--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "Plogic/equality.ma".
+
+ninductive True: Prop ≝
+I : True.
+
+default "true" cic:/matita/basics/connectives/True.ind.
+
+ninductive False: Prop ≝ .
+
+default "false" cic:/matita/basics/connectives/False.ind.
+
+ndefinition Not: Prop → Prop ≝
+λA. A → False.
+
+interpretation "logical not" 'not x = (Not x).
+
+ntheorem absurd : ∀ A,C:Prop. A → ¬A → C.
+#A; #C; #H; #Hn; nelim (Hn H).
+nqed.
+
+ntheorem not_to_not : ∀A,B:Prop. (A → B) → ¬B →¬A.
+/3/.
+nqed.
+
+ninductive And (A,B:Prop) : Prop ≝
+ conj : A → B → And A B.
+
+interpretation "logical and" 'and x y = (And x y).
+
+ntheorem proj1: ∀A,B:Prop. A ∧ B → A.
+#A; #B; #AB; nelim AB; //.
+nqed.
+
+ntheorem proj2: ∀ A,B:Prop. A ∧ B → B.
+#A; #B; #AB; nelim AB; //.
+nqed.
+
+ninductive Or (A,B:Prop) : Prop ≝
+ or_introl : A → (Or A B)
+ | or_intror : B → (Or A B).
+
+interpretation "logical or" 'or x y = (Or x y).
+
+ndefinition decidable : Prop → Prop ≝
+λ A:Prop. A ∨ ¬ A.
+
+ninductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
+ ex_intro: ∀ x:A. P x → ex A P.
+
+interpretation "exists" 'exists x = (ex ? x).
+
+ninductive ex2 (A:Type[0]) (P,Q:A \to Prop) : Prop ≝
+ ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
+
+ndefinition iff :=
+ λ A,B. (A → B) ∧ (B → A).
include "logic/pts.ma".
-ninductive peq (A:Type[3]) (x:A) : A \to Prop \def
- refl_peq : peq A x x.
+ninductive eq (A:Type[3]) (x:A) : A → Prop ≝
+ refl_eq : eq A x x.
+
+interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
-interpretation "leibnitz's equality" 'eq t x y = (peq t x y).
+nlemma eq_rect_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type. P a (refl_eq A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+nqed.
+
+nlemma eq_ind_r :
+ ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl_eq A a) → ∀x.∀p:eq ? x a.P x p.
+ #A; #a; #P; #p; #x0; #p0; napply (eq_rect_r ? ? ? p0); nassumption.
+nqed.
+
+(*
+nlemma eq_ind_r :
+ ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl_eq A a) → ∀x.∀p:eq ? x a.P x p.
+ #A; #a; #P; #p; #x0; #p0; ngeneralize in match p;
+ncases p0; #Heq; nassumption.
+nqed.
+*)
ntheorem rewrite_l: ∀A:Type[3].∀x.∀P:A → Prop. P x → ∀y. x = y → P y.
#A; #x; #P; #Hx; #y; #Heq;ncases Heq;nassumption.
nqed.
-ntheorem sym_peq: ∀A:Type[3].∀x,y:A. x = y → y = x.
+ntheorem sym_eq: ∀A:Type[3].∀x,y:A. x = y → y = x.
#A; #x; #y; #Heq; napply (rewrite_l A x (λz.z=x));
##[ @; ##| nassumption; ##]
nqed.
ntheorem rewrite_r: ∀A:Type[3].∀x.∀P:A → Prop. P x → ∀y. y = x → P y.
-#A; #x; #P; #Hx; #y; #Heq;ncases (sym_peq ? ? ?Heq);nassumption.
+#A; #x; #P; #Hx; #y; #Heq;ncases (sym_eq ? ? ?Heq);nassumption.
nqed.
ntheorem eq_coerc: ∀A,B:Type[2].A→(A=B)→B.
#A; #B; #Ha; #Heq;nelim Heq; nassumption.
nqed.
-
-(*
-theorem eq_rect':
- \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Type.
- P ? (refl_eq ? x) \to \forall y:A. \forall p:x=y. P y p.
- intros.
- exact
- (match p1 return \lambda y. \lambda p.P y p with
- [refl_eq \Rightarrow p]).
-qed.
-
-variant reflexive_eq : \forall A:Type. reflexive A (eq A)
-\def refl_eq.
-
-theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
-unfold symmetric.intros.elim H. apply refl_eq.
-qed.
-
-variant sym_eq : \forall A:Type.\forall x,y:A. x=y \to y=x
-\def symmetric_eq.
-
-theorem transitive_eq : \forall A:Type. transitive A (eq A).
-unfold transitive.intros.elim H1.assumption.
-qed.
-
-variant trans_eq : \forall A:Type.\forall x,y,z:A. x=y \to y=z \to x=z
-\def transitive_eq.
-
-theorem symmetric_not_eq: \forall A:Type. symmetric A (λx,y.x ≠ y).
-unfold symmetric.simplify.intros.unfold.intro.apply H.apply sym_eq.assumption.
-qed.
-
-variant sym_neq: ∀A:Type.∀x,y.x ≠ y →y ≠ x
-≝ symmetric_not_eq.
-
-theorem eq_elim_r:
- \forall A:Type.\forall x:A. \forall P: A \to Prop.
- P x \to \forall y:A. y=x \to P y.
-intros. elim (sym_eq ? ? ? H1).assumption.
-qed.
-
-theorem eq_elim_r':
- \forall A:Type.\forall x:A. \forall P: A \to Set.
- P x \to \forall y:A. y=x \to P y.
-intros. elim (sym_eq ? ? ? H).assumption.
-qed.
-
-theorem eq_elim_r'':
- \forall A:Type.\forall x:A. \forall P: A \to Type.
- P x \to \forall y:A. y=x \to P y.
-intros. elim (sym_eq ? ? ? H).assumption.
-qed.
-
-theorem eq_f: \forall A,B:Type.\forall f:A\to B.
-\forall x,y:A. x=y \to f x = f y.
-intros.elim H.apply refl_eq.
-qed.
-
-theorem eq_f': \forall A,B:Type.\forall f:A\to B.
-\forall x,y:A. x=y \to f y = f x.
-intros.elim H.apply refl_eq.
-qed.
-*)
-
-(*
-coercion sym_eq.
-coercion eq_f.
-
-
-default "equality"
- cic:/matita/logic/equality/eq.ind
- cic:/matita/logic/equality/sym_eq.con
- cic:/matita/logic/equality/transitive_eq.con
- cic:/matita/logic/equality/eq_ind.con
- cic:/matita/logic/equality/eq_elim_r.con
- cic:/matita/logic/equality/eq_rec.con
- cic:/matita/logic/equality/eq_elim_r'.con
- cic:/matita/logic/equality/eq_rect.con
- cic:/matita/logic/equality/eq_elim_r''.con
- cic:/matita/logic/equality/eq_f.con
-(* *)
- cic:/matita/logic/equality/eq_OF_eq.con.
-(* *)
-(*
- cic:/matita/logic/equality/eq_f'.con. (* \x.sym (eq_f x) *)
- *)
-
-theorem eq_f2: \forall A,B,C:Type.\forall f:A\to B \to C.
-\forall x1,x2:A. \forall y1,y2:B.
-x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
-intros.elim H1.elim H.reflexivity.
-qed.
-
-definition comp \def
- \lambda A.
- \lambda x,y,y':A.
- \lambda eq1:x=y.
- \lambda eq2:x=y'.
- eq_ind ? ? (\lambda a.a=y') eq2 ? eq1.
-
-lemma trans_sym_eq:
- \forall A.
- \forall x,y:A.
- \forall u:x=y.
- comp ? ? ? ? u u = refl_eq ? y.
- intros.
- apply (eq_rect' ? ? ? ? ? u).
- reflexivity.
-qed.
-
-definition nu \def
- \lambda A.
- \lambda H: \forall x,y:A. decidable (x=y).
- \lambda x,y. \lambda p:x=y.
- match H x y with
- [ (or_introl p') \Rightarrow p'
- | (or_intror K) \Rightarrow False_ind ? (K p) ].
-
-theorem nu_constant:
- \forall A.
- \forall H: \forall x,y:A. decidable (x=y).
- \forall x,y:A.
- \forall u,v:x=y.
- nu ? H ? ? u = nu ? H ? ? v.
- intros.
- unfold nu.
- unfold decidable in H.
- apply (Or_ind' ? ? ? ? ? (H x y)); simplify.
- intro; reflexivity.
- intro; elim (q u).
-qed.
-
-definition nu_inv \def
- \lambda A.
- \lambda H: \forall x,y:A. decidable (x=y).
- \lambda x,y:A.
- \lambda v:x=y.
- comp ? ? ? ? (nu ? H ? ? (refl_eq ? x)) v.
-
-theorem nu_left_inv:
- \forall A.
- \forall H: \forall x,y:A. decidable (x=y).
- \forall x,y:A.
- \forall u:x=y.
- nu_inv ? H ? ? (nu ? H ? ? u) = u.
- intros.
- apply (eq_rect' ? ? ? ? ? u).
- unfold nu_inv.
- apply trans_sym_eq.
-qed.
-
-theorem eq_to_eq_to_eq_p_q:
- \forall A. \forall x,y:A.
- (\forall x,y:A. decidable (x=y)) \to
- \forall p,q:x=y. p=q.
- intros.
- rewrite < (nu_left_inv ? H ? ? p).
- rewrite < (nu_left_inv ? H ? ? q).
- elim (nu_constant ? H ? ? q).
- reflexivity.
-qed.
-
-(*CSC: alternative proof that does not pollute the environment with
- technical lemmata. Unfortunately, it is a pain to do without proper
- support for let-ins.
-theorem eq_to_eq_to_eq_p_q:
- \forall A. \forall x,y:A.
- (\forall x,y:A. decidable (x=y)) \to
- \forall p,q:x=y. p=q.
-intros.
-letin nu \def
- (\lambda x,y. \lambda p:x=y.
- match H x y with
- [ (or_introl p') \Rightarrow p'
- | (or_intror K) \Rightarrow False_ind ? (K p) ]).
-cut
- (\forall q:x=y.
- eq_ind ? ? (\lambda z. z=y) (nu ? ? q) ? (nu ? ? (refl_eq ? x))
- = q).
-focus 8.
- clear q; clear p.
- intro.
- apply (eq_rect' ? ? ? ? ? q);
- fold simplify (nu ? ? (refl_eq ? x)).
- generalize in match (nu ? ? (refl_eq ? x)); intro.
- apply
- (eq_rect' A x
- (\lambda y. \lambda u.
- eq_ind A x (\lambda a.a=y) u y u = refl_eq ? y)
- ? x H1).
- reflexivity.
-unfocus.
-rewrite < (Hcut p); fold simplify (nu ? ? p).
-rewrite < (Hcut q); fold simplify (nu ? ? q).
-apply (Or_ind' (x=x) (x \neq x)
- (\lambda p:decidable (x=x). eq_ind A x (\lambda z.z=y) (nu x y p) x
- ([\lambda H1.eq A x x]
- match p with
- [(or_introl p') \Rightarrow p'
- |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]) =
- eq_ind A x (\lambda z.z=y) (nu x y q) x
- ([\lambda H1.eq A x x]
- match p with
- [(or_introl p') \Rightarrow p'
- |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]))
- ? ? (H x x)).
-intro; simplify; reflexivity.
-intro q; elim (q (refl_eq ? x)).
-qed.
-*)
-
-(*
-theorem a:\forall x.x=x\land True.
-[
-2:intros;
- split;
- [
- exact (refl_eq Prop x);
- |
- exact I;
- ]
-1:
- skip
-]
-qed.
-*) *)
-
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