include "logic/equality.ma".
inductive Or (A,B:CProp) : CProp ≝
- Left : A → Or A B
+ | Left : A → Or A B
| Right : B → Or A B.
interpretation "constructive or" 'or x y = (Or x y).
inductive And3 (A,B,C:CProp) : CProp ≝
| Conj3 : A → B → C → And3 A B C.
-notation < "a ∧ b ∧ c" left associative with precedence 60 for @{'and3 $a $b $c}.
+notation < "a ∧ b ∧ c" left associative with precedence 35 for @{'and3 $a $b $c}.
interpretation "constructive ternary and" 'and3 x y z = (Conj3 x y z).
-inductive And4 (A,B,C:CProp) : CProp ≝
- | Conj4 : A → B → C → And4 A B C.
+inductive And4 (A,B,C,D:CProp) : CProp ≝
+ | Conj4 : A → B → C → D → And4 A B C D.
-notation < "a ∧ b ∧ c ∧ d" left associative with precedence 60 for @{'and3 $a $b $c $d}.
+notation < "a ∧ b ∧ c ∧ d" left associative with precedence 35 for @{'and4 $a $b $c $d}.
interpretation "constructive quaternary and" 'and4 x y z t = (Conj4 x y z t).
inductive exT23 (A:Type) (P:A→CProp) (Q:A→CProp) (R:A→A→CProp) : CProp ≝
ex_introT23: ∀w,p:A. P w → Q p → R w p → exT23 A P Q R.
-notation < "'fst' \nbsp x" non associative with precedence 50 for @{'pi1 $x}.
-notation < "'snd' \nbsp 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}.
-notation < "'fst' \nbsp x \nbsp y" non associative with precedence 50 for @{'pi12 $x $y}.
-notation < "'snd' \nbsp x \nbsp y" non associative with precedence 50 for @{'pi22 $x $y}.
+notation < "'fst' \nbsp x" non associative with precedence 90 for @{'pi1a $x}.
+notation < "'snd' \nbsp x" non associative with precedence 90 for @{'pi2a $x}.
+notation < "'fst' \nbsp x \nbsp y" non associative with precedence 90 for @{'pi1b $x $y}.
+notation < "'snd' \nbsp x \nbsp y" non associative with precedence 90 for @{'pi2b $x $y}.
+notation > "'fst'" non associative with precedence 90 for @{'pi1}.
+notation > "'snd'" non associative with precedence 90 for @{'pi2}.
definition pi1exT ≝ λA,P.λx:exT A P.match x with [ex_introT x _ ⇒ x].
+definition pi2exT ≝
+ λA,P.λx:exT A P.match x return λx.P (pi1exT ?? x) with [ex_introT _ p ⇒ p].
-interpretation "exT fst" 'pi1 x = (pi1exT _ _ x).
-interpretation "exT fst 2" 'pi12 x y = (pi1exT _ _ x y).
+interpretation "exT fst" 'pi1 = (pi1exT _ _).
+interpretation "exT fst" 'pi1a x = (pi1exT _ _ x).
+interpretation "exT fst" 'pi1b x y = (pi1exT _ _ x y).
+interpretation "exT snd" 'pi2 = (pi2exT _ _).
+interpretation "exT snd" 'pi2a x = (pi2exT _ _ x).
+interpretation "exT snd" 'pi2b x y = (pi2exT _ _ x y).
definition pi1exT23 ≝
λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 x _ _ _ _ ⇒ x].
definition pi2exT23 ≝
λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 _ x _ _ _ ⇒ x].
-
-interpretation "exT2 fst" 'pi1 x = (pi1exT23 _ _ _ _ x).
-interpretation "exT2 snd" 'pi2 x = (pi2exT23 _ _ _ _ x).
-interpretation "exT2 fst 2" 'pi12 x y = (pi1exT23 _ _ _ _ x y).
-interpretation "exT2 snd 2" 'pi22 x y = (pi2exT23 _ _ _ _ x y).
+interpretation "exT2 fst" 'pi1 = (pi1exT23 _ _ _ _).
+interpretation "exT2 snd" 'pi2 = (pi2exT23 _ _ _ _).
+interpretation "exT2 fst" 'pi1a x = (pi1exT23 _ _ _ _ x).
+interpretation "exT2 snd" 'pi2a x = (pi2exT23 _ _ _ _ x).
+interpretation "exT2 fst" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
+interpretation "exT2 snd" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
definition Not : CProp → Prop ≝ λx:CProp.x → False.
(* *)
(**************************************************************************)
-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))).
+ (λl,u.∀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 (\circ \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).
+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).
+(*
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].
+definition pi1sig : ∀A.∀P.sigma A P → A ≝ λA,P,s.match s with [sig_in x _ ⇒ x].
-interpretation "sigma pi1" 'pi1 x = (pi1 _ _ x).
+interpretation "sigma pi1" 'pi1a x = (pi1sig _ _ x).
-interpretation "Type exists" 'exists \eta.x =
- (cic:/matita/dama/supremum/sigma.ind#xpointer(1/1) _ x).
+interpretation "Type exists" 'exists \eta.x = (sigma _ x).
+*)
lemma segment_ordered_set:
∀O:ordered_set.∀u,v:O.ordered_set.
|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;
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;
|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 pi1" 'pi1 = (first _ _).
+interpretation "pair pi2" 'pi2 = (second _ _).
+interpretation "pair pi1" 'pi1a x = (first _ _ x).
+interpretation "pair pi2" 'pi2a x = (second _ _ x).
+interpretation "pair pi1" 'pi1b x y = (first _ _ x y).
+interpretation "pair pi2" 'pi2b x y = (second _ _ x y).
+
+notation "hvbox(\langle a, break b\rangle)" left associative with precedence 70 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));
|2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
qed.
-notation < "s 2 \atop \nleq" non associative with precedence 90
+notation < "s 2 \atop \nleq" non associative with precedence 90
for @{ 'square $s }.
notation > "s 'square'" non associative with precedence 90
for @{ 'square $s }.
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))).
+ 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 *)
for @{ 'subset $a $u }.
interpretation "Bishop subset" 'subset a b = (subset _ a b).
-notation "hvbox({ ident x : t | break p })" non associative with precedence 80
- for @{ 'explicitset (\lambda ${ident x} : $t . $p) }.
-definition mk_set ≝ λT:bishop_set.λx:T→Prop.x.
-interpretation "explicit set" 'explicitset t = (mk_set _ t).
-
notation < "s 2 \atop \neq" non associative with precedence 90
for @{ 'square2 $s }.
notation > "s 'square'" non associative with precedence 90
interpretation "bishop set square" 'square x = (square_bishop_set x).
interpretation "bishop set square" 'square2 x = (square_bishop_set x).
-
alias symbol "exists" = "exists".
alias symbol "and" = "logical and".
definition compose_relations ≝
λC:bishop_set.λU,V:C square → Prop.
λx:C square.∃y:C. U 〈fst x,y〉 ∧ V 〈y,snd x〉.
-notation "a \circ b" left associative with precedence 60
+notation "a \circ b" left associative with precedence 70
for @{'compose $a $b}.
interpretation "relations composition" 'compose a b = (compose_relations _ a b).
-notation "hvbox(x \in break a \circ break b)" non associative with precedence 50
- for @{'compose2 $a $b $x}.
-interpretation "relations composition" 'compose2 a b x =
- (compose_relations _ a b x).
definition invert_relation ≝
λC:bishop_set.λU:C square → Prop.
λx:C square. U 〈snd x,fst x〉.
-notation < "s \sup (-1)" non associative with precedence 90
+notation < "s \sup (-1)" left associative with precedence 70
for @{ 'invert $s }.
-notation < "s \sup (-1) x" non associative with precedence 90
+notation < "s \sup (-1) x" left associative with precedence 70
for @{ 'invert2 $s $x}.
-notation > "'inv' s" non associative with precedence 90
- for @{ 'invert $s }.
+notation > "'inv'" right associative with precedence 70
+ for @{ 'invert0 }.
interpretation "relation invertion" 'invert a = (invert_relation _ a).
+interpretation "relation invertion" 'invert0 = (invert_relation _).
interpretation "relation invertion" 'invert2 a x = (invert_relation _ a x).
alias symbol "exists" = "CProp exists".
-alias symbol "and" (instance 18) = "constructive and".
-alias symbol "and" (instance 10) = "constructive and".
+alias symbol "and" (instance 21) = "constructive and".
+alias symbol "and" (instance 16) = "constructive and".
+alias symbol "and" (instance 9) = "constructive and".
record uniform_space : Type ≝ {
us_carr:> bishop_set;
us_unifbase: (us_carr square → Prop) → CProp;
us_phi1: ∀U:us_carr square → Prop. us_unifbase U →
- {x:us_carr square|fst x ≈ snd x} ⊆ U;
+ (λx:us_carr square.fst x ≈ snd x) ⊆ U;
us_phi2: ∀U,V:us_carr square → Prop. us_unifbase U → us_unifbase V →
- ∃W:us_carr square → Prop.us_unifbase W ∧ (W ⊆ {x:?|U x ∧ V x});
+ ∃W:us_carr square → Prop.us_unifbase W ∧ (W ⊆ (λx.U x ∧ V x));
us_phi3: ∀U:us_carr square → Prop. us_unifbase U →
∃W:us_carr square → Prop.us_unifbase W ∧ (W ∘ W) ⊆ U;
us_phi4: ∀U:us_carr square → Prop. us_unifbase U → ∀x.(U x → (inv U) x) ∧ ((inv U) x → U x)
cases (us_phi4 ?? Hv 〈a i,x〉) (P1 P2); apply P2;
apply (Hn ? H1);
qed.
-
-(* Definition 2.17 *)
-definition mk_big_set ≝
- λP:CProp.λF:P→CProp.F.
-interpretation "explicit big set" 'explicitset t = (mk_big_set _ t).
-
-definition restrict_uniformity ≝
- λC:uniform_space.λX:C→Prop.
- {U:C square → Prop| (U ⊆ {x:C square|X (fst x) ∧ X(snd x)}) ∧ us_unifbase C U}.
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