X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flibrary%2Fdama%2Fmodels%2Fincreasing_supremum_stabilizes.ma;fp=matita%2Fmatita%2Flibrary%2Fdama%2Fmodels%2Fincreasing_supremum_stabilizes.ma;h=fc4424e5ec7cde8616c68e0d96d237cfc37baa4e;hb=2c01ff6094173915e7023076ea48b5804dca7778;hp=0000000000000000000000000000000000000000;hpb=a050e3f80d7ea084ce0184279af98e8251c7d2a6;p=helm.git diff --git a/matita/matita/library/dama/models/increasing_supremum_stabilizes.ma b/matita/matita/library/dama/models/increasing_supremum_stabilizes.ma new file mode 100644 index 000000000..fc4424e5e --- /dev/null +++ b/matita/matita/library/dama/models/increasing_supremum_stabilizes.ma @@ -0,0 +1,140 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "dama/models/nat_uniform.ma". +include "dama/supremum.ma". +include "nat/le_arith.ma". +include "dama/russell_support.ma". + +lemma hint1: + ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s)) + → sequence (hos_carr (os_l (segment_ordered_set nat_ordered_set s))). +intros; assumption; +qed. + +coercion hint1 nocomposites. + +alias symbol "pi1" = "exT \fst". +alias symbol "N" = "ordered set N". +alias symbol "dependent_pair" = "dependent pair". +lemma increasing_supremum_stabilizes: + ∀sg:‡ℕ.∀a:sequence {[sg]}. + a is_increasing → + ∀X.X is_supremum a → ∃i.∀j.i ≤ j → \fst X = \fst (a j). +intros 4; cases X (x Hx); clear X; letin X ≝ ≪x,Hx≫; +fold normalize X; intros; cases H1; +alias symbol "N" = "Natural numbers". +letin spec ≝ (λi,j:ℕ.(𝕦_ sg ≤ i ∧ x = \fst (a j)) ∨ (i < 𝕦_ sg ∧ x + i ≤ 𝕦_ sg + \fst (a j))); +(* x - aj <= max 0 (u - i) *) +letin m ≝ (hide ? ( + let rec aux i ≝ + match i with + [ O ⇒ O + | S m ⇒ + let pred ≝ aux m in + let apred ≝ a pred in + match cmp_nat x (\fst apred) with + [ cmp_le _ ⇒ pred + | cmp_gt nP ⇒ \fst (H3 apred ?)]] + in aux + : + ∀i:nat.∃j:nat.spec i j));[whd; apply nP;] unfold spec in aux ⊢ %; +[3: unfold X in H2; clear H4 n aux spec H3 H1 H X; + cases (cases_in_segment ??? Hx); + elim 𝕦_ sg in H1 ⊢ %; intros (a Hs H); + [1: left; split; [apply le_n] + generalize in match H; + generalize in match Hx; + rewrite > (?:x = O); + [2: cases Hx; lapply (os_le_to_nat_le ?? H1); + apply (symmetric_eq nat O x ?).apply (le_n_O_to_eq x ?).apply (Hletin). + |1: intros; unfold Type_OF_ordered_set in sg a; whd in a:(? %); + lapply (H2 O) as K; lapply (sl2l_ ?? (a O) ≪x,Hx≫ K) as P; + simplify in P:(???%); lapply (le_transitive ??? P H1) as W; + lapply (os_le_to_nat_le ?? W) as R; apply (le_n_O_to_eq (\fst (a O)) R);] + |2: right; cases Hx; rewrite > (sym_plus x O); split; [apply le_S_S; apply le_O_n]; + apply (trans_le ??? (os_le_to_nat_le ?? H3)); + apply le_plus_n_r;] +|2: clear H6; cut (x = \fst (a (aux n1))); [2: + cases (le_to_or_lt_eq ?? H5); [2: assumption] + cases (?:False); apply (H2 (aux n1) H6);] clear H5; + generalize in match Hcut; clear Hcut; intro H5; +|1: clear H6] +[2,1: + cases (aux n1) in H5 ⊢ %; intros; + change in match (a ≪w,H5≫) in H6 ⊢ % with (a w); + cases H5; clear H5; cases H7; clear H7; + [1: left; split; [ apply (le_S ?? H5); | assumption] + |3: cases (?:False); rewrite < H8 in H6; apply (not_le_Sn_n ? H6); + |*: cases (cmp_nat 𝕦_ sg (S n1)); + [1,3: left; split; [1,3: assumption |2: assumption] + cut (𝕦_ sg = S n1); [2: apply le_to_le_to_eq; assumption ] + clear H7 H5 H4;rewrite > Hcut in H8:(? ? (? % ?)); clear Hcut; + cut (x = S (\fst (a w))); + [2: apply le_to_le_to_eq; [2: assumption] + change in H8 with (x + n1 ≤ S (n1 + \fst (a w))); + rewrite > plus_n_Sm in H8; rewrite > sym_plus in H8; + apply (le_plus_to_le ??? H8);] + cases (H3 (a w) H6); + change with (x = \fst (a w1)); + change in H4 with (\fst (a w) < \fst (a w1)); + apply le_to_le_to_eq; [ rewrite > Hcut; assumption ] + apply (os_le_to_nat_le (\fst (a w1)) x (H2 w1)); + |*: right; split; try assumption; + [1: rewrite > sym_plus in ⊢ (? ? %); + rewrite < H6; apply le_plus_r; assumption; + |2: cases (H3 (a w) H6); + change with (x + S n1 ≤ 𝕦_ sg + \fst (a w1));rewrite < plus_n_Sm; + apply (trans_le ??? (le_S_S ?? H8)); rewrite > plus_n_Sm; + apply (le_plus ???? (le_n ?) H9);]]]] +clearbody m; unfold spec in m; clear spec; +alias symbol "exists" = "CProp exists". +letin find ≝ ( + let rec find i u on u : nat ≝ + match u with + [ O ⇒ (m i:nat) + | S w ⇒ match eqb (\fst (a (m i))) x with + [ true ⇒ (m i:nat) + | false ⇒ find (S i) w]] + in find + : + ∀i,bound.∃j.i + bound = 𝕦_ sg → x = \fst (a j)); +[1: cases (find (S n) n2); intro; change with (x = \fst (a w)); + apply H6; rewrite < H7; simplify; apply plus_n_Sm; +|2: intros; rewrite > (eqb_true_to_eq ?? H5); reflexivity +|3: intros; rewrite > sym_plus in H5; rewrite > H5; clear H5 H4 n n1; + cases (m 𝕦_ sg); cases H4; clear H4; cases H5; clear H5; [assumption] + cases (not_le_Sn_n ? H4)] +clearbody find; cases (find O 𝕦_ sg); +exists [apply w]; intros; change with (x = \fst (a j)); +rewrite > (H4 ?); [2: reflexivity] +apply le_to_le_to_eq; +[1: apply os_le_to_nat_le; + apply (trans_increasing a H ? ? (nat_le_to_os_le ?? H5)); +|2: apply (trans_le ? x ?);[apply os_le_to_nat_le; apply (H2 j);] + rewrite < (H4 ?); [2: reflexivity] apply le_n;] +qed. + +lemma hint2: + ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s)) + → sequence (hos_carr (os_r (segment_ordered_set nat_ordered_set s))). +intros; assumption; +qed. + +coercion hint2 nocomposites. + +alias symbol "N" = "ordered set N". +axiom increasing_supremum_stabilizes_r: + ∀s:‡ℕ.∀a:sequence {[s]}.a is_decreasing → + ∀x.x is_infimum a → ∃i.∀j.i ≤ j → \fst x = \fst (a j).