(* *)
(**************************************************************************)
+
+include "datatypes/constructors.ma".
+include "nat/plus.ma".
+include "nat_ordered_set.ma".
include "sequence.ma".
-include "ordered_set.ma".
(* Definition 2.4 *)
-definition upper_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition upper_bound ≝
+ λO:half_ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤≤ u.
-definition strong_sup ≝
- λO:ordered_set.λs:sequence O.λx.upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
+definition supremum ≝
+ λO:half_ordered_set.λs:sequence O.λx.
+ upper_bound ? s x ∧ (∀y:O.x ≰≰ y → ∃n.s n ≰≰ y).
-definition increasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
+definition increasing ≝
+ λO:half_ordered_set.λa:sequence O.∀n:nat.a n ≤≤ a (S n).
-notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 50
+notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 45
for @{'upper_bound $s $x}.
-notation < "s \nbsp 'is_increasing'" non associative with precedence 50
+notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 45
+ for @{'lower_bound $s $x}.
+notation < "s \nbsp 'is_increasing'" non associative with precedence 45
for @{'increasing $s}.
-notation < "x \nbsp 'is_strong_sup' \nbsp s" non associative with precedence 50
- for @{'strong_sup $s $x}.
-
-notation > "x 'is_upper_bound' s" non associative with precedence 50
+notation < "s \nbsp 'is_decreasing'" non associative with precedence 45
+ for @{'decreasing $s}.
+notation < "x \nbsp 'is_supremum' \nbsp s" non associative with precedence 45
+ for @{'supremum $s $x}.
+notation < "x \nbsp 'is_infimum' \nbsp s" non associative with precedence 45
+ for @{'infimum $s $x}.
+notation > "x 'is_upper_bound' s" non associative with precedence 45
for @{'upper_bound $s $x}.
-notation > "s 'is_increasing'" non associative with precedence 50
+notation > "x 'is_lower_bound' s" non associative with precedence 45
+ for @{'lower_bound $s $x}.
+notation > "s 'is_increasing'" non associative with precedence 45
for @{'increasing $s}.
-notation > "x 'is_strong_sup' s" non associative with precedence 50
- for @{'strong_sup $s $x}.
+notation > "s 'is_decreasing'" non associative with precedence 45
+ for @{'decreasing $s}.
+notation > "x 'is_supremum' s" non associative with precedence 45
+ for @{'supremum $s $x}.
+notation > "x 'is_infimum' s" non associative with precedence 45
+ for @{'infimum $s $x}.
+
+interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound (os_l _) s x).
+interpretation "Ordered set lower bound" 'lower_bound s x = (upper_bound (os_r _) s x).
+
+interpretation "Ordered set increasing" 'increasing s = (increasing (os_l _) s).
+interpretation "Ordered set decreasing" 'decreasing s = (increasing (os_r _) s).
+
+interpretation "Ordered set strong sup" 'supremum s x = (supremum (os_l _) s x).
+interpretation "Ordered set strong inf" 'infimum s x = (supremum (os_r _) s x).
+
+(* Fact 2.5 *)
+lemma h_supremum_is_upper_bound:
+ ∀C:half_ordered_set.∀a:sequence C.∀u:C.
+ supremum ? a u → ∀v.upper_bound ? a v → u ≤≤ v.
+intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
+cases (H1 ? H) (w Hw); apply Hv; [apply w] assumption;
+qed.
-interpretation "Ordered set upper bound" 'upper_bound s x =
- (cic:/matita/dama/supremum/upper_bound.con _ s x).
-interpretation "Ordered set increasing" 'increasing s =
- (cic:/matita/dama/supremum/increasing.con _ s).
-interpretation "Ordered set strong sup" 'strong_sup s x =
- (cic:/matita/dama/supremum/strong_sup.con _ s x).
+notation "'supremum_is_upper_bound'" non associative with precedence 90 for @{'supremum_is_upper_bound}.
+notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'infimum_is_lower_bound}.
-include "bishop_set.ma".
+interpretation "supremum_is_upper_bound" 'supremum_is_upper_bound = (h_supremum_is_upper_bound (os_l _)).
+interpretation "infimum_is_lower_bound" 'infimum_is_lower_bound = (h_supremum_is_upper_bound (os_r _)).
+
+(* Lemma 2.6 *)
+definition strictly_increasing ≝
+ λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n.
+
+notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 45
+ for @{'strictly_increasing $s}.
+notation > "s 'is_strictly_increasing'" non associative with precedence 45
+ for @{'strictly_increasing $s}.
+interpretation "Ordered set strict increasing" 'strictly_increasing s =
+ (strictly_increasing (os_l _) s).
-lemma uniq_supremum:
- ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
- t1 is_upper_bound s → t2 is_upper_bound s → t1 ≈ t2.
-intros (O s t1 t2 Ht1 Ht2); apply le_le_eq; cases Ht1; cases Ht2;
+notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 45
+ for @{'strictly_decreasing $s}.
+notation > "s 'is_strictly_decreasing'" non associative with precedence 45
+ for @{'strictly_decreasing $s}.
+interpretation "Ordered set strict decreasing" 'strictly_decreasing s =
+ (strictly_increasing (os_r _) s).
+
+definition uparrow ≝
+ λC:half_ordered_set.λs:sequence C.λu:C.
+ increasing ? s ∧ supremum ? s u.
+
+interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
+interpretation "Ordered set downarrow" 'fintersects s u = (uparrow (os_r _) s u).
+
+lemma h_trans_increasing:
+ ∀C:half_ordered_set.∀a:sequence C.increasing ? a →
+ ∀n,m:nat_ordered_set. n ≤ m → a n ≤≤ a m.
+intros 5 (C a Hs n m); elim m; [
+ rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
+ intro X; cases (hos_coreflexive ? (a n) X);]
+cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
+[2: rewrite > H2; intro; cases (hos_coreflexive ? (a (S n1)) H1);
+|1: apply (hle_transitive ???? (H ?) (Hs ?));
+ intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
+qed.
+
+notation "'trans_increasing'" non associative with precedence 90 for @{'trans_increasing}.
+notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_decreasing}.
+
+interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)).
+interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)).
+
+lemma hint_nat :
+ Type_of_ordered_set nat_ordered_set →
+ hos_carr (os_l (nat_ordered_set)).
+intros; assumption;
+qed.
+
+coercion hint_nat nocomposites.
+
+lemma h_trans_increasing_exc:
+ ∀C:half_ordered_set.∀a:sequence C.increasing ? a →
+ ∀n,m:nat_ordered_set. m ≰≰ n → a n ≤≤ a m.
+intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
+intro; apply H;
+[1: change in n1 with (hos_carr (os_l nat_ordered_set));
+ change with (n<n1);
+ cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
+ cases (Hs n); rewrite < H3 in H2; assumption;
+|2: cases (hos_cotransitive ? (a n) (a (S n1)) (a n1) H2); [assumption]
+ cases (Hs n1); assumption;]
+qed.
+
+notation "'trans_increasing_exc'" non associative with precedence 90 for @{'trans_increasing_exc}.
+notation "'trans_decreasing_exc'" non associative with precedence 90 for @{'trans_decreasing_exc}.
+
+interpretation "trans_increasing_exc" 'trans_increasing_exc = (h_trans_increasing_exc (os_l _)).
+interpretation "trans_decreasing_exc" 'trans_decreasing_exc = (h_trans_increasing_exc (os_r _)).
+
+alias symbol "exists" = "CProp exists".
+lemma nat_strictly_increasing_reaches:
+ ∀m:sequence nat_ordered_set.
+ m is_strictly_increasing → ∀w.∃t.m t ≰ w.
+intros; elim w;
+[1: cases (nat_discriminable O (m O)); [2: cases (not_le_Sn_n O (ltn_to_ltO ?? H1))]
+ cases H1; [exists [apply O] apply H2;]
+ exists [apply (S O)] lapply (H O) as H3; rewrite < H2 in H3; assumption
+|2: cases H1 (p Hp); cases (nat_discriminable (S n) (m p));
+ [1: cases H2; clear H2;
+ [1: exists [apply p]; assumption;
+ |2: exists [apply (S p)]; rewrite > H3; apply H;]
+ |2: cases (?:False); change in Hp with (n<m p);
+ apply (not_le_Sn_n (m p));
+ apply (transitive_le ??? H2 Hp);]]
+qed.
+
+lemma h_selection_uparrow:
+ ∀C:half_ordered_set.∀m:sequence nat_ordered_set.
+ m is_strictly_increasing →
+ ∀a:sequence C.∀u.uparrow ? a u → uparrow ? ⌊x,a (m x)⌋ u.
+intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split;
+[1: intro n; simplify; apply (h_trans_increasing_exc ? a Ia); apply (Hm n);
+|2: intro n; simplify; apply Uu;
+|3: intros (y Hy); simplify; cases (Hu ? Hy);
+ cases (nat_strictly_increasing_reaches ? Hm w);
+ exists [apply w1]; cases (hos_cotransitive ? (a w) y (a (m w1)) H); [2:assumption]
+ cases (h_trans_increasing_exc ?? Ia w (m w1) H1); assumption;]
+qed.
+
+notation "'selection_uparrow'" non associative with precedence 90 for @{'selection_uparrow}.
+notation "'selection_downarrow'" non associative with precedence 90 for @{'selection_downarrow}.
+
+interpretation "selection_uparrow" 'selection_uparrow = (h_selection_uparrow (os_l _)).
+interpretation "selection_downarrow" 'selection_downarrow = (h_selection_uparrow (os_r _)).
+
+(* Definition 2.7 *)
+definition order_converge ≝
+ λO:ordered_set.λa:sequence O.λx:O.
+ exT23 (sequence O) (λl.l ↑ x) (λu.u ↓ x)
+ (λl,u:sequence O.∀i:nat. (l i) is_infimum ⌊w,a (w+i)⌋ ∧
+ (u i) is_supremum ⌊w,a (w+i)⌋).
+
+notation < "a \nbsp (\cir \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 45
+ for @{'order_converge $a $x}.
+notation > "a 'order_converges' x" non associative with precedence 45
+ for @{'order_converge $a $x}.
+interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).
+(* Definition 2.8 *)
+record segment (O : Type) : Type ≝ {
+ seg_l_ : O;
+ seg_u_ : O
+}.
+
+notation > "𝕦_term 90 s p" non associative with precedence 45 for @{'upp $s $p}.
+notation "𝕦 \sub term 90 s p" non associative with precedence 45 for @{'upp $s $p}.
+notation > "𝕝_term 90 s p" non associative with precedence 45 for @{'low $s $p}.
+notation "𝕝 \sub term 90 s p" non associative with precedence 45 for @{'low $s $p}.
+
+definition seg_u ≝
+ λO:half_ordered_set.λs:segment O.λP: O → CProp.
+ wloss O ? (λl,u.P u) (seg_l_ ? s) (seg_u_ ? s).
+definition seg_l ≝
+ λO:half_ordered_set.λs:segment O.λP: O → CProp.
+ wloss O ? (λl,u.P u) (seg_u_ ? s) (seg_l_ ? s).
+
+interpretation "uppper" 'upp s P = (seg_u (os_l _) s P).
+interpretation "lower" 'low s P = (seg_l (os_l _) s P).
+interpretation "uppper dual" 'upp s P = (seg_l (os_r _) s P).
+interpretation "lower dual" 'low s P = (seg_u (os_r _) s P).
+
+definition in_segment ≝
+ λO:half_ordered_set.λs:segment O.λx:O.
+ wloss O ? (λp1,p2.p1 ∧ p2) (seg_u ? s (λu.u ≤≤ x)) (seg_l ? s (λl.x ≤≤ l)).
+
+notation "‡O" non associative with precedence 90 for @{'segment $O}.
+interpretation "Ordered set sergment" 'segment x = (segment x).
+
+interpretation "Ordered set sergment in" 'mem x s = (in_segment _ s x).
+
+definition segment_ordered_set_carr ≝
+ λO:half_ordered_set.λs:‡O.∃x.x ∈ s.
+definition segment_ordered_set_exc ≝
+ λO:half_ordered_set.λs:‡O.
+ λx,y:segment_ordered_set_carr O s.hos_excess_ O (\fst x) (\fst y).
+lemma segment_ordered_set_corefl:
+ ∀O,s. coreflexive ? (wloss O ? (segment_ordered_set_exc O s)).
+intros 3; cases x; cases (wloss_prop O);
+generalize in match (hos_coreflexive O w);
+rewrite < (H1 ? (segment_ordered_set_exc O s));
+rewrite < (H1 ? (hos_excess_ O)); intros; assumption;
+qed.
+lemma segment_ordered_set_cotrans :
+ ∀O,s. cotransitive ? (wloss O ? (segment_ordered_set_exc O s)).
+intros 5 (O s x y z); cases x; cases y ; cases z; clear x y z;
+generalize in match (hos_cotransitive O w w1 w2);
+cases (wloss_prop O);
+do 3 rewrite < (H3 ? (segment_ordered_set_exc O s));
+do 3 rewrite < (H3 ? (hos_excess_ O)); intros; apply H4; assumption;
+qed.
+
+lemma half_segment_ordered_set:
+ ∀O:half_ordered_set.∀s:segment O.half_ordered_set.
+intros (O a); constructor 1;
+[ apply (segment_ordered_set_carr O a);
+| apply (wloss O);
+| apply (wloss_prop O);
+| apply (segment_ordered_set_exc O a);
+| apply (segment_ordered_set_corefl O a);
+| apply (segment_ordered_set_cotrans ??);
+]
+qed.
+
+lemma segment_ordered_set:
+ ∀O:ordered_set.∀s:‡O.ordered_set.
+intros (O s);
+apply half2full; apply (half_segment_ordered_set (os_l O) s);
+qed.
+
+notation "{[ term 19 s ]}" non associative with precedence 90 for @{'segset $s}.
+interpretation "Ordered set segment" 'segset s = (segment_ordered_set _ s).
+
+(* test :
+ ∀O:ordered_set.∀s: segment (os_l O).∀x:O.
+ in_segment (os_l O) s x
+ =
+ in_segment (os_r O) s x.
+intros; try reflexivity;
+*)
+
+definition hint_sequence:
+ ∀C:ordered_set.
+ sequence (hos_carr (os_l C)) → sequence (Type_of_ordered_set C).
+intros;assumption;
+qed.
+
+definition hint_sequence1:
+ ∀C:ordered_set.
+ sequence (hos_carr (os_r C)) → sequence (Type_of_ordered_set_dual C).
+intros;assumption;
+qed.
+
+definition hint_sequence2:
+ ∀C:ordered_set.
+ sequence (Type_of_ordered_set C) → sequence (hos_carr (os_l C)).
+intros;assumption;
+qed.
+
+definition hint_sequence3:
+ ∀C:ordered_set.
+ sequence (Type_of_ordered_set_dual C) → sequence (hos_carr (os_r C)).
+intros;assumption;
+qed.
+
+coercion hint_sequence nocomposites.
+coercion hint_sequence1 nocomposites.
+coercion hint_sequence2 nocomposites.
+coercion hint_sequence3 nocomposites.
+
+(* Lemma 2.9 - non easily dualizable *)
+
+lemma x2sx:
+ ∀O:half_ordered_set.
+ ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
+ \fst x ≰≰ \fst y → x ≰≰ y.
+intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
+whd in ⊢ (?→? (% ? ?) ? ? ? ?); simplify in ⊢ (?→%);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma sx2x:
+ ∀O:half_ordered_set.
+ ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
+ x ≰≰ y → \fst x ≰≰ \fst y.
+intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
+whd in ⊢ (? (% ? ?) ? ? ? ? → ?); simplify in ⊢ (% → ?);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma h_segment_preserves_supremum:
+ ∀O:half_ordered_set.∀s:segment O.
+ ∀a:sequence (half_segment_ordered_set ? s).
+ ∀x:half_segment_ordered_set ? s.
+ increasing ? ⌊n,\fst (a n)⌋ ∧
+ supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
+intros; split; cases H; clear H;
+[1: intro n; lapply (H1 n) as K; clear H1 H2;
+ intro; apply K; clear K; apply (sx2x ???? H);
+|2: cases H2; split; clear H2;
+ [1: intro n; lapply (H n) as K; intro W; apply K;
+ apply (sx2x ???? W);
+ |2: clear H1 H; intros (y0 Hy0); cases (H3 (\fst y0));[exists[apply w]]
+ [1: change in H with (\fst (a w) ≰≰ \fst y0); apply (x2sx ???? H);
+ |2: apply (sx2x ???? Hy0);]]]
+qed.
+
+notation "'segment_preserves_supremum'" non associative with precedence 90 for @{'segment_preserves_supremum}.
+notation "'segment_preserves_infimum'" non associative with precedence 90 for @{'segment_preserves_infimum}.
+
+interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
+interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
+
+(* TEST, ma quanto godo! *)
+lemma segment_preserves_infimum2:
+ ∀O:ordered_set.∀s:‡O.∀a:sequence {[s]}.∀x:{[s]}.
+ ⌊n,\fst (a n)⌋ is_decreasing ∧
+ (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
+intros; apply (segment_preserves_infimum s a x H);
+qed.
+*)
+
+(* Definition 2.10 *)
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+definition square_segment ≝
+ λO:ordered_set.λa,b:O.λx: O squareO.
+ And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
+
+definition convex ≝
+ λO:ordered_set.λU:O squareO → Prop.
+ ∀p.U p → \fst p ≤ \snd p → ∀y.
+ square_segment O (\fst p) (\snd p) y → U y.
+
+(* Definition 2.11 *)
+definition upper_located ≝
+ λO:half_ordered_set.λa:sequence O.∀x,y:O. y ≰≰ x →
+ (∃i:nat.a i ≰≰ x) ∨ (∃b:O.y ≰≰ b ∧ ∀i:nat.a i ≤≤ b).
+
+notation < "s \nbsp 'is_upper_located'" non associative with precedence 45
+ for @{'upper_located $s}.
+notation > "s 'is_upper_located'" non associative with precedence 45
+ for @{'upper_located $s}.
+interpretation "Ordered set upper locatedness" 'upper_located s =
+ (upper_located (os_l _) s).
+
+notation < "s \nbsp 'is_lower_located'" non associative with precedence 45
+ for @{'lower_located $s}.
+notation > "s 'is_lower_located'" non associative with precedence 45
+ for @{'lower_located $s}.
+interpretation "Ordered set lower locatedness" 'lower_located s =
+ (upper_located (os_r _) s).
+
+(* Lemma 2.12 *)
+lemma h_uparrow_upperlocated:
+ ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a.
+intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
+cases H3 (H4 H5); clear H3; cases (hos_cotransitive ??? u Hxy) (W W);
+[2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+|1: right; exists [apply u]; split; [apply W|apply H4]]
+qed.
+
+notation "'uparrow_upperlocated'" non associative with precedence 90 for @{'uparrow_upperlocated}.
+notation "'downarrow_lowerlocated'" non associative with precedence 90 for @{'downarrow_lowerlocated}.
+
+interpretation "uparrow_upperlocated" 'uparrow_upperlocated = (h_uparrow_upperlocated (os_l _)).
+interpretation "downarrow_lowerlocated" 'downarrow_lowerlocated = (h_uparrow_upperlocated (os_r _)).
projects / helm.git / blobdiff