ok

[helm.git] / helm / software / matita / dama / ordered_divisible_group.ma

diff --git a/helm/software/matita/dama/ordered_divisible_group.ma b/helm/software/matita/dama/ordered_divisible_group.ma
index 34bd25a237149beffc813b51c56c7ad43d6f97b2..0b84d9add806f09caface73ae3800317d672bc2e 100644 (file)
--- a/helm/software/matita/dama/ordered_divisible_group.ma
+++ b/helm/software/matita/dama/ordered_divisible_group.ma
@@ -50,21 +50,102 @@ apply (le_rewl ??? (0+(x\sup n1)) (zero_neutral ??));
 apply fle_plusr; assumption;
 qed. 
 
-lemma ge_pow: ∀G:todgroup.∀x:G.∀n.0 < pow ? x n → 0 < x.
+lemma gt_pow: ∀G:todgroup.∀x:G.∀n.0 < pow ? x n → 0 < x.
 intros 3; elim n; [
   simplify in l; cases (lt_coreflexive ?? l);]
 simplify in l; 
 cut (0+0<x+(x)\sup(n1));[2:
   apply (lt_rewl ??? 0 (zero_neutral ??)); assumption].
-cases (pippo4 ????? Hcut); [assumption]
+cases (ltplus_orlt ????? Hcut); [assumption]
 apply f; assumption;
 qed.
 
+lemma gt_pow2: ∀G:dgroup.∀x,y:G.∀n.pow ? x n + pow ? y n ≈ pow ? (x+y) n.
+intros (G x y n); elim n; [apply (Eq≈ 0 (zero_neutral ??)); apply eq_reflexive]
+simplify; apply (Eq≈ (x+y+((x)\sup(n1)+(y)\sup(n1)))); [
+  apply (Eq≈ (x+((x)\sup(n1)+(y+(y)\sup(n1))))); [
+    apply eq_sym; apply plus_assoc;]
+  apply (Eq≈ (x+((x)\sup(n1)+y+(y)\sup(n1)))); [
+    apply feq_plusl; apply plus_assoc;]
+  apply (Eq≈ (x+(y+(x)\sup(n1)+(y)\sup(n1)))); [
+    apply feq_plusl; apply feq_plusr; apply plus_comm;] 
+  apply (Eq≈ (x+(y+((x)\sup(n1)+(y)\sup(n1))))); [
+    apply feq_plusl; apply eq_sym; apply plus_assoc;]
+  apply plus_assoc;]
+apply feq_plusl; assumption;
+qed. 
+  
+lemma xxxx: ∀E:abelian_group.∀x,a,y,b:E.x + a # y + b → x # y ∨ a # b.
+intros; cases (ap_cotransitive ??? (y+a) a1); [left|right]
+[apply (plus_cancr_ap ??? a)|apply (plus_cancl_ap ??? y)]
+assumption;
+qed.
+   
+lemma pow_gt0: ∀G:todgroup.∀y:G.∀n.0 < y → 0 < pow ? y (S n).
+intros (G y n H); elim n; [apply (lt_rewr ??? (0+y) (plus_comm ???)); apply (lt_rewr ??? y (zero_neutral ??)); apply H]
+simplify; apply (lt_rewl ? 0 ? (0+0) (zero_neutral ? 0));
+apply ltplus; assumption;
+qed.
+
 lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0<e → 0 < e/n.
 intros; elim n; [apply (lt_rewr ???? (div1 ??));assumption]
 unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
-apply (ge_pow ?? (S (S n1))); apply (lt_rewr ???? f); assumption;
+apply (gt_pow ?? (S (S n1))); apply (lt_rewr ???? f); assumption;
+qed.
+
+
+lemma bar1: ∀G:togroup.∀x,y:G.∀n.pow ? x (S n) # pow ? y (S n) → x # y.
+intros 4 (G x y n); elim n; [2:
+  simplify in a;
+  cases (xxxx ????? a); [assumption]
+  apply f; assumption;]
+apply (plus_cancr_ap ??? 0); assumption;
+qed.
+
+
+lemma foo: ∀G:todgroup.∀x,y:G.∀n.
+x < y → x\sup (S n) < y\sup (S n).
+intros; elim n; [simplify; apply flt_plusr; assumption]
+simplify; apply (ltplus); [assumption] assumption;
+qed.
+
+lemma foo1: ∀G:todgroup.∀x,y:G.∀n.
+x\sup (S n) < y\sup (S n) → x < y.
+intros 4; elim n; [apply (plus_cancr_lt ??? 0); assumption]
+simplify in l; cases (ltplus_orlt ????? l); [assumption]
+apply f; assumption;
+qed.
+
+alias num (instance 0) = "natural number".
+lemma foo3: ∀G:todgroup.∀x,y:G.
+   0<x → 0<y → x\sup 3 ≈ y\sup 4 → y < x.
+intros (G x y H1 H2 H3); apply (foo1 ??? 2); apply (lt_rewr ???? H3);
+simplify; repeat apply flt_plusl; apply (lt_rewr ???? (plus_comm ???));
+apply (lt_rewr ???? (zero_neutral ??)); assumption;
 qed.
 
 alias num (instance 0) = "natural number".
-axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
+lemma core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
+intro G; cases G; unfold divide; intro e;
+cases (dg_prop (mk_todgroup todg_order todg_division_ H) e 3) 0;
+cases (dg_prop (mk_todgroup todg_order todg_division_ H) e 2) 0; simplify;
+intro H3;
+cut (0<w1\sup 3); [2: apply (lt_rewr ???? H2); assumption]
+cut (0<w\sup 4); [2: apply (lt_rewr ???? H1); assumption]
+lapply (gt_pow ??? Hcut) as H4;
+lapply (gt_pow ??? Hcut1) as H5;
+(* elim (eq_le_le ??? H1); elim (eq_le_le ??? H2); *)
+cut (w<w1);[2: apply foo3; try assumption; apply (Eq≈ ? H2 H1);]
+apply (plus_cancr_lt ??? w1);
+apply (lt_rewl ??? (w+e)); [
+  apply (Eq≈ (w+w1\sup 3) ? H2);
+  apply (Eq≈ (w+w1+(w1+w1)) (plus_assoc ??w1 w1));
+  apply (Eq≈ (w+(w1+(w1+w1))) (plus_assoc ?w w1 ?));
+  simplify; repeat apply feq_plusl; apply eq_sym; 
+  apply (Eq≈ ? (plus_comm ???)); apply zero_neutral;]
+apply (lt_rewl ???? (plus_comm ???));
+apply flt_plusl; assumption;
+qed.
+  
+  
+