(* *)
(**************************************************************************)
-include "sequence.ma".
-include "ordered_set.ma".
+
include "datatypes/constructors.ma".
+include "nat/plus.ma".
+include "nat_ordered_set.ma".
+include "sequence.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 = (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).
-
-include "bishop_set.ma".
+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 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
λC:ordered_set.λs:sequence C.λu:C.
s is_decreasing ∧ u is_infimum s.
-notation < "a \uparrow \nbsp u" non associative with precedence 50 for @{'sup_inc $a $u}.
-notation > "a \uparrow u" non associative with precedence 50 for @{'sup_inc $a $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" 'sup_inc s u = (uparrow _ s u).
-notation < "a \downarrow \nbsp u" non associative with precedence 50 for @{'inf_dec $a $u}.
-notation > "a \downarrow u" non associative with precedence 50 for @{'inf_dec $a $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. n ≤ m → 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);]
qed.
lemma trans_decreasing:
- ∀C:ordered_set.∀a:sequence C.a is_decreasing → ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
+ ∀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);]
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.
+ ∀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 *)
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 .
+ ∀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 *)
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.
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_exc; [assumption] apply (Hm n);
|2: intro n; simplify; apply Uu;
lemma selection_downarrow:
∀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_decreasing_exc; [assumption] apply (Hm n);
|2: intro n; simplify; apply Uu;
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.
+qed.
(* Definition 2.7 *)
-alias id "ExT23" = "cic:/matita/dama/cprop_connectives/exT23.ind#xpointer(1/1)".
definition order_converge ≝
λO:ordered_set.λa:sequence O.λx:O.
- ExT23 (sequence O) (λl.l ↑ x) (λu.u ↓ x)
- (λl,u.∀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 = (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}.
+notation "[a,b]" left associative with precedence 70 for @{'segment $a $b}.
interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
-notation "hvbox(x \in break [a,b])" non associative with precedence 50
- for @{'segment2 $a $b $x}.
-interpretation "Ordered set sergment in" 'segment2 a b x= (segment _ 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].
-
-interpretation "sigma pi1" 'pi1 x = (pi1 _ _ x).
-
-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 "hvbox({[a, break b]})" non associative with precedence 80
+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).
+ (segment_ordered_set _ a b).
(* Lemma 2.9 *)
lemma segment_preserves_supremum:
∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
- (λn.fst (a n)) is_increasing ∧
- (fst x) is_supremum (λn.fst (a n)) → a ↑ x.
+ ⌊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.
lemma segment_preserves_infimum:
∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
- (λn.fst (a n)) is_decreasing ∧
- (fst x) is_infimum (λn.fst (a n)) → a ↓ x.
+ ⌊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));]]
+ |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 = (first _ _ x).
-interpretation "pair pi2" 'pi2 x = (second _ _ x).
-
-notation "hvbox(\langle a, break b\rangle)" non associative with precedence 91 for @{ 'pair $a $b}.
-interpretation "pair" 'pair a b = (prod _ _ a b).
-
-interpretation "prod" 'product a b = (pair a b).
-
-lemma square_ordered_set: ordered_set → ordered_set.
-intro O;
-apply (mk_ordered_set (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]]
-qed.
-
-notation < "s 2 \atop \nleq" non associative with precedence 90
- for @{ 'square $s }.
-notation > "s 'square'" non associative with precedence 90
- for @{ 'square $s }.
-interpretation "ordered set square" 'square s = (square_ordered_set s).
-
+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:O square → Prop.
- ∀p.U p → fst p ≤ snd p → ∀y. square_segment ? (fst p) (snd p) y → U y.
+ ∀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. 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 50
+notation < "s \nbsp 'is_upper_located'" non associative with precedence 45
for @{'upper_located $s}.
-notation > "s 'is_upper_located'" non associative with precedence 50
+notation > "s 'is_upper_located'" non associative with precedence 45
for @{'upper_located $s}.
-interpretation "Ordered set upper locatedness" 'upper_located s =
+interpretation "Ordered set upper locatedness" 'upper_located s =
(upper_located _ s).
-notation < "s \nbsp 'is_lower_located'" non associative with precedence 50
+notation < "s \nbsp 'is_lower_located'" non associative with precedence 45
for @{'lower_located $s}.
-notation > "s 'is_lower_located'" non associative with precedence 50
+notation > "s 'is_lower_located'" non associative with precedence 45
for @{'lower_located $s}.
-interpretation "Ordered set lower locatedness" 'lower_located s =
+interpretation "Ordered set lower locatedness" 'lower_located s =
(lower_located _ s).
(* Lemma 2.12 *)
projects / helm.git / blobdiff