-alias symbol "leq" = "ordered sets less or equal than".
-alias symbol "and" = "constructive and".
-alias symbol "exists" = "constructive exists (Type)".
-lemma carabinieri: (* non trova la coercion .... *)
- ∀R.∀ml:mlattice R.∀an,bn,xn:sequence (pordered_set_of_excedence ml).
- (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
- ∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
- tends0 ? (d2s ? ml xn x).
-intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
-intros (e He);
-elim (Ha ? He) (n1 H1); clear Ha; elim (Hb e He) (n2 H2); clear Hb;
-constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
-cut (n1<n3) [2:
- generalize in match Hn3; elim n2; [rewrite > sym_plus in H3; assumption]
- apply H3; rewrite > sym_plus in H4; simplify in H4; apply lt_S_to_lt;
- rewrite > sym_plus in H4; assumption;]
-elim (H1 ? Hcut) (H3 H4); clear Hcut;
-cut (n2<n3) [2:
- generalize in match Hn3; elim n1; [assumption]
- apply H5; simplify in H6; apply lt_S_to_lt; assumption]
-elim (H2 ? Hcut) (H5 H6); clear Hcut;
-elim (H n3) (H7 H8); clear H H1 H2;
-lapply (le_to_eqm ??? H7) as Dxm; lapply (le_to_eqj ??? H7) as Dym;
-cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x); [2:
- apply (le_transitive ???? (mtineq ???? (an n3)));
- lapply (le_mtri ????? H7 H8);
- lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? Hletin);
- cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))); [2:
- apply (Eq≈ (0 + δ(an n3) (xn n3)) ? (zero_neutral ??));
- apply (Eq≈ (δ(an n3) (xn n3) + 0) ? (plus_comm ???));
- apply (Eq≈ (δ(an n3) (xn n3) + (-δ(xn n3) (bn n3) + δ(xn n3) (bn n3))) ? (opp_inverse ??));
- apply (Eq≈ (δ(an n3) (xn n3) + (δ(xn n3) (bn n3) + -δ(xn n3) (bn n3))) ? (plus_comm ?? (δ(xn n3) (bn n3))));
- apply (Eq≈ ? ? (eq_sym ??? (plus_assoc ????))); assumption;] clear Hletin1;
- apply (le_rewl ??? ( δ(an n3) (xn n3)+ δ(an n3) x));[
- apply feq_plusr; apply msymmetric;]
- apply (le_rewl ??? (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x));[
- apply feq_plusr; assumption;]
- ]
-[2: split; [
- apply (lt_le_transitive ????? (mpositive ????));
- split; elim He; [apply le_zero_x_to_le_opp_x_zero|
- cases t1;
- [left; apply exc_zero_opp_x_to_exc_x_zero;
- apply (Ex≫ ? (eq_opp_opp_x_x ??));
- |right; apply exc_opp_x_zero_to_exc_zero_x;
- apply (Ex≪ ? (eq_opp_opp_x_x ??));]] assumption;]
-clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
-
-