include "logic/pts.ma".
-ninductive peq (A:Type[0]) (x:A) : A \to Prop \def
- refl_peq : peq A x x.
+ninductive eq (A:Type[2]) (x:A) : A → Prop ≝
+ refl: eq A x x.
-interpretation "leibnitz's equality" 'eq t x y = (peq t x y).
-
-ntheorem rewrite_l: ∀A.∀x.∀P:A → Prop. P x → ∀y. x = y → P y.
-#A; #x; #P; #Hx; #y; #Heq;ncases Heq;nassumption.
-nqed.
+interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
-ntheorem sym_peq: ∀A.∀x,y:A. x = y → y = x.
-#A; #x; #y; #Heq; napply (rewrite_l A x (λz.z=x));
-##[ @; ##| nassumption; ##]
+nlemma eq_rect_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
nqed.
-ntheorem rewrite_r: ∀A.∀x.∀P:A → Prop. P x → ∀y. y = x → P y.
-#A; #x; #P; #Hx; #y; #Heq;ncases (sym_peq ? ? ?Heq);nassumption.
+nlemma eq_ind_r :
+ ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ #A; #a; #P; #p; #x0; #p0; napply (eq_rect_r ? ? ? p0); 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.
+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.
*)
-(*
-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) ].
+ntheorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Prop. P x → ∀y. x = y → P y.
+#A; #x; #P; #Hx; #y; #Heq;ncases Heq;nassumption.
+nqed.
-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.
+ntheorem sym_eq: ∀A:Type[2].∀x,y:A. x = y → y = x.
+#A; #x; #y; #Heq; napply (rewrite_l A x (λz.z=x));
+##[ @; ##| nassumption; ##]
+nqed.
-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.
+ntheorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Prop. P x → ∀y. y = x → P y.
+#A; #x; #P; #Hx; #y; #Heq;ncases (sym_eq ? ? ?Heq);nassumption.
+nqed.
-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.
+ntheorem eq_coerc: ∀A,B:Type[1].A→(A=B)→B.
+#A; #B; #Ha; #Heq;nelim Heq; nassumption.
+nqed.
-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.
+ndefinition R0 ≝ λT:Type[0].λt:T.t.
+
+ndefinition R1 ≝ eq_rect_Type0.
+
+ndefinition R2 :
+ ∀T0:Type[0].
+ ∀a0:T0.
+ ∀T1:∀x0:T0. a0=x0 → Type[0].
+ ∀a1:T1 a0 (refl ? a0).
+ ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
+ ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
+ ∀b0:T0.
+ ∀e0:a0 = b0.
+ ∀b1: T1 b0 e0.
+ ∀e1:R1 ?? T1 a1 ? e0 = b1.
+ T2 b0 e0 b1 e1.
+#T0;#a0;#T1;#a1;#T2;#a2;#b0;#e0;#b1;#e1;
+napply (eq_rect_Type0 ????? e1);
+napply (R1 ?? ? ?? e0);
+napply a2;
+nqed.
-(*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.
-*)
+ndefinition R3 :
+ ∀T0:Type[0].
+ ∀a0:T0.
+ ∀T1:∀x0:T0. a0=x0 → Type[0].
+ ∀a1:T1 a0 (refl ? a0).
+ ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
+ ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
+ ∀T3:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0.∀p1:R1 ?? T1 a1 ? p0 = x1.
+ ∀x2:T2 x0 p0 x1 p1.R2 ???? T2 a2 x0 p0 ? p1 = x2 → Type[0].
+ ∀a3:T3 a0 (refl ? a0) a1 (refl ? a1) a2 (refl ? a2).
+ ∀b0:T0.
+ ∀e0:a0 = b0.
+ ∀b1: T1 b0 e0.
+ ∀e1:R1 ?? T1 a1 ? e0 = b1.
+ ∀b2: T2 b0 e0 b1 e1.
+ ∀e2:R2 ???? T2 a2 b0 e0 ? e1 = b2.
+ T3 b0 e0 b1 e1 b2 e2.
+#T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#b0;#e0;#b1;#e1;#b2;#e2;
+napply (eq_rect_Type0 ????? e2);
+napply (R2 ?? ? ???? e0 ? e1);
+napply a3;
+nqed.
-(*
-theorem a:\forall x.x=x\land True.
-[
-2:intros;
- split;
- [
- exact (refl_eq Prop x);
- |
- exact I;
- ]
-1:
- skip
-]
-qed.
-*) *)
+ndefinition R4 :
+ ∀T0:Type[0].
+ ∀a0:T0.
+ ∀T1:∀x0:T0. eq T0 a0 x0 → Type[0].
+ ∀a1:T1 a0 (refl T0 a0).
+ ∀T2:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1 → Type[0].
+ ∀a2:T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1).
+ ∀T3:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
+ ∀x2:T2 x0 p0 x1 p1.eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
+ ∀a3:T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
+ a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2).
+ ∀T4:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
+ ∀x2:T2 x0 p0 x1 p1.∀p2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
+ ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3.
+ Type[0].
+ ∀a4:T4 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
+ a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2)
+ a3 (refl (T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
+ a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2))
+ a3).
+ ∀b0:T0.
+ ∀e0:eq (T0 …) a0 b0.
+ ∀b1: T1 b0 e0.
+ ∀e1:eq (T1 …) (R1 T0 a0 T1 a1 b0 e0) b1.
+ ∀b2: T2 b0 e0 b1 e1.
+ ∀e2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
+ ∀b3: T3 b0 e0 b1 e1 b2 e2.
+ ∀e3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
+ T4 b0 e0 b1 e1 b2 e2 b3 e3.
+#T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#T4;#a4;#b0;#e0;#b1;#e1;#b2;#e2;#b3;#e3;
+napply (eq_rect_Type0 ????? e3);
+napply (R3 ????????? e0 ? e1 ? e2);
+napply a4;
+nqed.
+naxiom streicherK : ∀T:Type[0].∀t:T.∀P:t = t → Type[2].P (refl ? t) → ∀p.P p.
\ No newline at end of file
projects / helm.git / blobdiff