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).
-(* se non faccio il bs_of_hos perdo dualità qui *)
-lemma uniq_supremum:
- ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
- t1 is_supremum s → t2 is_supremum s → t1 ≈ t2.
-intros (O s t1 t2 Ht1 Ht2); cases Ht1 (U1 H1); cases Ht2 (U2 H2);
-apply le_le_eq; intro X;
-[1: cases (H1 ? X); apply (U2 w); assumption
-|2: cases (H2 ? X); apply (U1 w); assumption]
-qed.
-
(* Fact 2.5 *)
lemma h_supremum_is_upper_bound:
∀C:half_ordered_set.∀a:sequence C.∀u:C.
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 _)).
-(* TEST DUALITY
-lemma test_infimum_is_lower_bound_duality:
- ∀C:ordered_set.∀a:sequence C.∀u:C.
- u is_infimum a → ∀v.v is_lower_bound a → u ≥ v.
-intros; lapply (infimum_is_lower_bound a u H v H1); assumption;
-qed.
-*)
-
(* Lemma 2.6 *)
definition strictly_increasing ≝
λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n.
definition uparrow ≝
λC:half_ordered_set.λs:sequence C.λu:C.
increasing ? s ∧ supremum ? s u.
-(*
-notation < "a \uparrow \nbsp u" non associative with precedence 45 for @{'sup_inc $a $u}.
-notation > "a \uparrow u" non associative with precedence 45 for @{'sup_inc $a $u}.
-*)
-interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
-(*
-notation < "a \downarrow \nbsp u" non associative with precedence 45 for @{'inf_dec $a $u}.
-notation > "a \downarrow u" non associative with precedence 45 for @{'inf_dec $a $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:
∀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 ?? X);]
+ 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 ?? 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.
interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)).
interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)).
-(* TEST DUALITY
-lemma test_trans_decreasing_duality:
- ∀C:ordered_set.∀a:sequence C.a is_decreasing →
- ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
-intros; apply (trans_decreasing ? H ?? H1); qed.
-*)
+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.
+ ∀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 n1) H2); [assumption]
+|2: cases (hos_cotransitive ? (a n) (a (S n1)) (a n1) H2); [assumption]
cases (Hs n1); assumption;]
qed.
|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 (m w1)) H); [2:assumption]
- cases (h_trans_increasing_exc ?? Ia ?? H1); assumption;]
+ 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}.
interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).
(* Definition 2.8 *)
-definition segment ≝ λO:half_ordered_set.λa,b:O.λx:O.(x ≤≤ b) ∧ (a ≤≤ x).
+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 "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b = (segment (os_l _) a b).
+notation "‡O" non associative with precedence 90 for @{'segment $O}.
+interpretation "Ordered set sergment" 'segment x = (segment x).
-notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45
- for @{'segment_in $a $b $x}.
-interpretation "Ordered set sergment in" 'segment_in a b x= (segment (os_l _) a b x).
+interpretation "Ordered set sergment in" 'mem x s = (in_segment _ s x).
definition segment_ordered_set_carr ≝
- λO:half_ordered_set.λu,v:O.∃x.segment ? u v x.
+ λO:half_ordered_set.λs:‡O.∃x.x ∈ s.
definition segment_ordered_set_exc ≝
- λO:half_ordered_set.λu,v:O.
- λx,y:segment_ordered_set_carr ? u v.\fst x ≰≰ \fst y.
+ 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,u,v. coreflexive ? (segment_ordered_set_exc O u v).
-intros 4; cases x; simplify; apply hos_coreflexive; qed.
+ ∀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,u,v. cotransitive ? (segment_ordered_set_exc O u v).
-intros 6 (O u v x y z); cases x; cases y ; cases z; simplify; apply hos_cotransitive;
+ ∀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.∀u,v:O.half_ordered_set.
-intros (O u v); apply (mk_half_ordered_set ?? (segment_ordered_set_corefl O u v) (segment_ordered_set_cotrans ???));
+ ∀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.∀u,v:O.ordered_set.
-intros (O u v); letin hos ≝ (half_segment_ordered_set (os_l O) u v);
-constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity]
+ ∀O:ordered_set.∀s:‡O.ordered_set.
+intros (O s);
+apply half2full; apply (half_segment_ordered_set (os_l O) s);
qed.
-notation < "hvbox({[a, break b]/})" non associative with precedence 90
- for @{'h_segment_set $a $b}.
-notation > "hvbox({[a, break b]/})" non associative with precedence 90
- for @{'h_segment_set $a $b}.
-interpretation "Half ordered set segment" 'h_segment_set a b =
- (half_segment_ordered_set _ a b).
+notation "{[ term 19 s ]}" non associative with precedence 90 for @{'segset $s}.
+interpretation "Ordered set segment" 'segset s = (segment_ordered_set _ s).
-notation < "hvbox({[a, break b]})" non associative with precedence 90
- for @{'segment_set $a $b}.
-notation > "hvbox({[a, break b]})" non associative with precedence 90
- for @{'segment_set $a $b}.
-interpretation "Ordered set segment" 'segment_set a b =
- (segment_ordered_set _ a b).
+(* 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 2.9 *)
lemma h_segment_preserves_supremum:
- ∀O:half_ordered_set.∀l,u:O.∀a:sequence {[l,u]/}.∀x:{[l,u]/}.
- increasing ? ⌊n,\fst (a n)⌋ ∧
- supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
+ ∀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: apply H1;
+[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: apply H;
- |2: clear H; intro y0; apply (H3 (\fst y0));]]
+ [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}.
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".
projects / helm.git / blobdiff