coercion cic:/matita/ordered_divisible_group/todg_division.con.
-lemma pow_lt: ∀G:todgroup.∀x:G.∀n.0 < x → 0 < x + pow ? x n.
-intros (G x n H); elim n; [
- simplify; apply (lt_rewr ???? (plus_comm ???));
- apply (lt_rewr ???x (zero_neutral ??)); assumption]
-simplify; apply (lt_transitive ?? (x+(x)\sup(n1))); [assumption]
-apply flt_plusl; apply (lt_rewr ???? (plus_comm ???));
-apply (lt_rewl ??? (0 + (x \sup n1)) (eq_sym ??? (zero_neutral ??)));
-apply (lt_rewl ???? (plus_comm ???));
-apply flt_plusl; assumption;
-qed.
-
-lemma pow_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ pow ? x n.
+lemma mul_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ n * x.
intros (G x n); elim n; simplify; [apply le_reflexive]
apply (le_transitive ???? H1);
-apply (le_rewl ??? (0+(x\sup n1)) (zero_neutral ??));
+apply (le_rewl ??? (0+(n1*x)) (zero_neutral ??));
apply fle_plusr; assumption;
qed.
-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 (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 (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 lt_suplt: ∀G:todgroup.∀x,y:G.∀n.
-x < y → x\sup (S n) < y\sup (S n).
+lemma lt_ltmul: ∀G:todgroup.∀x,y:G.∀n. x < y → S n * x < S n * y.
intros; elim n; [simplify; apply flt_plusr; assumption]
simplify; apply (ltplus); [assumption] assumption;
qed.
-lemma suplt_lt: ∀G:todgroup.∀x,y:G.∀n. x\sup (S n) < y\sup (S n) → x < y.
+lemma ltmul_lt: ∀G:todgroup.∀x,y:G.∀n. S n * x < S n * y → x < y.
intros 4; elim n; [apply (plus_cancr_lt ??? 0); assumption]
simplify in l; cases (ltplus_orlt ????? l); [assumption]
apply f; assumption;
qed.
-lemma supeqplus_lt: ∀G:todgroup.∀x,y:G.∀n,m.
- 0<x → 0<y → x\sup (S n) ≈ y\sup (S (n+S m)) → y < x.
-intros (G x y n m H1 H2 H3); apply (suplt_lt ??? n); apply (lt_rewr ???? H3);
+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 (ltmul_lt ??? (S n1)); simplify; apply (lt_rewr ???? f);
+apply (lt_rewl ???? (zero_neutral ??));
+apply (lt_rewl ???? (zero_neutral ??));
+apply (lt_rewl ???? (mulzero ?n1));
+assumption;
+qed.
+
+lemma muleqplus_lt: ∀G:todgroup.∀x,y:G.∀n,m.
+ 0<x → 0<y → S n * x ≈ S (n + S m) * y → y < x.
+intros (G x y n m H1 H2 H3); apply (ltmul_lt ??? n); apply (lt_rewr ???? H3);
clear H3; elim m; [
- rewrite > sym_plus; simplify; apply (lt_rewl ??? (0+(y+y\sup n))); [
+ rewrite > sym_plus; simplify; apply (lt_rewl ??? (0+(y+n*y))); [
apply eq_sym; apply zero_neutral]
apply flt_plusr; assumption;]
apply (lt_transitive ???? l); rewrite > sym_plus; simplify;
rewrite > (sym_plus n); simplify; repeat apply flt_plusl;
-apply (lt_rewl ???(0+y\sup(n1+n))); [apply eq_sym; apply zero_neutral]
+apply (lt_rewl ???(0+(n1+n)*y)); [apply eq_sym; apply zero_neutral]
apply flt_plusr; assumption;
qed.
-alias num (instance 0) = "natural number".
-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;
-cut (w<w1);[2: apply (supeqplus_lt ??? 2 O); 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.
-
-
-
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