "num", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neq0 x : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> x%:num != 0 :> R.

Proof. case: i x => [//| [l u] [x /= Px]]; apply: contra => /eqP x0 /=. move: Px; rewrite x0 => /and3P[_ /= l0 u0]; apply/andP; split. - by case: l l0 => [[] l /= |//]; rewrite !bnd_simp ?lerz0 ?ltrz0. - by case: u u0 => [[] u /= |//]; rewrite !bnd_simp ?ler0z ?ltr0z. Qed.

Lemma

neq0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "Px", "apply", "bnd_simp", "l0", "ler0z", "lerz0", "ltr0z", "ltrz0", "num", "split", "sub", "unify" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq0F x : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (x%:num == 0 :> R) = false.

Proof. by move=> u; apply/negbTE/neq0. Qed.

Lemma

eq0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "neq0", "num", "sub", "unify" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neqn0 (n : nat_def i) : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (n%:num != 0)%N.

Proof. case: i n => [//| [l u] [n /= Pn]]; apply: contra => /eqP n0 /=. move: Pn; rewrite n0 => /andP[l0 u0]; apply/andP; split. - by case: l l0 => [[] l /= |//]; rewrite !bnd_simp ?lerz0 ?ltrz0. - by case: u u0 => [[] u /= |//]; rewrite !bnd_simp ?ler0z ?ltr0z. Qed.

Lemma

neqn0

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "bnd_simp", "l0", "ler0z", "lerz0", "ltr0z", "ltrz0", "nat_def", "num", "split", "sub", "unify" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqn0F (n : nat_def i) : unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i -> (n%:num == 0)%N = false.

Proof. by move=> u; apply/negbTE/neqn0. Qed.

Lemma

eqn0F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "nat_def", "neqn0", "num", "sub", "unify" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt1 x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> x%:num < 1 :> R.

Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.

Lemma

lt1

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge1F x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> (1 <= x%:num :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /lt_geF. Qed.

Lemma

ge1F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "lt_geF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le1 x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> x%:num <= 1 :> R.

Proof. by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv. Qed.

Lemma

le1

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt1F x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> (1 < x%:num :> R) = false.

Proof. case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=. by rewrite in_itv/= => /le_gtF. Qed.

Lemma

gt1F

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "le_gtF", "num", "num_spec_sub", "unify_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_itv_subproof x i' : Itv.sub i i' -> num_spec i' x%:num.

Proof. by case: x => x /= /[swap] /num_spec_sub; apply. Qed.

Lemma

widen_itv_subproof

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "num", "num_spec", "num_spec_sub", "sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_itv x i' (uni : unify_itv i i')

:= Itv.mk (widen_itv_subproof x uni).

Definition

widen_itv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "unify_itv", "widen_itv_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_itvE x (uni : unify_itv i i) : @widen_itv x i uni = x.

Proof. exact/val_inj. Qed.

Lemma

widen_itvE

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "unify_itv", "val_inj", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

posE x (uni : unify_itv i (Itv.Real `]0%Z, +oo[)) : (widen_itv x%:num%:itv uni)%:num = x%:num.

Proof. by []. Qed.

Lemma

posE

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "num", "unify_itv", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nngE x (uni : unify_itv i (Itv.Real `[0%Z, +oo[)) : (widen_itv x%:num%:itv uni)%:num = x%:num.

Proof. by []. Qed.

Lemma

nngE

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "num", "unify_itv", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_natitv_subproof n i' : Itv.sub i i' -> nat_spec i' n%:num.

Proof. by case: n => n /= /[swap] /nat_spec_sub; apply. Qed.

Lemma

widen_natitv_subproof

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "nat_spec", "nat_spec_sub", "num", "sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_natitv n i' (uni : unify_itv i i')

:= Itv.mk (widen_natitv_subproof n uni).

Definition

widen_natitv

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "unify_itv", "widen_natitv_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_natitvE n (uni : unify_itv i i) : @widen_natitv n i uni = n.

Proof. exact/val_inj. Qed.

Lemma

widen_natitvE

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "unify_itv", "val_inj", "widen_natitv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'gt0' 'of' x ]"

:= (ltac:(refine (gt0 x%:itv))) (only parsing).

Notation

[ 'gt0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "gt0", "itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'lt0' 'of' x ]"

:= (ltac:(refine (lt0 x%:itv))) (only parsing).

Notation

[ 'lt0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "lt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'ge0' 'of' x ]"

:= (ltac:(refine (ge0 x%:itv))) (only parsing).

Notation

[ 'ge0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "ge0", "itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'le0' 'of' x ]"

:= (ltac:(refine (le0 x%:itv))) (only parsing).

Notation

[ 'le0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "le0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'cmp0' 'of' x ]"

:= (ltac:(refine (cmp0 x%:itv))) (only parsing).

Notation

[ 'cmp0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "cmp0", "itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'neq0' 'of' x ]"

:= (ltac:(refine (neq0 x%:itv))) (only parsing).

Notation

[ 'neq0' 'of' x ]

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:i01"

:= (widen_itv x%:itv : {i01 _}) (only parsing) : ring_scope.

Notation

x %:i01

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:i01"

:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0, 1]%Z) _) (only printing) : ring_scope.

Notation

x %:i01

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:n01"

:= (widen_natitv x%N%:itv : {i01 nat}) (only parsing) : ring_scope.

Notation

x %:n01

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "nat", "widen_natitv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:n01"

:= (@widen_natitv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0, 1]%Z) _) (only printing) : ring_scope.

Notation

x %:n01

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from", "widen_natitv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:pos"

:= (widen_itv x%:itv : {posnum _}) (only parsing) : ring_scope.

Notation

x %:pos

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "posnum", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:pos"

:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `]0%Z, +oo[) _) (only printing) : ring_scope.

Notation

x %:pos

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:nng"

:= (widen_itv x%:itv : {nonneg _}) (only parsing) : ring_scope.

Notation

x %:nng

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "nonneg", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:nng"

:= (@widen_itv _ _ (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0%Z, +oo[) _) (only printing) : ring_scope.

Notation

x %:nng

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from", "widen_itv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:posnat"

:= (widen_natitv x%N%:itv : {posnum nat}) (only parsing) : ring_scope.

Notation

x %:posnat

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "itv", "nat", "posnum", "widen_natitv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x %:posnat"

:= (@widen_natitv (@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `]0, +oo[) _) (only printing) : type_scope.

Notation

x %:posnat

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "from", "widen_natitv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec_zero : num_spec (Itv.Real `[0, 0]) (0 : R).

Proof. by apply/andP; split; [exact: real0 | rewrite /= in_itv/= lexx]. Qed.

Lemma

num_spec_zero

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "lexx", "num_spec", "real0", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

zero_inum

:= Itv.mk num_spec_zero.

Canonical

zero_inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "num_spec_zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec_one : num_spec (Itv.Real `[1, 1]) (1 : R).

Proof. by apply/andP; split; [exact: real1 | rewrite /= in_itv/= lexx]. Qed.

Lemma

num_spec_one

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "in_itv", "lexx", "num_spec", "real1", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

one_inum

:= Itv.mk num_spec_one.

Canonical

one_inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "num_spec_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_boundr (x : R) b : (BRight (- x)%R <= num_itv_bound R (opp_bound b))%O = (num_itv_bound R b <= BLeft x)%O.

Proof. by case: b => [[] b | []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2. Qed.

Lemma

opp_boundr

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "BRight", "bnd_simp", "lerN2", "ltrN2", "mulrNz", "num_itv_bound", "opp_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_boundl (x : R) b : (num_itv_bound R (opp_bound b) <= BLeft (- x)%R)%O = (BRight x <= num_itv_bound R b)%O.

Proof. by case: b => [[] b | []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2. Qed.

Lemma

opp_boundl

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "BRight", "bnd_simp", "lerN2", "ltrN2", "mulrNz", "num_itv_bound", "opp_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec_opp (i : Itv.t) (x : num_def R i) (r := Itv.real1 opp i) : num_spec r (- x%:num).

Proof. apply: Itv.spec_real1 (Itv.P x). case: x => x /= _ [l u] /and3P[xr lx xu]. rewrite /Itv.num_sem/= realN xr/=; apply/andP. by rewrite opp_boundl opp_boundr. Qed.

Lemma

num_spec_opp

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "apply", "num", "num_def", "num_sem", "num_spec", "opp", "opp_boundl", "opp_boundr", "real1", "realN", "spec_real1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_inum (i : Itv.t) (x : num_def R i)

:= Itv.mk (num_spec_opp x).

Canonical

opp_inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "num_def", "num_spec_opp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_itv_add_boundl (x1 x2 : R) b1 b2 : (num_itv_bound R b1 <= BLeft x1)%O -> (num_itv_bound R b2 <= BLeft x2)%O -> (num_itv_bound R (add_boundl b1 b2) <= BLeft (x1 + x2)%R)%O.

Proof. case: b1 b2 => [bb1 b1 |//] [bb2 b2 |//]. case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr. - exact: lerD. - exact: ler_ltD. - exact: ltr_leD. - exact: ltrD. Qed.

Lemma

num_itv_add_boundl

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "add_boundl", "bnd_simp", "lerD", "ler_ltD", "ltrD", "ltr_leD", "mulrzDr", "num_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_itv_add_boundr (x1 x2 : R) b1 b2 : (BRight x1 <= num_itv_bound R b1)%O -> (BRight x2 <= num_itv_bound R b2)%O -> (BRight (x1 + x2)%R <= num_itv_bound R (add_boundr b1 b2))%O.

Proof. case: b1 b2 => [bb1 b1 |//] [bb2 b2 |//]. case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr. - exact: ltrD. - exact: ltr_leD. - exact: ler_ltD. - exact: lerD. Qed.

Lemma

num_itv_add_boundr

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BRight", "add_boundr", "bnd_simp", "lerD", "ler_ltD", "ltrD", "ltr_leD", "mulrzDr", "num_itv_bound" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

num_spec_add (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi) (r := Itv.real2 add xi yi) : num_spec r (x%:num + y%:num).

Proof. apply: Itv.spec_real2 (Itv.P x) (Itv.P y). case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=. move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]]. rewrite /Itv.num_sem realD//=; apply/andP. by rewrite num_itv_add_boundl ?num_itv_add_boundr. Qed.

Lemma

num_spec_add

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "add", "apply", "num", "num_def", "num_itv_add_boundl", "num_itv_add_boundr", "num_sem", "num_spec", "real2", "realD", "spec_real2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_inum (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)

:= Itv.mk (num_spec_add x y).

Canonical

add_inum

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "mk", "num_def", "num_spec_add" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sign_spec (l u : itv_bound int) (x : R) : signi -> Set

:= | ISignEqZero : l = BLeft 0 -> u = BRight 0 -> x = 0 -> sign_spec l u x (Known EqZero) | ISignNonNeg : (BLeft 0%:Z <= l)%O -> (BRight 0%:Z < u)%O -> 0 <= x -> sign_spec l u x (Known NonNeg) | ISignNonPos : (l < BLeft 0%:Z)%O -> (u <= BRight 0%:Z)%O -> x <= 0 -> ...

Variant

sign_spec

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "BRight", "int", "itv_bound", "real", "signi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signP (l u : itv_bound int) (x : R) : (num_itv_bound R l <= BLeft x)%O -> (BRight x <= num_itv_bound R u)%O -> x \in Num.real -> sign_spec l u x (sign (Interval l u)).

Proof. move=> + + xr; rewrite /sign/sign_boundl/sign_boundr. have [lneg|lpos|->] := ltgtP l; have [uneg|upos|->] := ltgtP u => lx xu. - apply: ISignNonPos => //; first exact: ltW. have:= le_trans xu (eqbRL (le_num_itv_bound _ _) (ltW uneg)). by rewrite bnd_simp. - exact: ISignBoth. - exact: ISignNonPos. - have:= @l...

Lemma

signP

algebra

algebra/interval_inference.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnat", "eqtype", "choice", "order", "interval", "ssralg", "orderedzmod", "numdomain", "numfield", "ssrint", "Order.TTheory", "Order.Syntax", "GRing.Theory", "Num.Theory", "Itv.Exports", "canonicals", "Ty...

[ "BLeft", "BRight", "apply", "bnd_simp", "int", "itv_bound", "leBRight_ltBLeft", "le_anti", "le_lt_trans", "le_num_itv_bound", "le_trans", "ltBRight_leBLeft", "ltW", "ltgtP", "ltxx", "num_itv_bound", "real", "sign", "sign_boundl", "sign_boundr", "sign_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d