+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "categories.ma".
-
-record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: A ⇒_1 CPROP }.
-interpretation "powerset low" 'powerset A = (powerset_carrier A).
-notation "hvbox(a break ∈. b)" non associative with precedence 45 for @{ 'mem_low $a $b }.
-interpretation "memlow" 'mem_low a S = (fun11 ?? (mem_operator ? S) a).
-
-definition subseteq_operator: ∀A: objs1 SET. Ω^A → Ω^A → CProp0 ≝
- λA:objs1 SET.λU,V.∀a:A. a ∈. U → a ∈. V.
-
-theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
- intros 6; intros 2;
- apply s1; apply s;
- assumption.
-qed.
-
-definition powerset_setoid1: SET → SET1.
- intros (T);
- constructor 1;
- [ apply (powerset_carrier T)
- | constructor 1;
- [ apply (λU,V. subseteq_operator ? U V ∧ subseteq_operator ? V U)
- | simplify; intros; split; intros 2; assumption
- | simplify; intros (x y H); cases H; split; assumption
- | simplify; intros (x y z H H1); cases H; cases H1; split;
- apply transitive_subseteq_operator; [1,4: apply y ]
- assumption ]]
-qed.
-
-interpretation "powerset" 'powerset A = (powerset_setoid1 A).
-
-interpretation "subset construction" 'subset \eta.x =
- (mk_powerset_carrier ? (mk_unary_morphism1 ? CPROP x ?)).
-
-definition mem: ∀A. A × Ω^A ⇒_1 CPROP.
- intros;
- constructor 1;
- [ apply (λx,S. mem_operator ? S x)
- | intros 5;
- cases b; clear b; cases b'; clear b'; simplify; intros;
- apply (trans1 ????? (prop11 ?? u ?? e));
- cases e1; whd in s s1;
- split; intro;
- [ apply s; assumption
- | apply s1; assumption]]
-qed.
-
-interpretation "mem" 'mem a S = (fun21 ??? (mem ?) a S).
-
-definition subseteq: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
- intros;
- constructor 1;
- [ apply (λU,V. subseteq_operator ? U V)
- | intros;
- cases e; cases e1;
- split; intros 1;
- [ apply (transitive_subseteq_operator ????? s2);
- apply (transitive_subseteq_operator ???? s1 s4)
- | apply (transitive_subseteq_operator ????? s3);
- apply (transitive_subseteq_operator ???? s s4) ]]
-qed.
-
-interpretation "subseteq" 'subseteq U V = (fun21 ??? (subseteq ?) U V).
-
-theorem subseteq_refl: ∀A.∀S:Ω^A.S ⊆ S.
- intros 4; assumption.
-qed.
-
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω^A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
- intros; apply transitive_subseteq_operator; [apply S2] assumption.
-qed.
-
-definition overlaps: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
- intros;
- constructor 1;
- [ apply (λA:objs1 SET.λU,V:Ω^A.(exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V) : CProp0))
- | intros;
- constructor 1; intro; cases e2; exists; [1,4: apply w]
- [ apply (. #‡e^-1); assumption
- | apply (. #‡e1^-1); assumption
- | apply (. #‡e); assumption;
- | apply (. #‡e1); assumption]]
-qed.
-
-interpretation "overlaps" 'overlaps U V = (fun21 ??? (overlaps ?) U V).
-
-definition intersects: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
- intros;
- constructor 1;
- [ apply rule (λU,V. {x | x ∈ U ∧ x ∈ V });
- intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1;
- | intros;
- split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
- | apply (. (#‡e)‡(#‡e1)); assumption]]
-qed.
-
-interpretation "intersects" 'intersects U V = (fun21 ??? (intersects ?) U V).
-
-definition union: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
- intros;
- constructor 1;
- [ apply (λU,V. {x | x ∈ U ∨ x ∈ V });
- intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1
- | intros;
- split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
- | apply (. (#‡e)‡(#‡e1)); assumption]]
-qed.
-
-interpretation "union" 'union U V = (fun21 ??? (union ?) U V).
-
-(* qua non riesco a mettere set *)
-definition singleton: ∀A:setoid. A ⇒_1 Ω^A.
- intros; constructor 1;
- [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
- intros; simplify;
- split; intro;
- apply (.= e1);
- [ apply e | apply (e \sup -1) ]
- | unfold setoid1_of_setoid; simplify;
- intros; split; intros 2; simplify in f ⊢ %; apply trans;
- [ apply a |4: apply a'] try assumption; apply sym; assumption]
-qed.
-
-interpretation "singleton" 'singl a = (fun11 ?? (singleton ?) a).
-notation > "{ term 19 a : S }" with precedence 90 for @{fun11 ?? (singleton $S) $a}.
-
-definition big_intersects: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
- intros; constructor 1;
- [ intro; whd; whd in I;
- apply ({x | ∀i:I. x ∈ c i});
- simplify; intros; split; intros; [ apply (. (e^-1‡#)); | apply (. e‡#); ]
- apply f;
- | intros; split; intros 2; simplify in f ⊢ %; intro;
- [ apply (. (#‡(e i)^-1)); apply f;
- | apply (. (#‡e i)); apply f]]
-qed.
-
-definition big_union: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
- intros; constructor 1;
- [ intro; whd; whd in A; whd in I;
- apply ({x | ∃i:I. x ∈ c i });
- simplify; intros; split; intros; cases e1; clear e1; exists; [1,3:apply w]
- [ apply (. (e^-1‡#)); | apply (. (e‡#)); ]
- apply x;
- | intros; split; intros 2; simplify in f ⊢ %; cases f; clear f; exists; [1,3:apply w]
- [ apply (. (#‡(e w)^-1)); apply x;
- | apply (. (#‡e w)); apply x]]
-qed.
-
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (\emsp) term 90 p)"
-non associative with precedence 50 for @{ 'bigcup $p }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (ident i ∈ I) break term 90 p)"
-non associative with precedence 50 for @{ 'bigcup_mk (λ${ident i}:$I.$p) }.
-notation > "hovbox(⋃ f)" non associative with precedence 60 for @{ 'bigcup $f }.
-
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (\emsp) term 90 p)"
-non associative with precedence 50 for @{ 'bigcap $p }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (ident i ∈ I) break term 90 p)"
-non associative with precedence 50 for @{ 'bigcap_mk (λ${ident i}:$I.$p) }.
-notation > "hovbox(⋂ f)" non associative with precedence 60 for @{ 'bigcap $f }.
-
-interpretation "bigcup" 'bigcup f = (fun12 ?? (big_union ??) f).
-interpretation "bigcap" 'bigcap f = (fun12 ?? (big_intersects ??) f).
-interpretation "bigcup mk" 'bigcup_mk f = (fun12 ?? (big_union ??) (mk_unary_morphism2 ?? f ?)).
-interpretation "bigcap mk" 'bigcap_mk f = (fun12 ?? (big_intersects ??) (mk_unary_morphism2 ?? f ?)).
projects / helm.git / blobdiff