include "ordered_sets.ma".
include "groups.ma".
-record pre_ordered_abelian_group : Type ≝
- { og_abelian_group_: abelian_group;
- og_tordered_set:> tordered_set;
- og_with: carr og_abelian_group_ = og_tordered_set
- }.
+record pre_ogroup : Type ≝ {
+ og_abelian_group_: abelian_group;
+ og_tordered_set:> tordered_set;
+ og_with: carr og_abelian_group_ = og_tordered_set
+}.
-lemma og_abelian_group: pre_ordered_abelian_group → abelian_group.
+lemma og_abelian_group: pre_ogroup → 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;
+unfold apartness_OF_pre_ogroup; 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_abelian_group.con.
-definition is_ordered_abelian_group ≝
- λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
+(* CSC: NO! Cosi' e' nel frammento negativo. Devi scriverlo con l'eccedenza!
+ Tutto il resto del file e' da rigirare nel frammento positivo.
+*)
+record ogroup : Type ≝ {
+ og_carr:> pre_ogroup;
+ exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
+}.
-record ordered_abelian_group : Type ≝
- { og_pre_ordered_abelian_group:> pre_ordered_abelian_group;
- og_ordered_abelian_group_properties:
- is_ordered_abelian_group og_pre_ordered_abelian_group
- }.
-
-lemma le_le_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
-intros 6 (E x y L1 L2 H); cases H; [apply (L1 H1)|apply (L2 H1)]
+lemma fexc_plusr:
+ ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
+intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
+apply (exc_rewl ??? (x + (z + -z)) (plus_assoc ????));
+apply (exc_rewl ??? (x + (-z + z)) (plus_comm ??z));
+apply (exc_rewl ??? (x+0) (opp_inverse ??));
+apply (exc_rewl ??? (0+x) (plus_comm ???));
+apply (exc_rewl ??? x (zero_neutral ??));
+apply (exc_rewr ??? (y + (z + -z)) (plus_assoc ????));
+apply (exc_rewr ??? (y + (-z + z)) (plus_comm ??z));
+apply (exc_rewr ??? (y+0) (opp_inverse ??));
+apply (exc_rewr ??? (0+y) (plus_comm ???));
+apply (exc_rewr ??? y (zero_neutral ??) L);
qed.
-lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
-unfold apart_of_excedence; unfold apart; simplify; intros; assumption;
-qed.
+coercion cic:/matita/ordered_groups/fexc_plusr.con nocomposites.
-lemma le_rewl: ∀E:excedence.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
-intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
-intro Xyz; apply Exy; apply unfold_apart; right; assumption;
+lemma exc_canc_plusl: ∀G:ogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
+intros 5 (G x y z L); apply (exc_canc_plusr ??? z);
+apply (exc_rewl ??? (z+x) (plus_comm ???));
+apply (exc_rewr ??? (z+y) (plus_comm ???) L);
qed.
-lemma le_rewr: ∀E:excedence.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
-intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
-intro Xyz; apply Exy; apply unfold_apart; left; assumption;
+lemma fexc_plusl:
+ ∀G:ogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
+intros 5 (G x y z L); apply (exc_canc_plusl ??? (-z));
+apply (exc_rewl ???? (plus_assoc ??z x));
+apply (exc_rewr ???? (plus_assoc ??z y));
+apply (exc_rewl ??? (0+x) (opp_inverse ??));
+apply (exc_rewr ??? (0+y) (opp_inverse ??));
+apply (exc_rewl ???? (zero_neutral ??));
+apply (exc_rewr ???? (zero_neutral ??) L);
qed.
+coercion cic:/matita/ordered_groups/fexc_plusl.con nocomposites.
+
lemma plus_cancr_le:
- ∀G:ordered_abelian_group.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
+ ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
intros 5 (G x y z L);
apply (le_rewl ??? (0+x) (zero_neutral ??));
apply (le_rewl ??? (x+0) (plus_comm ???));
-apply (le_rewl ??? (x+(-z+z))); [apply feq_plusl;apply opp_inverse;]
-apply (le_rewl ??? (x+(z+ -z))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewl ??? (x+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
+apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
+apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
+apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
apply (le_rewr ??? (0+y) (zero_neutral ??));
apply (le_rewr ??? (y+0) (plus_comm ???));
-apply (le_rewr ??? (y+(-z+z))); [apply feq_plusl;apply opp_inverse;]
-apply (le_rewr ??? (y+(z+ -z))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewr ??? (y+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
-apply (og_ordered_abelian_group_properties ??? (-z));
-assumption;
+apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
+apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
+apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
+intro H; apply L; clear L; apply (exc_canc_plusr ??? (-z) H);
qed.
+lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
+intros (G f g h);
+apply (plus_cancr_le ??? (-h));
+apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
+apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
+apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
+apply (le_rewl ??? (f+0) (opp_inverse ??));
+apply (le_rewl ??? (0+f) (plus_comm ???));
+apply (le_rewl ??? (f) (zero_neutral ??));
+apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
+apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
+apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
+apply (le_rewr ??? (g+0) (opp_inverse ??));
+apply (le_rewr ??? (0+g) (plus_comm ???));
+apply (le_rewr ??? (g) (zero_neutral ??) H);
+qed.
+
+lemma plus_cancl_le:
+ ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
+intros 5 (G x y z L);
+apply (le_rewl ??? (0+x) (zero_neutral ??));
+apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
+apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
+apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
+apply (fle_plusl ??? (-z) L);
+qed.
+
+lemma exc_opp_x_zero_to_exc_zero_x:
+ ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
+intros (G x H); apply (exc_canc_plusr ??? (-x));
+apply (exc_rewr ???? (plus_comm ???));
+apply (exc_rewr ???? (opp_inverse ??));
+apply (exc_rewl ???? (zero_neutral ??) H);
+qed.
+
lemma le_zero_x_to_le_opp_x_zero:
- ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
+ ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
intros (G x Px); apply (plus_cancr_le ??? x);
-apply (le_rewl ??? 0 (eq_symmetric ??? (opp_inverse ??)));
-apply (le_rewr ??? x (eq_symmetric ??? (zero_neutral ??)));
-assumption;
+apply (le_rewl ??? 0 (opp_inverse ??));
+apply (le_rewr ??? x (zero_neutral ??) Px);
+qed.
+
+lemma exc_zero_opp_x_to_exc_x_zero:
+ ∀G:ogroup.∀x:G. 0 ≰ -x → x ≰ 0.
+intros (G x H); apply (exc_canc_plusl ??? (-x));
+apply (exc_rewr ???? (plus_comm ???));
+apply (exc_rewl ???? (opp_inverse ??));
+apply (exc_rewr ???? (zero_neutral ??) H);
qed.
lemma le_x_zero_to_le_zero_opp_x:
- ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
+ ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
intros (G x Lx0); apply (plus_cancr_le ??? x);
-apply (le_rewr ??? 0 (eq_symmetric ??? (opp_inverse ??)));
-apply (le_rewl ??? x (eq_symmetric ??? (zero_neutral ??)));
+apply (le_rewr ??? 0 (opp_inverse ??));
+apply (le_rewl ??? x (zero_neutral ??));
assumption;
qed.
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