record powerset (A : Type) : Type ≝ { char : A → CProp }.
-interpretation "char" 'subset p = (mk_powerset _ p).
+interpretation "char" 'subset p = (mk_powerset ? p).
interpretation "pwset" 'powerset a = (powerset a).
-interpretation "in" 'mem a X = (char _ X a).
+interpretation "in" 'mem a X = (char ? X a).
definition subseteq ≝ λA.λu,v:\Omega \sup A.∀x.x ∈ u → x ∈ v.
-interpretation "subseteq" 'subseteq u v = (subseteq _ u v).
+interpretation "subseteq" 'subseteq u v = (subseteq ? u v).
definition overlaps ≝ λA.λU,V : Ω \sup A. exT2 ? (λx.x ∈ U) (λx.x ∈ V).
-interpretation "overlaps" 'overlaps u v = (overlaps _ u v).
+interpretation "overlaps" 'overlaps u v = (overlaps ? u v).
definition intersect ≝ λA.λu,v:Ω\sup A.{ y | y ∈ u ∧ y ∈ v }.
-interpretation "intersect" 'intersects u v = (intersect _ u v).
+interpretation "intersect" 'intersects u v = (intersect ? u v).
record axiom_set : Type ≝ {
A:> Type;
notation "hvbox(a break ◃ b)" non associative with precedence 45
for @{ 'covers $a $b }. (* a \ltri b *)
-interpretation "coversl" 'covers A U = (for_all _ U A (covers _ U)).
-interpretation "covers" 'covers a U = (covers _ U a).
+interpretation "coversl" 'covers A U = (for_all ? U A (covers ? U)).
+interpretation "covers" 'covers a U = (covers ? U a).
definition covers_elim ≝
λA:axiom_set.λU: \Omega \sup A.λP:\Omega \sup A.
notation "hvbox(a break ⋉ b)" non associative with precedence 45
for @{ 'fish $a $b }. (* a \ltimes b *)
-interpretation "fishl" 'fish A U = (ex_such _ U A (fish _ U)).
-interpretation "fish" 'fish a U = (fish _ U a).
+interpretation "fishl" 'fish A U = (ex_such ? U A (fish ? U)).
+interpretation "fish" 'fish a U = (fish ? U a).
let corec fish_rec (A:axiom_set) (U: \Omega \sup A)
(P: Ω \sup A) (H1: P ⊆ U)
simplify in H4 ⊢ %;
apply H1;
[ apply (C ? a1 j);
- | autobatch;
+ | autobatch;
| assumption;
| assumption]]
qed.
definition leq ≝ λA:axiom_set.λa,b:A. a ◃ {y|b=y}.
-interpretation "covered by one" 'leq a b = (leq _ a b).
+interpretation "covered by one" 'leq a b = (leq ? a b).
theorem leq_refl: ∀A:axiom_set.∀a:A. a ≤ a.
intros;
definition uparrow ≝ λA:axiom_set.λa:A.mk_powerset ? (λb:A. a ≤ b).
-interpretation "uparrow" 'uparrow a = (uparrow _ a).
+interpretation "uparrow" 'uparrow a = (uparrow ? a).
definition downarrow ≝ λA:axiom_set.λU:Ω \sup A.mk_powerset ? (λa:A. (↑a) ≬ U).
-interpretation "downarrow" 'downarrow a = (downarrow _ a).
+interpretation "downarrow" 'downarrow a = (downarrow ? a).
definition fintersects ≝ λA:axiom_set.λU,V:Ω \sup A.↓U ∩ ↓V.
-interpretation "fintersects" 'fintersects U V = (fintersects _ U V).
+interpretation "fintersects" 'fintersects U V = (fintersects ? U V).
record convergent_generated_topology : Type ≝
{ AA:> axiom_set;