(** * Generalities on the type of integers and integer arithmetic. Vladimir Voevodsky . Aug. - Sep. 2011. In this file we introduce the type [ hz ] of integers defined as the quotient set of [ dirprod nat nat ] by the standard equivalence relation and develop the main notions of the integer arithmetic using this definition . *) (** ** Preamble *) (** Settings *) Unset Kernel Term Sharing. (** Imports *) Require Export UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.PartA. Require Export UniMath.MoreFoundations.NegativePropositions. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Export UniMath.Algebra.RigsAndRings. Require Export UniMath.NumberSystems.NaturalNumbersAlgebra. (** Upstream *) (** ** The commutative ring [ hz ] of integres *) (** *** General definitions *) Definition hz : commring := commrigtocommring natcommrig . Definition hzaddabgr : abgr := hz . Definition hzmultabmonoid : abmonoid := ringmultabmonoid hz . Definition natnattohz : nat -> nat -> hz := λ n m, setquotpr _ ( make_dirprod n m ) . Definition hzplus : hz -> hz -> hz := @op1 hz. Definition hzsign : hz -> hz := grinv hzaddabgr . Definition hzminus : hz -> hz -> hz := λ x y, hzplus x ( hzsign y ) . Definition hzzero : hz := unel hzaddabgr . Definition hzmult : hz -> hz -> hz := @op2 hz . Definition hzone : hz := unel hzmultabmonoid . Declare Scope hz_scope. Bind Scope hz_scope with hz . Notation " x + y " := ( hzplus x y ) : hz_scope . Notation " 0 " := hzzero : hz_scope . Notation " 1 " := hzone : hz_scope . Notation " - x " := ( hzsign x ) : hz_scope . Notation " x - y " := ( hzminus x y ) : hz_scope . Notation " x * y " := ( hzmult x y ) : hz_scope . Delimit Scope hz_scope with hz . (** *** Properties of equlaity on [ hz ] *) Theorem isdeceqhz : isdeceq hz . Proof . change ( isdeceq ( abgrdiff ( rigaddabmonoid natcommrig ) ) ) . apply isdeceqabgrdiff . apply isinclnatplusr . apply isdeceqnat . Defined . Opaque isdeceqhz. Lemma isasethz : isaset hz . Proof . apply ( setproperty hzaddabgr ) . Defined . Opaque isasethz. Definition hzeq ( x y : hz ) : hProp := make_hProp ( x = y ) ( isasethz _ _ ) . Definition isdecrelhzeq : isdecrel hzeq := λ a b, isdeceqhz a b . Definition hzdeceq : decrel hz := make_decrel isdecrelhzeq . (* Canonical Structure hzdeceq. *) Definition hzbooleq := decreltobrel hzdeceq . Definition hzneq ( x y : hz ) : hProp := make_hProp ( neg ( x = y ) ) ( isapropneg _ ) . Definition isdecrelhzneq : isdecrel hzneq := isdecnegrel _ isdecrelhzeq . Definition hzdecneq : decrel hz := make_decrel isdecrelhzneq . (* Canonical Structure hzdecneq. *) Definition hzboolneq := decreltobrel hzdecneq . Local Open Scope hz_scope . (** *** [ hz ] is a non-zero ring *) Lemma isnonzerorighz : isnonzerorig hz . Proof . apply ( ct ( hzneq , isdecrelhzneq, 1 , 0 ) ) . Defined . (** *** Properties of addition and subtraction on [ hz ] *) Definition hzminuszero : ( - 0 ) = 0 := ringinvunel1 hz . Lemma hzplusr0 ( x : hz ) : ( x + 0 ) = x . Proof . apply ( ringrunax1 _ x ) . Defined . Lemma hzplusl0 ( x : hz ) : ( 0 + x ) = x . Proof . apply ( ringlunax1 _ x ) . Defined . Lemma hzplusassoc ( x y z : hz ) : ( ( x + y ) + z ) = ( x + ( y + z ) ) . Proof . intros . apply ( ringassoc1 hz x y z ) . Defined . Lemma hzpluscomm ( x y : hz ) : ( x + y ) = ( y + x ) . Proof . intros . apply ( ringcomm1 hz x y ) . Defined . Lemma hzlminus ( x : hz ) : ( -x + x ) = 0 . Proof . apply ( ringlinvax1 hz x ) . Defined . Lemma hzrminus ( x : hz ) : ( x - x ) = 0 . Proof . apply ( ringrinvax1 hz x ) . Defined . Lemma isinclhzplusr ( n : hz ) : isincl ( λ m : hz, m + n ) . Proof. apply ( pr2 ( weqtoincl ( weqrmultingr hzaddabgr n ) ) ) . Defined. Lemma isinclhzplusl ( n : hz ) : isincl ( λ m : hz, n + m ) . Proof. apply ( pr2 ( weqtoincl ( weqlmultingr hzaddabgr n ) ) ) . Defined . Lemma hzpluslcan ( a b c : hz ) ( is : ( c + a ) = ( c + b ) ) : a = b . Proof . intros . apply ( @grlcan hzaddabgr a b c is ) . Defined . Lemma hzplusrcan ( a b c : hz ) ( is : ( a + c ) = ( b + c ) ) : a = b . Proof . intros . apply ( @grrcan hzaddabgr a b c is ) . Defined . Definition hzinvmaponpathsminus { a b : hz } ( e : ( - a ) = ( - b ) ) : a = b := grinvmaponpathsinv hzaddabgr e . Lemma hzrplusminus (n m : hz) : n + m - m = n. Proof. unfold hzminus, hzplus, hzplus. rewrite ringassoc1. set (tmp := hzrminus m). unfold hzminus, hzplus in tmp. rewrite tmp. clear tmp. apply hzplusr0. Defined. Opaque hzrplusminus. Lemma hzrplusminus' (n m : hz) : n = n + m - m. Proof. apply pathsinv0. apply hzrplusminus. Defined. Opaque hzrplusminus'. Lemma hzrminusplus (n m : hz) : n - m + m = n. Proof. unfold hzplus, hzminus. rewrite ringassoc1. rewrite hzlminus. apply hzplusr0. Defined. Opaque hzrminusplus. Lemma hzrminusplus' (n m : hz) : n = n - m + m. Proof. apply pathsinv0. apply hzrminusplus. Defined. Opaque hzrminusplus'. (** *** Properties of multiplication on [ hz ] *) Lemma hzmultr1 ( x : hz ) : ( x * 1 ) = x . Proof . apply ( ringrunax2 _ x ) . Defined . Lemma hzmultl1 ( x : hz ) : ( 1 * x ) = x . Proof . apply ( ringlunax2 _ x ) . Defined . Lemma hzmult0x ( x : hz ) : ( 0 * x ) = 0 . Proof . apply ( ringmult0x _ x ) . Defined . Lemma hzmultx0 ( x : hz ) : ( x * 0 ) = 0 . Proof . apply ( ringmultx0 _ x ) . Defined . Lemma hzmultassoc ( x y z : hz ) : ( ( x * y ) * z ) = ( x * ( y * z ) ) . Proof . intros . apply ( ringassoc2 hz x y z ) . Defined . Lemma hzmultcomm ( x y : hz ) : ( x * y ) = ( y * x ) . Proof . intros . apply ( ringcomm2 hz x y ) . Defined . Definition hzneq0andmultlinv ( n m : hz ) ( isnm : hzneq ( n * m ) 0 ) : hzneq n 0 := ringneq0andmultlinv hz n m isnm . Definition hzneq0andmultrinv ( n m : hz ) ( isnm : hzneq ( n * m ) 0 ) : hzneq m 0 := ringneq0andmultrinv hz n m isnm . (** ** Definition and properties of "greater", "less", "greater or equal" and "less or equal" on [ hz ] . *) (** *** Definitions and notations *) Definition hzgth : hrel hz := rigtoringrel natcommrig isplushrelnatgth . Definition hzlth : hrel hz := λ a b, hzgth b a . Definition hzleh : hrel hz := λ a b, make_hProp ( neg ( hzgth a b ) ) ( isapropneg _ ) . Definition hzgeh : hrel hz := λ a b, make_hProp ( neg ( hzgth b a ) ) ( isapropneg _ ) . (** *** Decidability *) Lemma isdecrelhzgth : isdecrel hzgth . Proof . apply ( isdecrigtoringrel natcommrig isplushrelnatgth ) . apply isinvplushrelnatgth . apply isdecrelnatgth . Defined . Definition hzgthdec := make_decrel isdecrelhzgth . (* Canonical Structure hzgthdec . *) Definition isdecrelhzlth : isdecrel hzlth := λ x x', isdecrelhzgth x' x . Definition hzlthdec := make_decrel isdecrelhzlth . (* Canonical Structure hzlthdec . *) Definition isdecrelhzleh : isdecrel hzleh := isdecnegrel _ isdecrelhzgth . Definition hzlehdec := make_decrel isdecrelhzleh . (* Canonical Structure hzlehdec . *) Definition isdecrelhzgeh : isdecrel hzgeh := λ x x', isdecrelhzleh x' x . Definition hzgehdec := make_decrel isdecrelhzgeh . (* Canonical Structure hzgehdec . *) (** *** Properties of individual relations *) (** [ hzgth ] *) Lemma istranshzgth ( n m k : hz ) : hzgth n m -> hzgth m k -> hzgth n k . Proof. apply ( istransabgrdiffrel nataddabmonoid isplushrelnatgth ) . unfold istrans . apply istransnatgth . Defined. Lemma isirreflhzgth ( n : hz ) : neg ( hzgth n n ) . Proof. apply ( isirreflabgrdiffrel nataddabmonoid isplushrelnatgth ) . unfold isirrefl . apply isirreflnatgth . Defined . Lemma hzgthtoneq ( n m : hz ) ( g : hzgth n m ) : neg ( n = m ) . Proof . intros . intro e . rewrite e in g . apply ( isirreflhzgth _ g ) . Defined . Lemma isasymmhzgth ( n m : hz ) : hzgth n m -> hzgth m n -> empty . Proof. apply ( isasymmabgrdiffrel nataddabmonoid isplushrelnatgth ) . unfold isasymm . apply isasymmnatgth . Defined . Lemma isantisymmneghzgth ( n m : hz ) : neg ( hzgth n m ) -> neg ( hzgth m n ) -> n = m . Proof . apply ( isantisymmnegabgrdiffrel nataddabmonoid isplushrelnatgth ) . unfold isantisymmneg . apply isantisymmnegnatgth . Defined . Lemma isnegrelhzgth : isnegrel hzgth . Proof . apply isdecreltoisnegrel . apply isdecrelhzgth . Defined . Lemma iscoantisymmhzgth ( n m : hz ) : neg ( hzgth n m ) -> ( hzgth m n ) ⨿ ( n = m ) . Proof . revert n m. apply isantisymmnegtoiscoantisymm . apply isdecrelhzgth . intros n m . apply isantisymmneghzgth . Defined . Lemma iscotranshzgth ( n m k : hz ) : hzgth n k -> hdisj ( hzgth n m ) ( hzgth m k ) . Proof . intros gxz . destruct ( isdecrelhzgth n m ) as [ gxy | ngxy ] . apply ( hinhpr ( ii1 gxy ) ) . apply hinhpr . apply ii2 . destruct ( isdecrelhzgth m n ) as [ gyx | ngyx ] . apply ( istranshzgth _ _ _ gyx gxz ) . set ( e := isantisymmneghzgth _ _ ngxy ngyx ) . rewrite e in gxz . apply gxz . Defined . (** [ hzlth ] *) Definition istranshzlth ( n m k : hz ) : hzlth n m -> hzlth m k -> hzlth n k := λ lnm lmk, istranshzgth _ _ _ lmk lnm . Definition isirreflhzlth ( n : hz ) : neg ( hzlth n n ) := isirreflhzgth n . Lemma hzlthtoneq ( n m : hz ) ( g : hzlth n m ) : neg ( n = m ) . Proof . intros . intro e . rewrite e in g . apply ( isirreflhzlth _ g ) . Defined . Definition isasymmhzlth ( n m : hz ) : hzlth n m -> hzlth m n -> empty := λ lnm lmn, isasymmhzgth _ _ lmn lnm . Definition isantisymmneghztth ( n m : hz ) : neg ( hzlth n m ) -> neg ( hzlth m n ) -> n = m := λ nlnm nlmn, isantisymmneghzgth _ _ nlmn nlnm . Definition isnegrelhzlth : isnegrel hzlth := λ n m, isnegrelhzgth m n . Definition iscoantisymmhzlth ( n m : hz ) : neg ( hzlth n m ) -> ( hzlth m n ) ⨿ ( n = m ) . Proof . intros nlnm . destruct ( iscoantisymmhzgth m n nlnm ) as [ l | e ] . apply ( ii1 l ) . apply ( ii2 ( pathsinv0 e ) ) . Defined . Definition iscotranshzlth ( n m k : hz ) : hzlth n k -> hdisj ( hzlth n m ) ( hzlth m k ) . Proof . intros lnk . apply ( ( pr1 islogeqcommhdisj ) ( iscotranshzgth _ _ _ lnk ) ) . Defined . (** [ hzleh ] *) Definition istranshzleh ( n m k : hz ) : hzleh n m -> hzleh m k -> hzleh n k . Proof. apply istransnegrel . unfold iscotrans. apply iscotranshzgth . Defined. Definition isreflhzleh ( n : hz ) : hzleh n n := isirreflhzgth n . Definition isantisymmhzleh ( n m : hz ) : hzleh n m -> hzleh m n -> n = m := isantisymmneghzgth n m . Definition isnegrelhzleh : isnegrel hzleh . Proof . apply isdecreltoisnegrel . apply isdecrelhzleh . Defined . Definition iscoasymmhzleh ( n m : hz ) ( nl : neg ( hzleh n m ) ) : hzleh m n := negf ( isasymmhzgth _ _ ) nl . Definition istotalhzleh : istotal hzleh . Proof . intros x y . destruct ( isdecrelhzleh x y ) as [ lxy | lyx ] . apply ( hinhpr ( ii1 lxy ) ) . apply hinhpr . apply ii2 . apply ( iscoasymmhzleh _ _ lyx ) . Defined . (** [ hzgeh ] . *) Definition istranshzgeh ( n m k : hz ) : hzgeh n m -> hzgeh m k -> hzgeh n k := λ gnm gmk, istranshzleh _ _ _ gmk gnm . Definition isreflhzgeh ( n : hz ) : hzgeh n n := isreflhzleh _ . Definition isantisymmhzgeh ( n m : hz ) : hzgeh n m -> hzgeh m n -> n = m := λ gnm gmn, isantisymmhzleh _ _ gmn gnm . Definition isnegrelhzgeh : isnegrel hzgeh := λ n m, isnegrelhzleh m n . Definition iscoasymmhzgeh ( n m : hz ) ( nl : neg ( hzgeh n m ) ) : hzgeh m n := iscoasymmhzleh _ _ nl . Definition istotalhzgeh : istotal hzgeh := λ n m, istotalhzleh m n . (** *** Simple implications between comparisons *) Definition hzgthtogeh ( n m : hz ) : hzgth n m -> hzgeh n m . Proof. intros g . apply iscoasymmhzgeh . apply ( todneg _ g ) . Defined . Definition hzlthtoleh ( n m : hz ) : hzlth n m -> hzleh n m := hzgthtogeh _ _ . Definition hzlehtoneghzgth ( n m : hz ) : hzleh n m -> neg ( hzgth n m ) . Proof. intros is is' . apply ( is is' ) . Defined . Definition hzgthtoneghzleh ( n m : hz ) : hzgth n m -> neg ( hzleh n m ) := λ g l , hzlehtoneghzgth _ _ l g . Definition hzgehtoneghzlth ( n m : hz ) : hzgeh n m -> neg ( hzlth n m ) := λ gnm lnm, hzlehtoneghzgth _ _ gnm lnm . Definition hzlthtoneghzgeh ( n m : hz ) : hzlth n m -> neg ( hzgeh n m ) := λ gnm lnm, hzlehtoneghzgth _ _ lnm gnm . Definition neghzlehtogth ( n m : hz ) : neg ( hzleh n m ) -> hzgth n m := isnegrelhzgth n m . Definition neghzgehtolth ( n m : hz ) : neg ( hzgeh n m ) -> hzlth n m := isnegrelhzlth n m . Definition neghzgthtoleh ( n m : hz ) : neg ( hzgth n m ) -> hzleh n m . Proof . intros ng . destruct ( isdecrelhzleh n m ) as [ l | nl ] . apply l . destruct ( nl ng ) . Defined . Definition neghzlthtogeh ( n m : hz ) : neg ( hzlth n m ) -> hzgeh n m := λ nl, neghzgthtoleh _ _ nl . (** *** Comparison alternatives *) Definition hzgthorleh ( n m : hz ) : ( hzgth n m ) ⨿ ( hzleh n m ) . Proof . intros . apply ( isdecrelhzgth n m ) . Defined . Definition hzlthorgeh ( n m : hz ) : ( hzlth n m ) ⨿ ( hzgeh n m ) := hzgthorleh _ _ . Definition hzneqchoice ( n m : hz ) ( ne : neg ( n = m ) ) : ( hzgth n m ) ⨿ ( hzlth n m ) . Proof . intros . destruct ( hzgthorleh n m ) as [ g | l ] . destruct ( hzlthorgeh n m ) as [ g' | l' ] . destruct ( isasymmhzgth _ _ g g' ) . apply ( ii1 g ) . destruct ( hzlthorgeh n m ) as [ l' | g' ] . apply ( ii2 l' ) . destruct ( ne ( isantisymmhzleh _ _ l g' ) ) . Defined . Definition hzlehchoice ( n m : hz ) ( l : hzleh n m ) : ( hzlth n m ) ⨿ ( n = m ) . Proof . intros . destruct ( hzlthorgeh n m ) as [ l' | g ] . apply ( ii1 l' ) . apply ( ii2 ( isantisymmhzleh _ _ l g ) ) . Defined . Definition hzgehchoice ( n m : hz ) ( g : hzgeh n m ) : ( hzgth n m ) ⨿ ( n = m ) . Proof . intros . destruct ( hzgthorleh n m ) as [ g' | l ] . apply ( ii1 g' ) . apply ( ii2 ( isantisymmhzleh _ _ l g ) ) . Defined . (** *** Mixed transitivities *) Lemma hzgthgehtrans ( n m k : hz ) : hzgth n m -> hzgeh m k -> hzgth n k . Proof. intros gnm gmk . destruct ( hzgehchoice m k gmk ) as [ g' | e ] . apply ( istranshzgth _ _ _ gnm g' ) . rewrite e in gnm . apply gnm . Defined. Lemma hzgehgthtrans ( n m k : hz ) : hzgeh n m -> hzgth m k -> hzgth n k . Proof. intros gnm gmk . destruct ( hzgehchoice n m gnm ) as [ g' | e ] . apply ( istranshzgth _ _ _ g' gmk ) . rewrite e . apply gmk . Defined. Lemma hzlthlehtrans ( n m k : hz ) : hzlth n m -> hzleh m k -> hzlth n k . Proof . intros l1 l2 . apply ( hzgehgthtrans k m n l2 l1 ) . Defined . Lemma hzlehlthtrans ( n m k : hz ) : hzleh n m -> hzlth m k -> hzlth n k . Proof . intros l1 l2 . apply ( hzgthgehtrans k m n l2 l1 ) . Defined . (** *** Addition and comparisons *) (** [ hzgth ] *) Definition hzgthandplusl ( n m k : hz ) : hzgth n m -> hzgth ( k + n ) ( k + m ) . Proof. apply ( pr1 ( isbinopabgrdiffrel nataddabmonoid isplushrelnatgth ) ) . Defined . Definition hzgthandplusr ( n m k : hz ) : hzgth n m -> hzgth ( n + k ) ( m + k ) . Proof. apply ( pr2 ( isbinopabgrdiffrel nataddabmonoid isplushrelnatgth ) ) . Defined . Definition hzgthandpluslinv ( n m k : hz ) : hzgth ( k + n ) ( k + m ) -> hzgth n m . Proof. intros g . set ( g' := hzgthandplusl _ _ ( - k ) g ) . clearbody g' . rewrite ( pathsinv0 ( hzplusassoc _ _ n ) ) in g' . rewrite ( pathsinv0 ( hzplusassoc _ _ m ) ) in g' . rewrite ( hzlminus k ) in g' . rewrite ( hzplusl0 _ ) in g' . rewrite ( hzplusl0 _ ) in g' . apply g' . Defined . Definition hzgthandplusrinv ( n m k : hz ) : hzgth ( n + k ) ( m + k ) -> hzgth n m . Proof. intros l . rewrite ( hzpluscomm n k ) in l . rewrite ( hzpluscomm m k ) in l . apply ( hzgthandpluslinv _ _ _ l ) . Defined . Lemma hzgthsnn ( n : hz ) : hzgth ( n + 1 ) n . Proof . set ( int := hzgthandplusl _ _ n ( ct ( hzgth , isdecrelhzgth, 1 , 0 ) ) ) . clearbody int . rewrite ( hzplusr0 _ ) in int . apply int . Defined . (** [ hzlth ] *) Definition hzlthandplusl ( n m k : hz ) : hzlth n m -> hzlth ( k + n ) ( k + m ) := hzgthandplusl _ _ _ . Definition hzlthandplusr ( n m k : hz ) : hzlth n m -> hzlth ( n + k ) ( m + k ) := hzgthandplusr _ _ _ . Definition hzlthandpluslinv ( n m k : hz ) : hzlth ( k + n ) ( k + m ) -> hzlth n m := hzgthandpluslinv _ _ _ . Definition hzlthandplusrinv ( n m k : hz ) : hzlth ( n + k ) ( m + k ) -> hzlth n m := hzgthandplusrinv _ _ _ . Definition hzlthnsn ( n : hz ) : hzlth n ( n + 1 ) := hzgthsnn n . (** [ hzleh ] *) Definition hzlehandplusl ( n m k : hz ) : hzleh n m -> hzleh ( k + n ) ( k + m ) := negf ( hzgthandpluslinv n m k ) . Definition hzlehandplusr ( n m k : hz ) : hzleh n m -> hzleh ( n + k ) ( m + k ) := negf ( hzgthandplusrinv n m k ) . Definition hzlehandpluslinv ( n m k : hz ) : hzleh ( k + n ) ( k + m ) -> hzleh n m := negf ( hzgthandplusl n m k ) . Definition hzlehandplusrinv ( n m k : hz ) : hzleh ( n + k ) ( m + k ) -> hzleh n m := negf ( hzgthandplusr n m k ) . (** [ hzgeh ] *) Definition hzgehandplusl ( n m k : hz ) : hzgeh n m -> hzgeh ( k + n ) ( k + m ) := negf ( hzgthandpluslinv m n k ) . Definition hzgehandplusr ( n m k : hz ) : hzgeh n m -> hzgeh ( n + k ) ( m + k ) := negf ( hzgthandplusrinv m n k ) . Definition hzgehandpluslinv ( n m k : hz ) : hzgeh ( k + n ) ( k + m ) -> hzgeh n m := negf ( hzgthandplusl m n k ) . Definition hzgehandplusrinv ( n m k : hz ) : hzgeh ( n + k ) ( m + k ) -> hzgeh n m := negf ( hzgthandplusr m n k ) . (** *** Properties of [ hzgth ] in the terminology of algebra1.v (continued below) Note: at the moment we do not need properties of [ hzlth ] , [ hzleh ] and [ hzgeh ] in terminology of algebra1 since the corresponding relations on [ hq ] are bulid from [ hqgth ] . *) Lemma isplushrelhzgth : @isbinophrel hzaddabgr hzgth . Proof . split . apply hzgthandplusl . apply hzgthandplusr . Defined . Lemma isinvplushrelhzgth : @isinvbinophrel hzaddabgr hzgth . Proof . split . apply hzgthandpluslinv . apply hzgthandplusrinv . Defined . Lemma isringmulthzgth : isringmultgt _ hzgth . Proof . apply ( isringrigtoringmultgt natcommrig isplushrelnatgth isrigmultgtnatgth ) . Defined . Lemma isinvringmulthzgth : isinvringmultgt _ hzgth . Proof . apply ( isinvringrigtoringmultgt natcommrig isplushrelnatgth isinvplushrelnatgth isinvrigmultgtnatgth ) . Defined . (** *** Negation and comparisons *) (** [ hzgth ] *) Lemma hzgth0andminus { n : hz } ( is : hzgth n 0 ) : hzlth ( - n ) 0 . Proof . intros . apply ( ringfromgt0 hz isplushrelhzgth ) . apply is . Defined . Lemma hzminusandgth0 { n : hz } ( is : hzgth ( - n ) 0 ) : hzlth n 0 . Proof . intros . apply ( ringtolt0 hz isplushrelhzgth ) . apply is . Defined . (** [ hzlth ] *) Lemma hzlth0andminus { n : hz } ( is : hzlth n 0 ) : hzgth ( - n ) 0 . Proof . intros . apply ( ringfromlt0 hz isplushrelhzgth ) . apply is . Defined . Lemma hzminusandlth0 { n : hz } ( is : hzlth ( - n ) 0 ) : hzgth n 0 . Proof . intros . apply ( ringtogt0 hz isplushrelhzgth ) . apply is . Defined . (* ??? Coq slows down on the proofs of these two lemmas for no good reason. *) (** [ hzleh ] *) Lemma hzleh0andminus { n : hz } ( is : hzleh n 0 ) : hzgeh ( - n ) 0 . Proof . revert is. apply ( negf ( @hzminusandlth0 n ) ) . Defined . Lemma hzminusandleh0 { n : hz } ( is : hzleh ( - n ) 0 ) : hzgeh n 0 . Proof . revert is. apply ( negf ( @hzlth0andminus n ) ) . Defined . (** [ hzgeh ] *) Lemma hzgeh0andminus { n : hz } ( is : hzgeh n 0 ) : hzleh ( - n ) 0 . Proof . revert is . apply ( negf ( @hzminusandgth0 n ) ) . Defined . Lemma hzminusandgeh0 { n : hz } ( is : hzgeh ( - n ) 0 ) : hzleh n 0 . Proof . revert is . apply ( negf ( @hzgth0andminus n ) ) . Defined . (** *** Multiplication and comparisons *) (** [ hzgth ] *) Definition hzgthandmultl ( n m k : hz ) ( is : hzgth k hzzero ) : hzgth n m -> hzgth ( k * n ) ( k * m ) . Proof. revert n m k is. apply ( isringmultgttoislringmultgt _ isplushrelhzgth isringmulthzgth ) . Defined . Definition hzgthandmultr ( n m k : hz ) ( is : hzgth k hzzero ) : hzgth n m -> hzgth ( n * k ) ( m * k ) . Proof . revert n m k is. apply ( isringmultgttoisrringmultgt _ isplushrelhzgth isringmulthzgth ) . Defined . Definition hzgthandmultlinv ( n m k : hz ) ( is : hzgth k hzzero ) : hzgth ( k * n ) ( k * m ) -> hzgth n m . Proof . intros is' . apply ( isinvringmultgttoislinvringmultgt hz isplushrelhzgth isinvringmulthzgth n m k is is' ) . Defined . Definition hzgthandmultrinv ( n m k : hz ) ( is : hzgth k hzzero ) : hzgth ( n * k ) ( m * k ) -> hzgth n m . Proof. intros is' . apply ( isinvringmultgttoisrinvringmultgt hz isplushrelhzgth isinvringmulthzgth n m k is is' ) . Defined . (** [ hzlth ] *) Definition hzlthandmultl ( n m k : hz ) ( is : hzgth k 0 ) : hzlth n m -> hzlth ( k * n ) ( k * m ) := hzgthandmultl _ _ _ is . Definition hzlthandmultr ( n m k : hz ) ( is : hzgth k 0 ) : hzlth n m -> hzlth ( n * k ) ( m * k ) := hzgthandmultr _ _ _ is . Definition hzlthandmultlinv ( n m k : hz ) ( is : hzgth k 0 ) : hzlth ( k * n ) ( k * m ) -> hzlth n m := hzgthandmultlinv _ _ _ is . Definition hzlthandmultrinv ( n m k : hz ) ( is : hzgth k 0 ) : hzlth ( n * k ) ( m * k ) -> hzlth n m := hzgthandmultrinv _ _ _ is . (** [ hzleh ] *) Definition hzlehandmultl ( n m k : hz ) ( is : hzgth k 0 ) : hzleh n m -> hzleh ( k * n ) ( k * m ) := negf ( hzgthandmultlinv _ _ _ is ) . Definition hzlehandmultr ( n m k : hz ) ( is : hzgth k 0 ) : hzleh n m -> hzleh ( n * k ) ( m * k ) := negf ( hzgthandmultrinv _ _ _ is ) . Definition hzlehandmultlinv ( n m k : hz ) ( is : hzgth k 0 ) : hzleh ( k * n ) ( k * m ) -> hzleh n m := negf ( hzgthandmultl _ _ _ is ) . Definition hzlehandmultrinv ( n m k : hz ) ( is : hzgth k 0 ) : hzleh ( n * k ) ( m * k ) -> hzleh n m := negf ( hzgthandmultr _ _ _ is ) . (** [ hzgeh ] *) Definition hzgehandmultl ( n m k : hz ) ( is : hzgth k 0 ) : hzgeh n m -> hzgeh ( k * n ) ( k * m ) := negf ( hzgthandmultlinv _ _ _ is ) . Definition hzgehandmultr ( n m k : hz ) ( is : hzgth k 0 ) : hzgeh n m -> hzgeh ( n * k ) ( m * k ) := negf ( hzgthandmultrinv _ _ _ is ) . Definition hzgehandmultlinv ( n m k : hz ) ( is : hzgth k 0 ) : hzgeh ( k * n ) ( k * m ) -> hzgeh n m := negf ( hzgthandmultl _ _ _ is ) . Definition hzgehandmultrinv ( n m k : hz ) ( is : hzgth k 0 ) : hzgeh ( n * k ) ( m * k ) -> hzgeh n m := negf ( hzgthandmultr _ _ _ is ) . (** Multiplication of positive with positive, positive with negative, negative with positive, two negatives etc. *) Lemma hzmultgth0gth0 { m n : hz } ( ism : hzgth m 0 ) ( isn : hzgth n 0 ) : hzgth ( m * n ) 0 . Proof . intros . apply isringmulthzgth . apply ism . apply isn . Defined . Lemma hzmultgth0geh0 { m n : hz } ( ism : hzgth m 0 ) ( isn : hzgeh n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . destruct ( hzgehchoice _ _ isn ) as [ gn | en ] . apply ( hzgthtogeh _ _ ( hzmultgth0gth0 ism gn ) ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzgeh . Defined . Lemma hzmultgeh0gth0 { m n : hz } ( ism : hzgeh m 0 ) ( isn : hzgth n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . destruct ( hzgehchoice _ _ ism ) as [ gm | em ] . apply ( hzgthtogeh _ _ ( hzmultgth0gth0 gm isn ) ) . rewrite em . rewrite ( hzmult0x _ ) . apply isreflhzgeh . Defined . Lemma hzmultgeh0geh0 { m n : hz } ( ism : hzgeh m 0 ) ( isn : hzgeh n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . destruct ( hzgehchoice _ _ isn ) as [ gn | en ] . apply ( hzmultgeh0gth0 ism gn ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzgeh . Defined . Lemma hzmultgth0lth0 { m n : hz } ( ism : hzgth m 0 ) ( isn : hzlth n 0 ) : hzlth ( m * n ) 0 . Proof . intros . apply ( ringmultgt0lt0 hz isplushrelhzgth isringmulthzgth ) . apply ism . apply isn . Defined . Lemma hzmultgth0leh0 { m n : hz } ( ism : hzgth m 0 ) ( isn : hzleh n 0 ) : hzleh ( m * n ) 0 . Proof . intros . destruct ( hzlehchoice _ _ isn ) as [ ln | en ] . apply ( hzlthtoleh _ _ ( hzmultgth0lth0 ism ln ) ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzleh . Defined . Lemma hzmultgeh0lth0 { m n : hz } ( ism : hzgeh m 0 ) ( isn : hzlth n 0 ) : hzleh ( m * n ) 0 . Proof . intros . destruct ( hzlehchoice _ _ ism ) as [ lm | em ] . apply ( hzlthtoleh _ _ ( hzmultgth0lth0 lm isn ) ) . destruct em . rewrite ( hzmult0x _ ) . apply isreflhzleh . Defined . Lemma hzmultgeh0leh0 { m n : hz } ( ism : hzgeh m 0 ) ( isn : hzleh n 0 ) : hzleh ( m * n ) 0 . Proof . intros . destruct ( hzlehchoice _ _ isn ) as [ ln | en ] . apply ( hzmultgeh0lth0 ism ln ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzleh . Defined . Lemma hzmultlth0gth0 { m n : hz } ( ism : hzlth m 0 ) ( isn : hzgth n 0 ) : hzlth ( m * n ) 0 . Proof . intros . rewrite ( hzmultcomm ) . apply hzmultgth0lth0 . apply isn . apply ism . Defined . Lemma hzmultlth0geh0 { m n : hz } ( ism : hzlth m 0 ) ( isn : hzgeh n 0 ) : hzleh ( m * n ) 0 . Proof . intros . rewrite ( hzmultcomm ) . apply hzmultgeh0lth0 . apply isn . apply ism . Defined . Lemma hzmultleh0gth0 { m n : hz } ( ism : hzleh m 0 ) ( isn : hzgth n 0 ) : hzleh ( m * n ) 0 . Proof . intros . rewrite ( hzmultcomm ) . apply hzmultgth0leh0 . apply isn . apply ism . Defined . Lemma hzmultleh0geh0 { m n : hz } ( ism : hzleh m 0 ) ( isn : hzgeh n 0 ) : hzleh ( m * n ) 0 . Proof . intros . rewrite ( hzmultcomm ) . apply hzmultgeh0leh0 . apply isn . apply ism . Defined . Lemma hzmultlth0lth0 { m n : hz } ( ism : hzlth m 0 ) ( isn : hzlth n 0 ) : hzgth ( m * n ) 0 . Proof . intros . assert ( ism' := hzlth0andminus ism ) . assert ( isn' := hzlth0andminus isn ) . assert ( int := isringmulthzgth _ _ ism' isn' ) . rewrite ( ringmultminusminus hz ) in int . apply int . Defined . Lemma hzmultlth0leh0 { m n : hz } ( ism : hzlth m 0 ) ( isn : hzleh n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . intros . destruct ( hzlehchoice _ _ isn ) as [ ln | en ] . apply ( hzgthtogeh _ _ ( hzmultlth0lth0 ism ln ) ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzgeh . Defined . Lemma hzmultleh0lth0 { m n : hz } ( ism : hzleh m 0 ) ( isn : hzlth n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . destruct ( hzlehchoice _ _ ism ) as [ lm | em ] . apply ( hzgthtogeh _ _ ( hzmultlth0lth0 lm isn ) ) . rewrite em . rewrite ( hzmult0x _ ) . apply isreflhzgeh . Defined . Lemma hzmultleh0leh0 { m n : hz } ( ism : hzleh m 0 ) ( isn : hzleh n 0 ) : hzgeh ( m * n ) 0 . Proof . intros . destruct ( hzlehchoice _ _ isn ) as [ ln | en ] . apply ( hzmultleh0lth0 ism ln ) . rewrite en . rewrite ( hzmultx0 m ) . apply isreflhzgeh . Defined . (** *** [ hz ] as an integral domain *) Lemma isintdomhz : isintdom hz . Proof . split with isnonzerorighz . intros a b e0 . destruct ( isdeceqhz a 0 ) as [ ea | nea ] . apply ( hinhpr ( ii1 ea ) ) . destruct ( isdeceqhz b 0 ) as [ eb | neb ] . apply ( hinhpr ( ii2 eb ) ) . destruct ( hzneqchoice _ _ nea ) as [ ga | la ] . destruct ( hzneqchoice _ _ neb ) as [ gb | lb ] . destruct ( hzgthtoneq _ _ ( hzmultgth0gth0 ga gb ) e0 ) . destruct ( hzlthtoneq _ _ ( hzmultgth0lth0 ga lb ) e0 ) . destruct ( hzneqchoice _ _ neb ) as [ gb | lb ] . destruct ( hzlthtoneq _ _ ( hzmultlth0gth0 la gb ) e0 ) . destruct ( hzgthtoneq _ _ ( hzmultlth0lth0 la lb ) e0 ) . Defined . Definition hzintdom : intdom := tpair _ _ isintdomhz . Definition hzneq0andmult ( n m : hz ) ( isn : hzneq n 0 ) ( ism : hzneq m 0 ) : hzneq ( n * m ) 0 := intdomneq0andmult hzintdom n m isn ism . Lemma hzmultlcan ( a b c : hz ) ( ne : neg ( c = 0 ) ) ( e : ( c * a ) = ( c * b ) ) : a = b . Proof . intros . apply ( intdomlcan hzintdom _ _ _ ne e ) . Defined . Lemma hzmultrcan ( a b c : hz ) ( ne : neg ( c = 0 ) ) ( e : ( a * c ) = ( b * c ) ) : a = b . Proof . intros . apply ( intdomrcan hzintdom _ _ _ ne e ) . Defined . Lemma isinclhzmultl ( n : hz )( ne : neg ( n = 0 ) ) : isincl ( λ m : hz, n * m ) . Proof. intros . apply ( pr1 ( intdomiscancelable hzintdom n ne ) ) . Defined . Lemma isinclhzmultr ( n : hz )( ne : neg ( n = 0 ) ) : isincl ( λ m : hz, m * n ) . Proof. intros . apply ( pr2 ( intdomiscancelable hzintdom n ne ) ) . Defined. (** *** Comparisons and [ n -> n + 1 ] *) Definition hzgthtogths ( n m : hz ) : hzgth n m -> hzgth ( n + 1 ) m . Proof. intros is . apply ( istranshzgth _ _ _ ( hzgthsnn n ) is ) . Defined . Definition hzlthtolths ( n m : hz ) : hzlth n m -> hzlth n ( m + 1 ) := hzgthtogths _ _ . Definition hzlehtolehs ( n m : hz ) : hzleh n m -> hzleh n ( m + 1 ) . Proof . intros is . apply ( istranshzleh _ _ _ is ( hzlthtoleh _ _ ( hzlthnsn _ ) ) ) . Defined . Definition hzgehtogehs ( n m : hz ) : hzgeh n m -> hzgeh ( n + 1 ) m := hzlehtolehs _ _ . (** *** Two comparisons and [ n -> n + 1 ] *) Lemma hzgthtogehsn ( n m : hz ) : hzgth n m -> hzgeh n ( m + 1 ) . Proof. assert ( int : ∏ n m , isaprop ( hzgth n m -> hzgeh n ( m + 1 ) ) ) . { intros . apply impred . intro . apply ( pr2 _ ) . } unfold hzgth in * . apply ( setquotuniv2prop _ ( λ n m, make_hProp _ ( int n m ) ) ) . set ( R := abgrdiffrelint nataddabmonoid natgth ) . intros x x' . change ( R x x' -> ( neg ( R ( @op ( abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig) ) x' ( make_dirprod 1%nat 0%nat ) ) x ) ) ) . unfold R . unfold abgrdiffrelint . simpl . apply ( @hinhuniv _ (make_hProp ( neg ( ishinh_UU _ ) ) ( isapropneg _ ) ) ) . intro t2 . simpl . unfold neg . apply ( @hinhuniv _ ( make_hProp _ isapropempty ) ) . intro t2' . set ( x1 := pr1 x ) . set ( a1 := pr2 x ) . set ( x2 := pr1 x' ) . set ( a2 := pr2 x' ) . set ( c1 := pr1 t2 ) . assert ( r1 := pr2 t2 ) . change ( pr1 ( ( x1 + a2 + c1 ) > ( x2 + a1 + c1 ) ) ) in r1 . set ( c2 := pr1 t2' ) . assert ( r2 := pr2 t2' ) . change ( pr1 ( ( ( x2 + 1 ) + a1 + c2 ) > ( x1 + ( a2 + 0 ) + c2 ) ) ) in r2 . assert ( r1' := natgthandplusrinv _ _ c1 r1 ) . assert ( r2' := natgthandplusrinv _ _ c2 r2 ) . rewrite ( natplusr0 _ ) in r2' . rewrite ( natpluscomm _ 1 ) in r2' . rewrite ( natplusassoc _ _ _ ) in r2' . change (1 + (x2 + a1) > x1 + a2) with (x1 + a2 ≤ x2 + a1) in r2'. contradicts (natlehneggth r2') r1'. Defined . Lemma hzgthsntogeh ( n m : hz ) : hzgth ( n + 1 ) m -> hzgeh n m . Proof. intros a . apply (hzgehandplusrinv n m 1) . apply ( hzgthtogehsn ( n + 1 ) m a ) . Defined. (* PeWa *) Lemma hzlehsntolth ( n m : hz ) : hzleh ( n + 1 ) m -> hzlth n m . Proof. intros X . apply ( hzlthlehtrans _ _ _ ( hzlthnsn n ) X ) . Defined . Lemma hzlthtolehsn ( n m : hz ) : hzlth n m -> hzleh ( n + 1 ) m . Proof. intros X . apply ( hzgthtogehsn m n X ) . Defined . Lemma hzlthsntoleh ( n m : hz ) : hzlth n ( m + 1 ) -> hzleh n m . Proof. intros a . apply (hzlehandplusrinv n m 1) . apply ( hzlthtolehsn n ( m + 1 ) a ) . Defined. (* PeWa *) Lemma hzgehsntogth ( n m : hz ) : hzgeh n ( m + 1 ) -> hzgth n m . Proof. intros X . apply ( hzlehsntolth m n X ) . Defined . (** *** Comparsion alternatives and [ n -> n + 1 ] *) Lemma hzlehchoice2 ( n m : hz ) : hzleh n m -> coprod ( hzleh ( n + 1 ) m ) ( n = m ) . Proof . intros l . destruct ( hzlehchoice n m l ) as [ l' | e ] . apply ( ii1 ( hzlthtolehsn _ _ l' ) ) . apply ( ii2 e ) . Defined . Lemma hzgehchoice2 ( n m : hz ) : hzgeh n m -> coprod ( hzgeh n ( m + 1 ) ) ( n = m ) . Proof . intros g . destruct ( hzgehchoice n m g ) as [ g' | e ] . apply ( ii1 ( hzgthtogehsn _ _ g' ) ) . apply ( ii2 e ) . Defined . Lemma hzgthchoice2 ( n m : hz ) : hzgth n m -> coprod ( hzgth n ( m + 1 ) ) ( n = ( m + 1 ) ) . Proof. intros g . destruct ( hzgehchoice _ _ ( hzgthtogehsn _ _ g ) ) as [ g' | e ] . apply ( ii1 g' ) . apply ( ii2 e ) . Defined . Lemma hzlthchoice2 ( n m : hz ) : hzlth n m -> coprod ( hzlth ( n + 1 ) m ) ( ( n + 1 ) = m ) . Proof. intros l . destruct ( hzlehchoice _ _ ( hzlthtolehsn _ _ l ) ) as [ l' | e ] . apply ( ii1 l' ) . apply ( ii2 e ) . Defined . (** *** Operations and comparisons on [ hz ] and [ natnattohz ] *) Lemma natnattohzandgth ( xa1 xa2 : dirprod nat nat ) ( is : hzgth ( setquotpr _ xa1 ) ( setquotpr _ xa2 ) ) : natgth ( ( pr1 xa1 ) + ( pr2 xa2 ) ) ( ( pr1 xa2 ) + ( pr2 xa1 ) ) . Proof . intros . change ( ishinh_UU ( total2 ( λ a0, natgth (pr1 xa1 + pr2 xa2 + a0) (pr1 xa2 + pr2 xa1 + a0) ) ) ) in is . generalize is . apply @hinhuniv . intro t2 . set ( a0 := pr1 t2 ) . assert ( g := pr2 t2 ) . change ( pr1 ( natgth (pr1 xa1 + pr2 xa2 + a0) (pr1 xa2 + pr2 xa1 + a0) ) ) in g . apply ( natgthandplusrinv _ _ a0 g ) . Defined . Lemma natnattohzandlth ( xa1 xa2 : dirprod nat nat ) ( is : hzlth ( setquotpr _ xa1 ) ( setquotpr _ xa2 ) ) : natlth ( ( pr1 xa1 ) + ( pr2 xa2 ) ) ( ( pr1 xa2 ) + ( pr2 xa1 ) ) . Proof . intros . apply ( natnattohzandgth xa2 xa1 is ) . Defined . (** *** Canonical rig homomorphism from [ nat ] to [ hz ] *) Definition nattohz : nat -> hz := λ n, setquotpr _ ( make_dirprod n 0%nat ) . Definition isinclnattohz : isincl nattohz := isincltoringdiff natcommrig ( λ n, isinclnatplusr n ) . Definition nattohzandneq ( n m : nat ) ( is : natnegpaths n m ) : hzneq ( nattohz n ) ( nattohz m ) := negf ( invmaponpathsincl _ isinclnattohz n m ) is . Definition nattohzand0 : ( nattohz 0%nat ) = 0 := idpath _ . Definition nattohzandS ( n : nat ) : ( nattohz ( S n ) ) = ( 1 + nattohz n ) := isbinop1funtoringdiff natcommrig 1%nat n . Definition nattohzand1 : ( nattohz 1%nat ) = 1 := idpath _ . Lemma nattorig_nattohz : ∏ n : nat, nattorig (X := hz) n = nattohz n. Proof. induction n as [|n IHn]. - unfold nattorig, nattohz ; simpl. reflexivity. - rewrite nattorigS, IHn. apply pathsinv0, nattohzandS. Qed. Definition nattohzandplus ( n m : nat ) : ( nattohz ( n + m )%nat ) = ( nattohz n + nattohz m ) := isbinop1funtoringdiff natcommrig n m . Definition nattohzandminus ( n m : nat ) ( is : natgeh n m ) : ( nattohz ( n - m )%nat ) = ( nattohz n - nattohz m ) . Proof . intros . apply ( hzplusrcan _ _ ( nattohz m ) ) . unfold hzminus . rewrite ( hzplusassoc ( nattohz n ) ( - nattohz m ) ( nattohz m ) ) . rewrite ( hzlminus _ ) . rewrite hzplusr0 . rewrite ( pathsinv0 ( nattohzandplus _ _ ) ) . rewrite ( minusplusnmm _ _ is ) . apply idpath . Defined . Opaque nattohzandminus . Definition nattohzandmult ( n m : nat ) : ( nattohz ( n * m )%nat ) = ( nattohz n * nattohz m ) . Proof . intros . simpl . change nattohz with ( toringdiff natcommrig ) . apply ( isbinop2funtoringdiff natcommrig n m ) . Defined . Definition nattohzandgth ( n m : nat ) ( is : natgth n m ) : hzgth ( nattohz n ) ( nattohz m ) := iscomptoringdiff natcommrig isplushrelnatgth n m is . Definition nattohzandlth ( n m : nat ) ( is : natlth n m ) : hzlth ( nattohz n ) ( nattohz m ) := nattohzandgth m n is . Definition nattohzandleh ( n m : nat ) ( is : natleh n m ) : hzleh ( nattohz n ) ( nattohz m ) . Proof . intros . destruct ( natlehchoice _ _ is ) as [ l | e ] . apply ( hzlthtoleh _ _ ( nattohzandlth _ _ l ) ) . rewrite e . apply ( isreflhzleh ) . Defined . Definition nattohzandgeh ( n m : nat ) ( is : natgeh n m ) : hzgeh ( nattohz n ) ( nattohz m ) := nattohzandleh _ _ is . (** *** Addition and subtraction on [ nat ] and [ hz ] *) (** *** Absolute value on [ hz ] *) Definition hzabsvalint : ( dirprod nat nat ) -> nat . Proof . intro nm . destruct ( natgthorleh ( pr1 nm ) ( pr2 nm ) ) . apply ( sub ( pr1 nm ) ( pr2 nm ) ) . apply ( sub ( pr2 nm ) ( pr1 nm ) ) . Defined . Lemma hzabsvalintcomp : @iscomprelfun ( dirprod nat nat ) nat ( hrelabgrdiff nataddabmonoid ) hzabsvalint . Proof . unfold iscomprelfun . intros x x' . unfold hrelabgrdiff . simpl . apply ( @hinhuniv _ ( make_hProp _ ( isasetnat (hzabsvalint x) (hzabsvalint x') ) ) ) . unfold hzabsvalint . set ( n := ( pr1 x ) : nat ) . set ( m := ( pr2 x ) : nat ) . set ( n' := ( pr1 x' ) : nat ) . set ( m' := ( pr2 x' ) : nat ) . set ( int := natgthorleh n m ) . set ( int' := natgthorleh n' m' ) . intro tt0 . simpl . destruct tt0 as [ x0 eq ] . simpl in eq . assert ( e' := invmaponpathsincl _ ( isinclnatplusr x0 ) _ _ eq ) . destruct int as [isgt | isle ] . destruct int' as [ isgt' | isle' ] . apply ( invmaponpathsincl _ ( isinclnatplusr ( m + m' ) ) ) . rewrite ( pathsinv0 ( natplusassoc ( n - m ) m m' ) ) . rewrite ( natpluscomm m m' ) . rewrite ( pathsinv0 ( natplusassoc ( n' - m' ) m' m ) ) . rewrite ( minusplusnmm n m ( natgthtogeh _ _ isgt ) ) . rewrite ( minusplusnmm n' m' ( natgthtogeh _ _ isgt' ) ) . apply e' . assert ( e'' := natlehandplusl n' m' n isle' ) . assert ( e''' := natgthandplusr n m n' isgt ) . assert ( e'''' := natlthlehtrans _ _ _ e''' e'' ) . rewrite e' in e'''' . rewrite ( natpluscomm m n' ) in e'''' . destruct ( isirreflnatgth _ e'''' ) . destruct int' as [ isgt' | isle' ] . destruct ( natpluscomm m n') . set ( e'' := natlehandplusr n m m' isle ) . set ( e''' := natgthandplusl n' m' m isgt' ) . set ( e'''' := natlehlthtrans _ _ _ e'' e''' ) . rewrite e' in e'''' . destruct ( isirreflnatgth _ e'''' ) . apply ( invmaponpathsincl _ ( isinclnatplusr ( n + n') ) ) . rewrite ( pathsinv0 ( natplusassoc ( m - n ) n n' ) ) . rewrite ( natpluscomm n n' ) . rewrite ( pathsinv0 ( natplusassoc ( m' - n') n' n ) ) . rewrite ( minusplusnmm m n isle ) . rewrite ( minusplusnmm m' n' isle' ) . rewrite ( natpluscomm m' n ) . rewrite ( natpluscomm m n' ) . apply ( pathsinv0 e' ) . Defined . Definition hzabsval : hz -> nat := setquotuniv _ natset hzabsvalint hzabsvalintcomp . Lemma hzabsval0 : ( hzabsval 0 ) = 0%nat . Proof . apply idpath . Defined . Lemma hzabsvalgth0 { x : hz } ( is : hzgth x 0 ) : ( nattohz ( hzabsval x ) ) = x . Proof . revert x is. assert ( int : ∏ x : hz , isaprop ( hzgth x 0 -> ( nattohz ( hzabsval x ) ) = x ) ) . intro . apply impred . intro . apply ( setproperty hz ) . apply ( setquotunivprop _ ( λ x, make_hProp _ ( int x ) ) ) . intros xa g . simpl in xa . assert ( g' := natnattohzandgth _ _ g ) . simpl in g' . simpl . change (( setquotpr (eqrelabgrdiff (rigaddabmonoid natcommrig)) ( make_dirprod ( hzabsvalint xa ) 0%nat ) ) = ( setquotpr (eqrelabgrdiff (rigaddabmonoid natcommrig)) xa ) ) . apply weqpathsinsetquot . simpl . apply hinhpr . split with 0%nat . change ( pr1 ( natgth ( pr1 xa + 0%nat ) ( pr2 xa ) ) ) in g' . rewrite ( natplusr0 _ ) in g' . change ((hzabsvalint xa + pr2 xa + 0)%nat = (pr1 xa + 0 + 0)%nat ) . rewrite ( natplusr0 _ ) . rewrite ( natplusr0 _ ) . rewrite ( natplusr0 _ ) . unfold hzabsvalint . destruct ( natgthorleh (pr1 xa) (pr2 xa) ) as [ g'' | l ] . rewrite ( minusplusnmm _ _ ( natlthtoleh _ _ g'' ) ) . apply idpath . contradicts (natlehneggth l) g'. Defined . Opaque hzabsvalgth0 . Lemma hzabsvalgeh0 { x : hz } ( is : hzgeh x 0 ) : ( nattohz ( hzabsval x ) ) = x . Proof . intros . destruct ( hzgehchoice _ _ is ) as [ g | e ] . apply ( hzabsvalgth0 g ) . rewrite e . apply idpath . Defined . Lemma hzabsvallth0 { x : hz } ( is : hzlth x 0 ) : ( nattohz ( hzabsval x ) ) = ( - x ) . Proof . revert x is. assert ( int : ∏ x : hz , isaprop ( hzlth x 0 -> ( nattohz ( hzabsval x ) ) = ( - x ) ) ) . intro . apply impred . intro . apply ( setproperty hz ) . apply ( setquotunivprop _ ( λ x, make_hProp _ ( int x ) ) ) . intros xa l . simpl in xa . assert ( l' := natnattohzandlth _ _ l ) . simpl in l' . simpl . change (( setquotpr (eqrelabgrdiff (rigaddabmonoid natcommrig)) ( make_dirprod ( hzabsvalint xa ) 0%nat ) ) = ( setquotpr (eqrelabgrdiff (rigaddabmonoid natcommrig)) ( make_dirprod ( pr2 xa ) ( pr1 xa ) ) ) ) . apply weqpathsinsetquot . simpl . apply hinhpr . split with 0%nat . change ( pr1 ( natlth ( pr1 xa + 0%nat ) ( pr2 xa ) ) ) in l' . rewrite ( natplusr0 _ ) in l' . change ((hzabsvalint xa + pr1 xa + 0)%nat = (pr2 xa + 0 + 0)%nat). rewrite ( natplusr0 _ ) . rewrite ( natplusr0 _ ) . rewrite ( natplusr0 _ ) . unfold hzabsvalint . destruct ( natgthorleh (pr1 xa) (pr2 xa) ) as [ g | l'' ] . destruct ( isasymmnatgth _ _ g l' ) . rewrite ( minusplusnmm _ _ l'' ) . apply idpath . Defined . Opaque hzabsvallth0 . Lemma hzabsvalleh0 { x : hz } ( is : hzleh x 0 ) : ( nattohz ( hzabsval x ) ) = ( - x ) . Proof . intros . destruct ( hzlehchoice _ _ is ) as [ l | e ] . apply ( hzabsvallth0 l ) . rewrite e . apply idpath . Defined . Lemma hzabsvaleq0 { x : hz } ( e : ( hzabsval x ) = 0%nat ) : x = 0 . Proof . intros . destruct ( isdeceqhz x 0 ) as [ e0 | ne0 ] . apply e0 . destruct ( hzneqchoice _ _ ne0 ) as [ g | l ] . assert ( e' := hzabsvalgth0 g ) . rewrite e in e' . change ( 0 = x ) in e' . apply ( pathsinv0 e' ) . assert ( e' := hzabsvallth0 l ) . rewrite e in e' . change ( 0 = ( - x ) ) in e' . assert ( g := hzlth0andminus l ) . rewrite e' in g . destruct ( isirreflhzgth _ g ) . Defined . Definition hzabsvalneq0 { x : hz } ne := neg_to_negProp (nP := natneq _ _) (negf ( @hzabsvaleq0 x ) ne). Lemma hzabsvalandmult ( a b : hz ) : ( ( hzabsval a ) * ( hzabsval b ) )%nat = ( hzabsval ( a * b ) ) . Proof . intros . apply ( invmaponpathsincl _ isinclnattohz ) . rewrite ( nattohzandmult _ _ ) . destruct ( hzgthorleh a 0 ) as [ ga | lea ] . destruct ( hzgthorleh b 0 ) as [ gb | leb ] . rewrite ( hzabsvalgth0 ga ) . rewrite ( hzabsvalgth0 gb ) . rewrite ( hzabsvalgth0 ( hzmultgth0gth0 ga gb ) ) . apply idpath . rewrite ( hzabsvalgth0 ga ) . rewrite ( hzabsvalleh0 leb ) . rewrite ( hzabsvalleh0 ( hzmultgth0leh0 ga leb ) ) . apply ( ringrmultminus hz ) . destruct ( hzgthorleh b 0 ) as [ gb | leb ] . rewrite ( hzabsvalgth0 gb ) . rewrite ( hzabsvalleh0 lea ) . rewrite ( hzabsvalleh0 ( hzmultleh0gth0 lea gb ) ) . apply ( ringlmultminus hz ) . rewrite ( hzabsvalleh0 lea ) . rewrite ( hzabsvalleh0 leb ) . rewrite ( hzabsvalgeh0 ( hzmultleh0leh0 lea leb ) ) . apply (ringmultminusminus hz ) . Defined . (** *** Some common equalities on integers *) (** These lemmas are used for example in Complexes.v to construct complexes. *) Local Opaque hz isdecrelhzeq iscommringops. Lemma hzeqbooleqii (i : hz) : hzbooleq i i = true. Proof. unfold hzbooleq. unfold decreltobrel. induction (pr2 hzdeceq i i) as [T | F]. - apply idpath. - apply fromempty. apply F. apply idpath. Qed. Lemma hzbooleqisi (i : hz) : hzbooleq i (i + 1) = false. Proof. apply negrtopaths. apply (negf (λ e, hzpluslcan _ _ _ (! (hzplusr0 i @ e)))); clear i. confirm_not_equal isdecrelhzeq. Qed. Lemma hzbooleqisi' (i : hz) : hzbooleq i (i + 1) = false. Proof. (* prove it again, to illustrate how to avoid the tactic [confirm_not_equal] *) apply negrtopaths. apply (negf (λ e, hzpluslcan _ _ _ (! (hzplusr0 i @ e)))); clear i. simple refine (confirm_not_equal isdecrelhzeq _ _ _). reflexivity. Qed. Lemma hzbooleqissi (i : hz) : hzbooleq i (i + 1 + 1) = false. Proof. apply negrtopaths. rewrite hzplusassoc. apply (negf (λ e, hzpluslcan _ _ _ (! (hzplusr0 i @ e)))); clear i. confirm_not_equal isdecrelhzeq. Qed. Lemma hzeqeisi {i i0 : hz} (e : hzeq i i0) (e' : hzeq i (i0 + 1)) : empty. Proof. apply nopathstruetofalse. use (pathscomp0 _ (hzbooleqisi i0)). rewrite <- e'. rewrite <- e. apply (! (hzeqbooleqii i)). Qed. Lemma hzeqisi {i : hz} (e' : hzeq i (i + 1)) : empty. Proof. apply nopathstruetofalse. rewrite <- (hzbooleqisi i). rewrite <- e'. apply (! (hzeqbooleqii i)). Qed. Lemma hzeqissi {i : hz} (e : hzeq i (i + 1 + 1)) : empty. Proof. set (tmp := hzbooleqissi i). cbn in e. rewrite <- e in tmp. rewrite (hzeqbooleqii i) in tmp. apply nopathstruetofalse. apply tmp. Qed. Lemma hzeqeissi {i i0 : hz} (e : hzeq i i0) (e' : hzeq i (i0 + 1 + 1)) : empty. Proof. cbn in e. rewrite e in e'. apply (hzeqissi e'). Qed. Lemma hzeqsnmnsm {n m : hz} (e : hzeq (n + 1) m) (e' : hzeq n (m + 1)) : empty. Proof. cbn in e. rewrite <- e in e'. apply (hzeqissi e'). Qed. Lemma hzeqnmplusr {n m i : hz} (e : n = m) (e' : ¬ (n + i = m + i)) : empty. Proof. apply e'. exact (maponpaths_2 _ e _). Qed. Lemma hzeqnmplusr' {n m i : hz} (e : ¬ (n = m)) (e' : n + i = m + i) : empty. Proof. apply e. exact (hzplusrcan _ _ i e'). Qed. Lemma isdecrelhzeqi (i : hz) : isdecrelhzeq i i = ii1 (idpath _). Proof. induction (isdecrelhzeq i i) as [T | F]. - apply maponpaths. apply isasethz. - apply fromempty. apply F. apply idpath. Qed. Lemma isdecrelhzeqminusplus (i j : hz) : isdecrelhzeq i (i - j + j) = ii1 (hzrminusplus' i j). Proof. induction (isdecrelhzeq i (i - j + j)) as [T | F]. - apply maponpaths. apply isasethz. - apply fromempty. apply F. apply (hzrminusplus' i j). Qed. Lemma isdecrelhzeqminusplus' (i j : hz) : isdecrelhzeq (i - j + j) i = ii1 (hzrminusplus i j). Proof. induction (isdecrelhzeq (i - j + j) i) as [T | F]. - apply maponpaths. apply isasethz. - apply fromempty. apply F. apply (hzrminusplus i j). Qed. Lemma hzeqpii {i : hz} : i - 1 != i. Proof. apply (negf (λ e, hzpluslcan _ _ _ (e @ ! hzplusr0 i))); clear i. confirm_not_equal isdecrelhzeq. Qed. Lemma isdecrelhzeqpii (i : hz) : isdecrelhzeq (i - 1) i = ii2 (fun (e : hzeq (i - 1) i) => hzeqpii e). Proof. induction (isdecrelhzeq (i - 1) i) as [e | n]. - apply fromempty. apply (hzeqpii e). - apply maponpaths. apply funextfun. intros e. apply fromempty. apply n. apply e. Qed. Local Transparent hz isdecrelhzeq iscommringops. (** *** [hz] is an archimedean ring *) Local Open Scope hz_scope . Lemma isarchhz : isarchring (X := hz) hzgth. Proof. simple refine (isarchrigtoring _ _ _ _ _ _). - reflexivity. - intros n m. apply istransnatgth. - apply isarchrig_setquot_aux. + split. * apply natgthandpluslinv. * apply natgthandplusrinv. + apply isarchnat. Qed. Lemma isarchhz_one : ∏ x : hz, hzgth x 0 → ∃ n : nat, hzgth (nattohz n * x) 1. Proof. intros x Hx. generalize (isarchring_1 _ isarchhz x Hx). apply hinhfun. intros n. exists (pr1 n). rewrite <- nattorig_nattohz. exact (pr2 n). Qed. Lemma isarchhz_gt : ∏ x : hz, ∃ n : nat, hzgth (nattohz n) x. Proof. intros x. generalize (isarchring_2 _ isarchhz x). apply hinhfun. intros n. exists (pr1 n). rewrite <- nattorig_nattohz. exact (pr2 n). Qed. (** **** hz -> abgr, 1 ↦ x, n ↦ x + x + ... + x (n times), [hz_abmonoid_monoidfun] *) Definition nat_to_monoid_fun {X : monoid} (x : X) : natset -> X. Proof. intros n. induction n as [ | n IHn]. - exact (unel X). - exact (@op X x IHn). Defined. Lemma nat_to_monoid_fun_unel {X : monoid} (x : X) : nat_to_monoid_fun x O = (unel X). Proof. exact (idpath (unel X)). Defined. Lemma nat_to_monoid_fun_S {X : abmonoid} (x : X) (n : nat) : nat_to_monoid_fun x (S n) = (nat_to_monoid_fun x n * x)%multmonoid. Proof. induction n as [ | n IHn]. - exact (commax X x (unel X)). - cbn. rewrite (assocax X). use two_arg_paths. + use idpath. + exact (commax X x _). Qed. Lemma nat_to_abmonoid_fun_plus {X : monoid} (x : X) (n m : nat) : nat_to_monoid_fun x (n + m)%nat = @op X (nat_to_monoid_fun x n) (nat_to_monoid_fun x m). Proof. revert m. induction n as [ | n IHn]. - intros m. rewrite (lunax X). use idpath. - intros m. cbn. rewrite (assocax X). use two_arg_paths. + use idpath. + exact (IHn m). Qed. Definition nat_nat_to_monoid_fun {X : gr} (x : X) : natset × natset -> X. Proof. intros n. exact (@op X (nat_to_monoid_fun x (dirprod_pr1 n)) (nat_to_monoid_fun (grinv X x) (dirprod_pr2 n))). Defined. Lemma nat_to_monoid_unel' {X : abgr} (x : X) (n : nat) : ((nat_to_monoid_fun x n) * (nat_to_monoid_fun (grinv X x) n))%multmonoid = (unel X). Proof. induction n as [ | n IHn]. - use (runax X). - Opaque nat_to_monoid_fun. cbn in *. rewrite (@nat_to_monoid_fun_S X x). rewrite (@nat_to_monoid_fun_S X (grinv X x)). rewrite (commax X _ x). rewrite (assocax X). rewrite <- (assocax X (@nat_to_monoid_fun X x n)). use (pathscomp0 (maponpaths (λ xx : pr1 X, (x * (xx * (grinv X x))))%multmonoid IHn)). clear IHn. use (pathscomp0 _ (grrinvax X x)). use two_arg_paths. + use idpath. + use (lunax X). Qed. Transparent nat_to_monoid_fun. Lemma nat_nat_to_monoid1 {X : gr} (x : X) {n1 n2 m2 : nat} (e : n2 = m2) : nat_nat_to_monoid_fun x (make_dirprod n1 n2) = nat_nat_to_monoid_fun x (make_dirprod n1 m2). Proof. induction e. use idpath. Qed. Lemma nat_nat_to_monoid2 {X : gr} (x : X) {n1 m1 n2 : nat} (e : n1 = m1) : nat_nat_to_monoid_fun x (make_dirprod n1 n2) = nat_nat_to_monoid_fun x (make_dirprod m1 n2). Proof. induction e. use idpath. Qed. Definition nataddabmonoid_nataddabmonoid_to_monoid_fun {X : gr} (x : X) : abmonoiddirprod nataddabmonoid nataddabmonoid -> X := nat_nat_to_monoid_fun x. Opaque nat_to_monoid_fun. Lemma nat_nat_monoid_fun_isbinopfun {X : abgr} (x : X) : isbinopfun (nataddabmonoid_nataddabmonoid_to_monoid_fun x). Proof. use make_isbinopfun. intros n m. induction n as [n1 n2]. induction m as [m1 m2]. cbn. unfold nataddabmonoid_nataddabmonoid_to_monoid_fun. unfold nat_nat_to_monoid_fun. cbn. rewrite nat_to_abmonoid_fun_plus. rewrite nat_to_abmonoid_fun_plus. rewrite (assocax X). rewrite (assocax X). use two_arg_paths. - use idpath. - rewrite <- (assocax X). rewrite (commax X (nat_to_monoid_fun (grinv X x) n2) _). rewrite (assocax X). rewrite (assocax X). use two_arg_paths. + use idpath. + use (commax X). Qed. Transparent nat_to_monoid_fun. Lemma nat_nat_to_monoid_plus1 {X : abgr} (x : X) {n1 m1 m2: nat} (e : m2 = (m1 + n1)%nat) : nat_to_monoid_fun (grinv X x) n1 = (nat_to_monoid_fun x m1 * nat_to_monoid_fun (grinv X x) m2)%multmonoid. Proof. rewrite e. clear e. rewrite nat_to_abmonoid_fun_plus. rewrite <- (assocax X). use pathsinv0. use (pathscomp0 (maponpaths (λ xx : X, (xx * (nat_to_monoid_fun (grinv X x) n1))%multmonoid) (nat_to_monoid_unel' x m1))). use (lunax X). Qed. Lemma nat_nat_prod_abmonoid_fun_unel {X : abgr} (x : X) : (nataddabmonoid_nataddabmonoid_to_monoid_fun x) (unel (abmonoiddirprod nataddabmonoid nataddabmonoid)) = (unel X). Proof. use (pathscomp0 (lunax X _)). use idpath. Qed. Definition nat_nat_prod_abmonoid_monoidfun {X : abgr} (x : X) : monoidfun (abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig)) X. Proof. use monoidfunconstr. - exact (nataddabmonoid_nataddabmonoid_to_monoid_fun x). - use make_ismonoidfun. + exact (nat_nat_monoid_fun_isbinopfun x). + exact (nat_nat_prod_abmonoid_fun_unel x). Defined. Lemma hz_abmonoid_ismonoidfun : @ismonoidfun (abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig)) hzaddabgr (@setquotpr (abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig)) (binopeqrelabgrdiff (rigaddabmonoid natcommrig))). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. use idpath. - use idpath. Qed. Definition hz_abmonoid_monoidfun : monoidfun (abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig)) hzaddabgr. Proof. use monoidfunconstr. - use setquotpr. - exact hz_abmonoid_ismonoidfun. Defined. Definition nat_nat_fun_unel {X : abgr} (x : X) (n : nat) : nat_nat_to_monoid_fun x (make_dirprod n n) = unel X. Proof. exact (nat_to_monoid_unel' x n). Qed. Opaque nat_to_monoid_fun. Definition nat_nat_fun_ind {X : abgr} (x : X) (n m : nat) : nat_nat_to_monoid_fun x (make_dirprod (n + m)%nat m) = nat_nat_to_monoid_fun x (make_dirprod n O). Proof. use (pathscomp0 (nat_nat_monoid_fun_isbinopfun x (make_dirprod n O) (make_dirprod m m))). unfold nataddabmonoid_nataddabmonoid_to_monoid_fun. rewrite (nat_nat_fun_unel x m). rewrite (runax X). use idpath. Qed. Transparent nat_to_monoid_fun. Opaque nat_to_monoid_fun. Definition nat_nat_fun_ind2 {X : abgr} (x : X) (n1 n2 m k : nat) : nat_nat_to_monoid_fun x (make_dirprod n1 m) = nat_nat_to_monoid_fun x (make_dirprod n2 k) -> nat_nat_to_monoid_fun x (make_dirprod n1 (S m)) = nat_nat_to_monoid_fun x (make_dirprod n2 (S k)). Proof. intros H. unfold nat_nat_to_monoid_fun in *. cbn in *. rewrite (@nat_to_monoid_fun_S X (grinv X x)). rewrite (@nat_to_monoid_fun_S X (grinv X x)). rewrite <- (assocax X). rewrite <- (assocax X). use two_arg_paths. - exact H. - use idpath. Qed. Transparent nat_to_monoid_fun. Opaque nat_to_monoid_fun. Definition abgr_precategory_integer_fun_iscomprelfun {X : abgr} (x : X) : iscomprelfun (binopeqrelabgrdiff (rigaddabmonoid natcommrig)) (nat_nat_prod_abmonoid_monoidfun x). Proof. intros x1. induction x1 as [x1 e1]. unfold nat_nat_prod_abmonoid_monoidfun. cbn. unfold nataddabmonoid_nataddabmonoid_to_monoid_fun. unfold nat_nat_to_monoid_fun. cbn. induction x1 as [ | x1 IHx1]. - intros x2 H. use (squash_to_prop H (setproperty X _ _)). intros H'. cbn in H'. induction H' as [H1 H2]. clear H. induction x2 as [x2 e2]. apply natplusrcan in H2. rewrite nat_to_monoid_fun_unel. rewrite (lunax X). cbn. cbn in H2. exact (nat_nat_to_monoid_plus1 x H2). - intros x2 H. use (squash_to_prop H (setproperty X _ _)). intros H'. cbn in H'. induction H' as [H1 H2]. clear H. induction x2 as [x2 e2]. cbn in H2. cbn. use (pathscomp0 (maponpaths (λ xx : X, (xx * (nat_to_monoid_fun (grinv X x) e1))%multmonoid) (@nat_to_monoid_fun_S X x x1))). rewrite (commax X _ x). rewrite (assocax X). cbn. assert (HH : ishinh_UU(∑ x0 : nat, (x1 + (S e2) + x0)%nat = (x2 + e1 + x0)%nat)). { use hinhpr. use tpair. - exact O. - cbn. rewrite natplusr0. rewrite natplusr0. cbn. rewrite natplusassoc in H2. rewrite plus_n_Sm in H2. rewrite plus_n_Sm in H2. rewrite natplusnsm in H2. rewrite <- natplusassoc in H2. apply natplusrcan in H2. exact H2. } set (tmp := IHx1 (make_dirprod x2 (S e2)) HH). cbn in tmp. use (pathscomp0 (maponpaths (λ xx : X, (x * xx)%multmonoid) tmp)). clear tmp. clear HH. clear H2. clear IHx1. rewrite (commax X x). rewrite (assocax X). use two_arg_paths. + use idpath. + use (pathscomp0 (maponpaths (λ xx : X, (xx * x)%multmonoid) (@nat_to_monoid_fun_S X (grinv X x) e2))). rewrite (assocax X). rewrite (grlinvax X x). use (runax X). Qed. Transparent nat_to_monoid_fun. (** Construction of tha map \mathbb{Z} --> A, 1 ↦ x *) Definition hz_abgr_fun {X : abgr} (x : X) : hzaddabgr -> X. Proof. use setquotuniv. - exact (nat_nat_prod_abmonoid_monoidfun x). - exact (abgr_precategory_integer_fun_iscomprelfun x). Defined. (** Hide ismonoidfun behind Qed. *) Definition hz_abgr_fun_ismonoidfun {X : abgr} (x : X) : ismonoidfun (hz_abgr_fun x). Proof. use make_ismonoidfun. - use isbinopfun_twooutof3b. + use (abmonoiddirprod (rigaddabmonoid natcommrig) (rigaddabmonoid natcommrig)). + use (hz_abmonoid_monoidfun). + use issurjsetquotpr. + use binopfunisbinopfun. + use binopfunisbinopfun. - use (runax X). Qed. (** Construction of the monoidfun \mathbb{Z} --> A, 1 ↦ x *) Definition hz_abgr_fun_monoidfun {X : abgr} (x : X) : monoidfun hzaddabgr X. Proof. use monoidfunconstr. - exact (hz_abgr_fun x). - exact (hz_abgr_fun_ismonoidfun x). Defined. (** Commutativity of the following diagram nat × nat --- nat_nat_prod_abmonoid_monoidfun ---> X hz_abgr_fun_monoidfun | || hz -------- hz_abmonoid_monoidfun -------------> X *) Lemma abgr_natnat_hz_X_comm {X : abgr} (x : X) : monoidfuncomp hz_abmonoid_monoidfun (hz_abgr_fun_monoidfun x) = nat_nat_prod_abmonoid_monoidfun x. Proof. use monoidfun_paths. use funextfun. intros n. use setquotunivcomm. Qed. Opaque nat_to_monoid_fun. Lemma monoidfun_nat_to_monoid_fun {X Y : abgr} (f : monoidfun X Y) (x : X) (n : nat) : pr1 f (nat_to_monoid_fun x n) = nat_to_monoid_fun (f x) n. Proof. induction n as [ | n IHn]. - use monoidfununel. - use (pathscomp0 (maponpaths (pr1 f) (@nat_to_monoid_fun_S X x n))). use (pathscomp0 (binopfunisbinopfun f _ _)). use (pathscomp0 _ (! (@nat_to_monoid_fun_S Y (f x) n))). use two_arg_paths. + exact IHn. + use idpath. Qed. Transparent nat_to_monoid_fun. (** Some more facts about integers added by D. Grayson *) Definition ℤ := hzaddabgr. Definition toℤ (n:nat) : ℤ := nattohz n. Definition toℤneg (n:nat) : ℤ := natnattohz O n. Definition zero := toℤ 0. Definition one := toℤ 1. Definition hzabsvalnat n : hzabsval (natnattohz n 0) = n. (* move to hz.v *) Proof. intros. unfold hzabsval. unfold setquotuniv. simpl. unfold hzabsvalint. simpl. destruct (natgthorleh n 0). { apply natminuseqn. } { exact (! (natleh0tois0 h)). } Defined. Lemma hzsign_natnattohz m n : - natnattohz m n = natnattohz n m. (* move to hz.v *) Proof. reflexivity. (* don't change the proof *) Defined. Lemma hzsign_nattohz m : - nattohz m = natnattohz 0 m. (* move to hz.v *) Proof. reflexivity. (* don't change the proof *) Defined. Lemma hzsign_hzsign (i:hz) : - - i = i. Proof. apply (grinvinv ℤ). Defined. Definition hz_normal_form (i:ℤ) := coprod (∑ n, natnattohz n 0 = i) (∑ n, natnattohz 0 (S n) = i). Definition hznf_pos n := _,, inl (n,,idpath _) : total2 hz_normal_form. Definition hznf_neg n := _,, inr (n,,idpath _) : total2 hz_normal_form. Definition hznf_zero := hznf_pos 0. Definition hznf_neg_one := hznf_neg 0. Definition hz_to_normal_form (i:ℤ) : hz_normal_form i. Proof. intros. destruct (hzlthorgeh i 0) as [r|s]. { apply inr. assert (a := hzabsvallth0 r). assert (b := hzlthtoneq _ _ r). assert (c := hzabsvalneq0 b). assert (d := natneq0togth0 _ c). assert (f := natgthtogehsn _ _ d). assert (g := minusplusnmm _ _ f). rewrite natpluscomm in g. simpl in g. exists (hzabsval i - 1)%nat. rewrite g. apply hzinvmaponpathsminus. exact a. } { apply inl. exists (hzabsval i). exact (hzabsvalgeh0 s). } Defined. Lemma nattohz_inj {m n} : nattohz m = nattohz n -> m = n. Proof. revert m n; exact (invmaponpathsincl _ isinclnattohz). Defined. Lemma hzdichot {m n} : neg (nattohz m = - nattohz (S n)). Proof. intros. intro e. assert (d := maponpaths hzsign e); clear e. rewrite hzsign_hzsign in d. assert( f := maponpaths (λ i, nattohz m + i) d); simpl in f; clear d. change (nattohz m + - nattohz m) with (nattohz m - nattohz m) in f. rewrite hzrminus in f. change (0 = nattohz (m + S n)) in f. assert (g := nattohz_inj f); clear f. rewrite natpluscomm in g. exact (negpaths0sx _ g). Defined. Definition negpos' : isweq (@pr1 _ hz_normal_form). Proof. apply isweqpr1; intro i. exists (hz_to_normal_form i). generalize (hz_to_normal_form i) as s. intros [[m p]|[m p]] [[n q]|[n q]]. { apply (maponpaths (@ii1 (∑ n, natnattohz n 0 = i) (∑ n, natnattohz 0 (S n) = i))). apply (proofirrelevance _ (isinclnattohz i)). } { apply fromempty. assert (r := p@!q); clear p q. apply (hzdichot r). } { apply fromempty. assert (r := q@!p); clear p q. apply (hzdichot r). } { apply (maponpaths (@ii2 (∑ n, natnattohz n 0 = i) (∑ n, natnattohz 0 (S n) = i))). assert (p' := maponpaths hzsign p). assert (q' := maponpaths hzsign q). change (- natnattohz O (S m)) with (nattohz (S m)) in p'. change (- natnattohz O (S n)) with (nattohz (S n)) in q'. assert (c := proofirrelevance _ (isinclnattohz (-i)) (S m,,p') (S n,,q')). assert (d := maponpaths pr1 c); simpl in d. assert (e := invmaponpathsS _ _ d); clear d. apply subtypePath. - intro; apply setproperty. - exact (!e). } Defined. Definition negpos_weq := make_weq _ negpos' : weq (total2 hz_normal_form) ℤ. Definition negpos : weq (coprod nat nat) ℤ. (* ℤ = (-inf,-1) + (0,inf) *) Proof. simple refine (make_weq _ (isweq_iso _ _ _ _)). { intros [n'|n]. { exact (natnattohz 0 (S n')). } { exact (natnattohz n 0). } } { intro i. destruct (hz_to_normal_form i) as [[n p]|[m q]]. { exact (inr n). } { exact (inl m). } } { intros [n'|n]. { simpl. rewrite natminuseqn. reflexivity. } { simpl. rewrite hzabsvalnat. reflexivity. } } { simpl. intro i. destruct (hz_to_normal_form i) as [[n p]|[m q]]. { exact p. } { exact q. } } Defined. Lemma hzminusplus (x y:hz) : -(x+y) = (-x) + (-y). (* move to hz.v *) Proof. intros. apply (hzplusrcan _ _ (x+y)). rewrite hzlminus. rewrite (hzpluscomm (-x)). rewrite (hzplusassoc (-y)). rewrite <- (hzplusassoc (-x)). rewrite hzlminus. rewrite hzplusl0. rewrite hzlminus. reflexivity. Defined. Definition loop_power {Y} {y:Y} (l:y = y) (n:ℤ) : y = y. Proof. intros. assert (m := loop_power_nat l (hzabsval n)). destruct (hzlthorgeh n 0%hz). { exact (!m). } { exact m. } Defined.