definition transitive ≝
λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
+
+definition associative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z).
+
+definition commutative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).
cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)]
qed.
-definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+record apartness : Type ≝ {
+ ap_carr:> Type;
+ ap_apart: ap_carr → ap_carr → Type;
+ ap_coreflexive: coreflexive ? ap_apart;
+ ap_symmetric: symmetric ? ap_apart;
+ ap_cotransitive: cotransitive ? ap_apart
+}.
notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ a b).
+interpretation "axiomatic apartness" 'apart x y =
+ (cic:/matita/excedence/ap_apart.con _ x y).
-lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
-intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
-intros (H1); apply (H x); cases H1; assumption;
-qed.
+definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
-lemma apart_symmetric: ∀E.symmetric ? (apart E).
-intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+definition apart_of_excedence: excedence → apartness.
+intros (E); apply (mk_apartness E (apart E));
+[1: unfold; cases E; simplify; clear E; intros (x); unfold;
+ intros (H1); apply (H x); cases H1; assumption;
+|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
+ cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+ [left; left|right; left|right; right|left; right] assumption]
qed.
-lemma apart_cotrans: ∀E. cotransitive ? (apart E).
-intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
-cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
-[left; left|right; left|right; right|left; right] assumption.
-qed.
+coercion cic:/matita/excedence/apart_of_excedence.con.
-definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
+definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
interpretation "alikeness" 'napart a b =
(cic:/matita/excedence/eq.con _ a b).
lemma eq_reflexive:∀E. reflexive ? (eq E).
-intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
-apply (cRf x); cases H; assumption;
+intros (E); unfold; intros (x); apply ap_coreflexive;
qed.
lemma eq_symmetric:∀E.symmetric ? (eq E).
-intros (E); unfold; unfold eq; unfold Not;
-intros (x y H1 H2); apply H1; cases H2; [right|left] assumption;
+intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
+apply ap_symmetric; assumption;
qed.
lemma eq_transitive: ∀E.transitive ? (eq E).
-intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not;
-intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
-cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
-[1,3:left|*:right] assumption;
+(* bug. intros k deve fare whd quanto basta *)
+intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy);
+[apply Exy|apply Eyz] assumption.
qed.
-
-lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
-intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
+(* BUG: vedere se ricapita *)
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
+intros 5 (E x y Lxy Lyx); intro H;
cases H; [apply Lxy;|apply Lyx] assumption;
qed.
(cic:/matita/excedence/lt.con _ a b).
lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
-intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS);
-apply (apart_coreflexive ? x ABS);
+intros 2 (E x); intro H; cases H (_ ABS);
+apply (ap_coreflexive ? x ABS);
qed.
+(*
lemma lt_transitive: ∀E.transitive ? (lt E).
intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
intros (E a b Lab); cases Lab (LEab Aab);
cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)
qed.
+
+*)
\ No newline at end of file
set "baseuri" "cic:/matita/groups/".
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
-include "constructive_connectives.ma".
+include "excedence.ma".
-definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
-
-definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
-
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e.
+definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x.
+definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x.
+definition left_inverse ≝ λC:apartness.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x ≈ e.
+definition right_inverse ≝ λC:apartness.λop.λe:C.λ inv: C→ C. ∀x:C. op x (inv x) ≈ e.
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+(* ALLOW DEFINITION WITH SOME METAS *)
definition distributive_left ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f x (g y z) = g (f x y) (f x z).
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f x (g y z) ≈ g (f x y) (f x z).
definition distributive_right ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f (g x y) z = g (f x z) (f y z).
-
-record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
- { (* abelian additive semigroup properties *)
- plus_assoc_: associative ? plus;
- plus_comm_: symmetric ? plus;
- (* additive monoid properties *)
- zero_neutral_: left_neutral ? plus zero;
- (* additive group properties *)
- opp_inverse_: left_inverse ? plus zero opp
- }.
-
-record abelian_group : Type \def
- { carrier:> Type;
- plus: carrier → carrier → carrier;
- zero: carrier;
- opp: carrier → carrier;
- ag_abelian_group_properties: is_abelian_group ? plus zero opp
- }.
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f (g x y) z ≈ g (f x z) (f y z).
+
+record abelian_group : Type ≝
+ { carr:> apartness;
+ plus: carr → carr → carr;
+ zero: carr;
+ opp: carr → carr;
+ plus_assoc: associative ? plus (eq carr);
+ plus_comm: commutative ? plus (eq carr);
+ zero_neutral: left_neutral ? plus zero;
+ opp_inverse: left_inverse ? plus zero opp;
+ plus_strong_ext: ∀z.strong_ext ? (plus z)
+}.
-notation "0" with precedence 89
-for @{ 'zero }.
+notation "0" with precedence 89 for @{ 'zero }.
interpretation "Abelian group zero" 'zero =
(cic:/matita/groups/zero.con _).
interpretation "Abelian group minus" 'minus a b =
(cic:/matita/groups/minus.con _ a b).
-theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
- intro;
- apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
+lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
+intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
+cases (Exy (ap_symmetric ??? a));
qed.
-
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
+
+lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
+apply ap_symmetric; assumption;
qed.
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
+definition ext ≝ λA:apartness.λop:A→A. ∀x,y. x ≈ y → op x ≈ op y.
+
+lemma strong_ext_to_ext: ∀A:apartness.∀op:A→A. strong_ext ? op → ext ? op.
+intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption;
+qed.
+
+lemma f_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x));
+assumption;
+qed.
+
+lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z).
+intros 5 (G z x y A); simplify in A;
+lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2;
+lapply (ap_rewl ???? E1 A) as A1; lapply (ap_rewr ???? E2 A1) as A2;
+apply (plus_strong_ext ???? A2);
qed.
-
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
+
+lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → y+x ≈ z+x.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x));
+assumption;
+qed.
+
+lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z → x+y # x+z.
+intros (G x y z Ayz); apply (plus_strong_ext ? (-x));
+apply (ap_rewl ??? ((-x + x) + y));
+[1: apply plus_assoc;
+|2: apply (ap_rewr ??? ((-x +x) +z));
+ [1: apply plus_assoc;
+ |2: apply (ap_rewl ??? (0 + y));
+ [1: apply (feq_plusr ???? (opp_inverse ??));
+ |2: apply (ap_rewl ???? (zero_neutral ? y)); apply (ap_rewr ??? (0 + z));
+ [1: apply (feq_plusr ???? (opp_inverse ??));
+ |2: apply (ap_rewr ???? (zero_neutral ? z)); assumption;]]]]
qed.
-lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z.
-intros;
-generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
-intros; clear H;
-rewrite < plus_assoc in H1;
-rewrite < plus_assoc in H1;
-rewrite > opp_inverse in H1;
-rewrite > zero_neutral in H1;
-rewrite > zero_neutral in H1;
-assumption.
-qed.
+lemma plus_canc: ∀G:abelian_group.∀x,y,z:G. x+y ≈ x+z → y ≈ z.
+intros 6 (G x y z E Ayz); apply E; apply fap_plusl; assumption;
+qed.
+
+(*
theorem eq_opp_plus_plus_opp_opp: ∀G:abelian_group.∀x,y:G. -(x+y) = -x + -y.
intros;
rewrite > zero_neutral;
reflexivity
].
-qed.
\ No newline at end of file
+qed.
+
+*)
\ No newline at end of file
include "ordered_sets.ma".
record pre_ordered_abelian_group : Type ≝
- { og_abelian_group:> abelian_group;
- og_tordered_set_: tordered_set;
- og_with: exc_carr og_tordered_set_ = og_abelian_group
+ { og_abelian_group_: abelian_group;
+ og_tordered_set:> tordered_set;
+ og_with: carr og_abelian_group_ = og_tordered_set
}.
-lemma og_tordered_set: pre_ordered_abelian_group → tordered_set.
-intro G; apply mk_tordered_set;
-[1: apply mk_pordered_set;
- [1: apply (mk_excedence G);
- [1: cases G; clear G; simplify; rewrite < H; clear H;
- cases og_tordered_set_; clear og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify; assumption;
- |2: cases G; simplify; cases H; simplify; clear H;
- cases og_tordered_set_; simplify; clear og_tordered_set_;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply H;
- |3: cases G; simplify; cases H; simplify; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply c; assumption]
- |2: cases G; simplify;
- cases H; simplify; clear H; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; assumption;]
-|2: simplify; (* SLOW, senza la simplify il widget muore *)
- cases G; simplify;
- generalize in match (tos_totality og_tordered_set_);
- unfold total_order_property;
- cases H; simplify; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply f; assumption;]
+lemma og_abelian_group: pre_ordered_abelian_group → abelian_group.
+intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
+[apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
+unfold apartness_OF_pre_ordered_abelian_group; cases (og_with G); simplify;
+[apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
qed.
-coercion cic:/matita/ordered_groups/og_tordered_set.con.
+coercion cic:/matita/ordered_groups/og_abelian_group.con.
definition is_ordered_abelian_group ≝
λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
|2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
qed.
+(*
+
definition reverse_pordered_set: pordered_set → pordered_set.
intros (p); apply (mk_pordered_set (reverse_excedence p));
generalize in match (reverse_excedence p); intros (E); cases E (T f cRf cTf);
}.
*)
+*)
+
definition total_order_property : ∀E:excedence. Type ≝
λE:excedence. ∀a,b:E. a ≰ b → a < b.
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