-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 foo: ∀G:todgroup.∀x,y:G.∀n.
-x < y → x\sup (S n) < y\sup (S n).