(* *)
(**************************************************************************)
+
+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 increasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
definition decreasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a 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 < "x \nbsp 'is_lower_bound' \nbsp s" 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 50
+notation < "s \nbsp 'is_increasing'" non associative with precedence 45
for @{'increasing $s}.
-notation < "s \nbsp 'is_decreasing'" 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 50
+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 50
+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 50
+notation > "x 'is_upper_bound' s" non associative with precedence 45
for @{'upper_bound $s $x}.
-notation > "x 'is_lower_bound' s" 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 50
+notation > "s 'is_increasing'" non associative with precedence 45
for @{'increasing $s}.
-notation > "s 'is_decreasing'" non associative with precedence 50
+notation > "s 'is_decreasing'" non associative with precedence 45
for @{'decreasing $s}.
-notation > "x 'is_supremum' s" non associative with precedence 50
+notation > "x 'is_supremum' s" non associative with precedence 45
for @{'supremum $s $x}.
-notation > "x 'is_infimum' s" non associative with precedence 50
+notation > "x 'is_infimum' s" non associative with precedence 45
for @{'infimum $s $x}.
-interpretation "Ordered set upper bound" 'upper_bound s x =
- (cic:/matita/dama/supremum/upper_bound.con _ s x).
-interpretation "Ordered set lower bound" 'lower_bound s x =
- (cic:/matita/dama/supremum/lower_bound.con _ s x).
-interpretation "Ordered set increasing" 'increasing s =
- (cic:/matita/dama/supremum/increasing.con _ s).
-interpretation "Ordered set decreasing" 'decreasing s =
- (cic:/matita/dama/supremum/decreasing.con _ s).
-interpretation "Ordered set strong sup" 'supremum s x =
- (cic:/matita/dama/supremum/supremum.con _ s x).
-interpretation "Ordered set strong inf" 'infimum s x =
- (cic:/matita/dama/supremum/infimum.con _ s x).
-
-include "bishop_set.ma".
+interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound _ s x).
+interpretation "Ordered set lower bound" 'lower_bound s x = (lower_bound _ s x).
+interpretation "Ordered set increasing" 'increasing s = (increasing _ s).
+interpretation "Ordered set decreasing" 'decreasing s = (decreasing _ s).
+interpretation "Ordered set strong sup" 'supremum s x = (supremum _ s x).
+interpretation "Ordered set strong inf" 'infimum s x = (infimum _ s x).
lemma uniq_supremum:
∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
cases (H1 ? H) (w Hw); apply Hv; assumption;
qed.
+lemma infimum_is_lower_bound:
+ ∀C:ordered_set.∀a:sequence C.∀u:C.
+ u is_infimum a → ∀v.v is_lower_bound a → v ≤ u.
+intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
+cases (H1 ? H) (w Hw); apply Hv; assumption;
+qed.
+
(* Lemma 2.6 *)
definition strictly_increasing ≝
λC:ordered_set.λa:sequence C.∀n:nat.a (S n) ≰ a n.
definition strictly_decreasing ≝
λC:ordered_set.λa:sequence C.∀n:nat.a n ≰ a (S n).
-
notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 50
for @{'strictly_increasing $s}.
notation > "s 'is_strictly_increasing'" non associative with precedence 50
for @{'strictly_increasing $s}.
interpretation "Ordered set strict increasing" 'strictly_increasing s =
- (cic:/matita/dama/supremum/strictly_increasing.con _ s).
+ (strictly_increasing _ s).
notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 50
for @{'strictly_decreasing $s}.
notation > "s 'is_strictly_decreasing'" non associative with precedence 50
for @{'strictly_decreasing $s}.
interpretation "Ordered set strict decreasing" 'strictly_decreasing s =
- (cic:/matita/dama/supremum/strictly_decreasing.con _ s).
-
-notation "a \uparrow u" non associative with precedence 50 for @{'sup_inc $a $u}.
-interpretation "Ordered set supremum of increasing" 'sup_inc s u =
- (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1)
- (cic:/matita/dama/supremum/increasing.con _ s)
- (cic:/matita/dama/supremum/supremum.con _ s u)).
-notation "a \downarrow u" non associative with precedence 50 for @{'inf_dec $a $u}.
-interpretation "Ordered set supremum of increasing" 'inf_dec s u =
- (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1)
- (cic:/matita/dama/supremum/decreasing.con _ s)
- (cic:/matita/dama/supremum/infimum.con _ s u)).
+ (strictly_decreasing _ s).
+
+definition uparrow ≝
+ λC:ordered_set.λs:sequence C.λu:C.
+ s is_increasing ∧ u is_supremum s.
+
+definition downarrow ≝
+ λC:ordered_set.λs:sequence C.λu:C.
+ s is_decreasing ∧ u is_infimum s.
+
+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" 'sup_inc s u = (uparrow _ 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 downarrow" 'inf_dec s u = (downarrow _ s u).
-include "nat/plus.ma".
-include "nat_ordered_set.ma".
-
-alias symbol "nleq" = "Ordered set excess".
-alias symbol "leq" = "Ordered set less or equal than".
lemma trans_increasing:
- ∀C:ordered_set.∀a:sequence C.a is_increasing → ∀n,m:nat_ordered_set. m ≰ n → a n ≤ a m.
+ ∀C:ordered_set.∀a:sequence C.a is_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 (os_coreflexive ?? X);]
+cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
+[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
+|1: apply (le_transitive ???? (H ?) (Hs ?));
+ intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
+qed.
+
+lemma trans_decreasing:
+ ∀C:ordered_set.∀a:sequence C.a is_decreasing →
+ ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
+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 (os_coreflexive ?? X);]
+cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
+[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
+|1: apply (le_transitive ???? (Hs ?) (H ?));
+ intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
+qed.
+
+lemma trans_increasing_exc:
+ ∀C:ordered_set.∀a:sequence C.a is_increasing →
+ ∀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 (os_carr nat_ordered_set); (* canonical structures *)
cases (Hs n1); assumption;]
qed.
+lemma trans_decreasing_exc:
+ ∀C:ordered_set.∀a:sequence C.a is_decreasing →
+ ∀n,m:nat_ordered_set. m ≰ n → a m ≤ a n .
+intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
+intro; apply H;
+[1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *)
+ change with (n<n1); (* is sort elimination not allowed preserved by delta? *)
+ cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
+ cases (Hs n); rewrite < H3 in H2; assumption (* ogni goal di tipo Prop non è anche di tipo CProp *)
+|2: cases (os_cotransitive ??? (a n1) H2); [2:assumption]
+ cases (Hs n1); assumption;]
+qed.
+
+alias symbol "exists" = "CProp exists".
lemma strictly_increasing_reaches:
∀C:ordered_set.∀m:sequence nat_ordered_set.
m is_strictly_increasing → ∀w.∃t.m t ≰ w.
apply (transitive_le ??? H2 Hp);]]
qed.
-lemma selection:
+lemma selection_uparrow:
∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
- ∀a:sequence C.∀u.a ↑ u → (λx.a (m x)) ↑ u.
+ ∀a:sequence C.∀u.a ↑ u → ⌊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 trans_increasing; [assumption] apply (Hm n);
+[1: intro n; simplify; apply trans_increasing_exc; [assumption] apply (Hm n);
|2: intro n; simplify; apply Uu;
|3: intros (y Hy); simplify; cases (Hu ? Hy);
cases (strictly_increasing_reaches C ? Hm w);
exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [2:assumption]
- cases (trans_increasing C ? Ia ?? H1); assumption;]
+ cases (trans_increasing_exc C ? Ia ?? H1); assumption;]
qed.
-
+
+lemma selection_downarrow:
+ ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
+ ∀a:sequence C.∀u.a ↓ u → ⌊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 trans_decreasing_exc; [assumption] apply (Hm n);
+|2: intro n; simplify; apply Uu;
+|3: intros (y Hy); simplify; cases (Hu ? Hy);
+ cases (strictly_increasing_reaches C ? Hm w);
+ exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [assumption]
+ cases (trans_decreasing_exc C ? Ia ?? H1); assumption;]
+qed.
+
(* Definition 2.7 *)
-alias symbol "exists" = "CProp exists".
-alias symbol "and" = "constructive and".
definition order_converge ≝
λO:ordered_set.λa:sequence O.λx:O.
- ∃l:sequence O.∃u:sequence O.
- l is_increasing ∧ u is_decreasing ∧ l ↑ x ∧ u ↓ x ∧
- ∀i:nat. (l i) is_infimum (λw.a (w+i)) ∧ (u i) is_supremum (λw.a (w+i)).
+ 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 (\circ \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 50
+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 50
+notation > "a 'order_converges' x" non associative with precedence 45
for @{'order_converge $a $x}.
-interpretation "Order convergence" 'order_converge s u =
- (cic:/matita/dama/supremum/order_converge.con _ s u).
+interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).
(* Definition 2.8 *)
+alias symbol "and" = "constructive and".
+definition segment ≝ λO:ordered_set.λa,b:O.λx:O.(x ≤ b) ∧ (a ≤ x).
-definition segment ≝ λO:ordered_set.λa,b:O.λx:O.
- (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (x ≤ b) (a ≤ x)).
-
-notation "[a,b]" non associative with precedence 50
- for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b =
- (cic:/matita/dama/supremum/segment.con _ a b).
-
-notation "x \in [a,b]" non associative with precedence 50
- for @{'segment2 $a $b $x}.
-interpretation "Ordered set sergment in" 'segment2 a b x=
- (cic:/matita/dama/supremum/segment.con _ a b x).
-
-coinductive sigma (A:Type) (P:A→Prop) : Type ≝ sig_in : ∀x.P x → sigma A P.
-
-definition pi1 : ∀A.∀P.sigma A P → A ≝ λA,P,s.match s with [sig_in x _ ⇒ x].
-
-notation < "\pi \sub 1 x" non associative with precedence 50 for @{'pi1 $x}.
-notation < "\pi \sub 2 x" non associative with precedence 50 for @{'pi2 $x}.
-notation > "'fst' x" non associative with precedence 50 for @{'pi1 $x}.
-notation > "'snd' x" non associative with precedence 50 for @{'pi2 $x}.
-interpretation "sigma pi1" 'pi1 x =
- (cic:/matita/dama/supremum/pi1.con _ _ x).
+notation "[a,b]" left associative with precedence 70 for @{'segment $a $b}.
+interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
-interpretation "Type exists" 'exists \eta.x =
- (cic:/matita/dama/supremum/sigma.ind#xpointer(1/1) _ x).
+notation "hvbox(x \in break [a,b])" non associative with precedence 45
+ for @{'segment_in $a $b $x}.
+interpretation "Ordered set sergment in" 'segment_in a b x= (segment _ a b x).
lemma segment_ordered_set:
∀O:ordered_set.∀u,v:O.ordered_set.
intros (O u v); apply (mk_ordered_set (∃x.x ∈ [u,v]));
-[1: intros (x y); apply (fst x ≰ fst y);
+[1: intros (x y); apply (\fst x ≰ \fst y);
|2: intro x; cases x; simplify; apply os_coreflexive;
|3: intros 3 (x y z); cases x; cases y ; cases z; simplify; apply os_cotransitive]
qed.
-notation < "{\x|\x \in [a,b]}" non associative with precedence 90
+notation "hvbox({[a, break b]})" non associative with precedence 90
for @{'segment_set $a $b}.
interpretation "Ordered set segment" 'segment_set a b =
- (cic:/matita/dama/supremum/segment_ordered_set.con _ a b).
+ (segment_ordered_set _ a b).
(* Lemma 2.9 *)
lemma segment_preserves_supremum:
- ∀O:ordered_set.∀l,u:O.∀a:sequence (segment_ordered_set ? l u).
- ∀x:(segment_ordered_set ? l u).
- (λn.fst (a n)) is_increasing ∧
- (fst x) is_supremum (λn.fst (a n)) → a ↑ x.
+ ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
+ ⌊n,\fst (a n)⌋ is_increasing ∧
+ (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x.
intros; split; cases H; clear H;
[1: apply H1;
|2: cases H2; split; clear H2;
[1: apply H;
- |2: clear H; intro y0; apply (H3 (fst y0));]]
+ |2: clear H; intro y0; apply (H3 (\fst y0));]]
qed.
-(* Definition 2.10 *)
-coinductive pair (A,B:Type) : Type ≝ prod : ∀a:A.∀b:B.pair A B.
-definition first : ∀A.∀P.pair A P → A ≝ λA,P,s.match s with [prod x _ ⇒ x].
-definition second : ∀A.∀P.pair A P → P ≝ λA,P,s.match s with [prod _ y ⇒ y].
-
-interpretation "pair pi1" 'pi1 x =
- (cic:/matita/dama/supremum/first.con _ _ x).
-interpretation "pair pi2" 'pi2 x =
- (cic:/matita/dama/supremum/second.con _ _ x).
-
-lemma square_ordered_set: ordered_set → ordered_set.
-intro O; apply (mk_ordered_set (pair O O));
-[1: intros (x y); apply (fst x ≰ fst y ∨ snd x ≰ snd y);
-|2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
- cases H (X X); apply (os_coreflexive ?? X);
-|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2);
- clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
- [1: cases (os_cotransitive ??? z1 H1); [left; left|right;left]assumption;
- |2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+lemma segment_preserves_infimum:
+ ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
+ ⌊n,\fst (a n)⌋ is_decreasing ∧
+ (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
+intros; split; cases H; clear H;
+[1: apply H1;
+|2: cases H2; split; clear H2;
+ [1: apply H;
+ |2: clear H; intro y0; apply (H3 (\fst y0));]]
qed.
-
+
+(* Definition 2.10 *)
+alias symbol "square" = "ordered set square".
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
definition square_segment ≝
- λO:ordered_set.λa,b:O.λx:square_ordered_set O.
- (cic:/matita/logic/connectives/And.ind#xpointer(1/1)
- (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (fst x ≤ b) (a ≤ fst x))
- (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (snd x ≤ b) (a ≤ snd x))).
+ λO:ordered_set.λa,b:O.λx:O square.
+ And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
definition convex ≝
- λO:ordered_set.λU:square_ordered_set O → Prop.
- ∀p.U p → fst p ≤ snd p → ∀y. square_segment ? (fst p) (snd p) y → U y.
+ λO:ordered_set.λU:O square → Prop.
+ ∀p.U p → \fst p ≤ \snd p → ∀y. square_segment ? (\fst p) (\snd p) y → U y.
(* Definition 2.11 *)
definition upper_located ≝
λO:ordered_set.λa:sequence O.∀x,y:O. y ≰ x →
(∃i:nat.a i ≰ x) ∨ (∃b:O.y≰b ∧ ∀i:nat.a i ≤ b).
+
+definition lower_located ≝
+ λO:ordered_set.λa:sequence O.∀x,y:O. x ≰ y →
+ (∃i:nat.x ≰ a i) ∨ (∃b:O.b≰y ∧ ∀i:nat.b ≤ a i).
+
+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 _ 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 =
+ (lower_located _ s).
(* Lemma 2.12 *)
-lemma uparrow_located:
- ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → upper_located ? a.
+lemma uparrow_upperlocated:
+ ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → a is_upper_located.
intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
cases H3 (H4 H5); clear H3; cases (os_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.
-
-
\ No newline at end of file
+lemma downarrow_lowerlocated:
+ ∀C:ordered_set.∀a:sequence C.∀u:C.a ↓ u → a is_lower_located.
+intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
+cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
+[1: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+|2: right; exists [apply u]; split; [apply W|apply H4]]
+qed.
projects / helm.git / blobdiff