property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
sequence.ma nat/nat.ma
supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
-property_sigma.ma ordered_uniform.ma russell_support.ma
bishop_set_rewrite.ma bishop_set.ma
+property_sigma.ma ordered_uniform.ma russell_support.ma
models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma logic/cprop_connectives.ma
+models/increasing_supremum_stabilizes.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+ordered_set.ma datatypes/constructors.ma logic/cprop_connectives.ma
sandwich.ma ordered_uniform.ma
nat_ordered_set.ma bishop_set.ma nat/compare.ma
-ordered_set.ma datatypes/constructors.ma logic/cprop_connectives.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/q_bars.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma russell_support.ma
-models/list_support.ma list/list.ma logic/cprop_connectives.ma nat/le_arith.ma nat/minus.ma
+models/list_support.ma list/list.ma logic/cprop_connectives.ma nat/minus.ma
+models/q_bars.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
+.unnamed.ma
+lebesgue.ma ordered_set.ma property_exhaustivity.ma sandwich.ma
bishop_set.ma ordered_set.ma
-lebesgue.ma property_exhaustivity.ma sandwich.ma
-models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
+models/nat_order_continuous.ma models/increasing_supremum_stabilizes.ma models/nat_ordered_uniform.ma
models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
russell_support.ma logic/cprop_connectives.ma nat/nat.ma
models/q_copy.ma models/q_bars.ma
-models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
-uniform.ma supremum.ma
ordered_uniform.ma uniform.ma
+uniform.ma supremum.ma
models/q_rebase.ma Q/q/qtimes.ma models/q_copy.ma russell_support.ma
Q/q/qplus.ma
Q/q/qtimes.ma
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "models/nat_uniform.ma".
+include "supremum.ma".
+include "nat/le_arith.ma".
+include "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".
+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;
+ 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;
+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 ? 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).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "models/nat_uniform.ma".
-include "supremum.ma".
-include "nat/le_arith.ma".
-include "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".
-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;
- 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;
-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 ? 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).
-(**************************************************************************)
+ (**************************************************************************)
(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* *)
(**************************************************************************)
-include "models/nat_dedekind_sigma_complete.ma".
+include "models/increasing_supremum_stabilizes.ma".
include "models/nat_ordered_uniform.ma".
lemma nat_us_is_oc: ∀s:‡ℕ.order_continuity (segment_ordered_uniform_space ℕ s).
intros 3; split; intros 3; cases H1; cases H2; clear H2 H1; cases H; clear H;
-[1: cases (nat_dedekind_sigma_complete s a H1 ? H2);
+[1: cases (increasing_supremum_stabilizes s a H1 ? H2);
exists [apply w1]; intros;
apply (restrict nat_ordered_uniform_space s ??? H4);
generalize in match (H ? H5);
generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X;
intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify;
apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space));
-|2: cases (nat_dedekind_sigma_complete_r s a H1 ? H2);
+|2: cases (increasing_supremum_stabilizes_r s a H1 ? H2);
exists [apply w1]; intros;
apply (restrict nat_ordered_uniform_space s ??? H4);
generalize in match (H ? H5);
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