-lemma sandwich_ineq: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
-intro G; cases G; unfold divide; intro e;
-alias num (instance 0) = "natural number".
-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) as H4; [2:
- apply (ltpow_lt ??? 2); apply (lt_rewr ???? H2);
- apply (lt_rewl ???? (powzero ? 3)); assumption]
-cut (0<w) as H5; [2:
- apply (ltpow_lt ??? 3); apply (lt_rewr ???? H1);
- apply (lt_rewl ???? (powzero ? 4)); assumption]
-cut (w<w1) as H6; [2:
- apply (poweqplus_lt ??? 2 O); try assumption; apply (Eq≈ ? H2 H1);]
-apply (plus_cancr_lt ??? w1);
-apply (lt_rewl ??? (w+e)); [
- apply (Eq≈ (w+3*w1) ? 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.
-