1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/logic/equality/".
17 include "higher_order_defs/relations.ma".
19 inductive eq (A:Type) (x:A) : A \to Prop \def
22 (*CSC: the URI must disappear: there is a bug now *)
23 interpretation "leibnitz's equality"
24 'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
25 (*CSC: the URI must disappear: there is a bug now *)
26 interpretation "leibnitz's non-equality"
27 'neq x y = (cic:/matita/logic/connectives/Not.con
28 (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y)).
31 \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Type.
32 P ? (refl_eq ? x) \to \forall y:A. \forall p:x=y. P y p.
35 (match p1 return \lambda y. \lambda p.P y p with
36 [refl_eq \Rightarrow p]).
39 variant reflexive_eq : \forall A:Type. reflexive A (eq A)
41 (* simplify.intros.apply refl_eq. *)
43 theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
44 unfold symmetric.intros.elim H. apply refl_eq.
47 variant sym_eq : \forall A:Type.\forall x,y:A. x=y \to y=x
50 theorem transitive_eq : \forall A:Type. transitive A (eq A).
51 unfold transitive.intros.elim H1.assumption.
54 variant trans_eq : \forall A:Type.\forall x,y,z:A. x=y \to y=z \to x=z
58 \forall A:Type.\forall x:A. \forall P: A \to Prop.
59 P x \to \forall y:A. y=x \to P y.
60 intros. elim (sym_eq ? ? ? H1).assumption.
63 theorem eq_f: \forall A,B:Type.\forall f:A\to B.
64 \forall x,y:A. x=y \to f x = f y.
65 intros.elim H.apply refl_eq.
68 theorem eq_f': \forall A,B:Type.\forall f:A\to B.
69 \forall x,y:A. x=y \to f y = f x.
70 intros.elim H.apply refl_eq.
74 coercion cic:/matita/logic/equality/sym_eq.con.
75 coercion cic:/matita/logic/equality/eq_f.con.
79 cic:/matita/logic/equality/eq.ind
80 cic:/matita/logic/equality/sym_eq.con
81 cic:/matita/logic/equality/transitive_eq.con
82 cic:/matita/logic/equality/eq_ind.con
83 cic:/matita/logic/equality/eq_elim_r.con
84 cic:/matita/logic/equality/eq_f.con
86 cic:/matita/logic/equality/eq_OF_eq.con.
89 cic:/matita/logic/equality/eq_f'.con. (* \x.sym (eq_f x) *)
92 theorem eq_f2: \forall A,B,C:Type.\forall f:A\to B \to C.
93 \forall x1,x2:A. \forall y1,y2:B.
94 x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
95 intros.elim H1.elim H.reflexivity.
103 eq_ind ? ? (\lambda a.a=y') eq2 ? eq1.
109 comp ? ? ? ? u u = refl_eq ? y.
111 apply (eq_rect' ? ? ? ? ? u).
117 \lambda H: \forall x,y:A. decidable (x=y).
118 \lambda x,y. \lambda p:x=y.
120 [ (or_introl p') \Rightarrow p'
121 | (or_intror K) \Rightarrow False_ind ? (K p) ].
125 \forall H: \forall x,y:A. decidable (x=y).
128 nu ? H ? ? u = nu ? H ? ? v.
131 unfold decidable in H.
132 apply (Or_ind' ? ? ? ? ? (H x y)); simplify.
137 definition nu_inv \def
139 \lambda H: \forall x,y:A. decidable (x=y).
142 comp ? ? ? ? (nu ? H ? ? (refl_eq ? x)) v.
146 \forall H: \forall x,y:A. decidable (x=y).
149 nu_inv ? H ? ? (nu ? H ? ? u) = u.
151 apply (eq_rect' ? ? ? ? ? u).
156 theorem eq_to_eq_to_eq_p_q:
157 \forall A. \forall x,y:A.
158 (\forall x,y:A. decidable (x=y)) \to
159 \forall p,q:x=y. p=q.
161 rewrite < (nu_left_inv ? H ? ? p).
162 rewrite < (nu_left_inv ? H ? ? q).
163 elim (nu_constant ? H ? ? q).
167 (*CSC: alternative proof that does not pollute the environment with
168 technical lemmata. Unfortunately, it is a pain to do without proper
170 theorem eq_to_eq_to_eq_p_q:
171 \forall A. \forall x,y:A.
172 (\forall x,y:A. decidable (x=y)) \to
173 \forall p,q:x=y. p=q.
176 (\lambda x,y. \lambda p:x=y.
178 [ (or_introl p') \Rightarrow p'
179 | (or_intror K) \Rightarrow False_ind ? (K p) ]).
182 eq_ind ? ? (\lambda z. z=y) (nu ? ? q) ? (nu ? ? (refl_eq ? x))
187 apply (eq_rect' ? ? ? ? ? q);
188 fold simplify (nu ? ? (refl_eq ? x)).
189 generalize in match (nu ? ? (refl_eq ? x)); intro.
192 (\lambda y. \lambda u.
193 eq_ind A x (\lambda a.a=y) u y u = refl_eq ? y)
197 rewrite < (Hcut p); fold simplify (nu ? ? p).
198 rewrite < (Hcut q); fold simplify (nu ? ? q).
199 apply (Or_ind' (x=x) (x \neq x)
200 (\lambda p:decidable (x=x). eq_ind A x (\lambda z.z=y) (nu x y p) x
201 ([\lambda H1.eq A x x]
203 [(or_introl p') \Rightarrow p'
204 |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]) =
205 eq_ind A x (\lambda z.z=y) (nu x y q) x
206 ([\lambda H1.eq A x x]
208 [(or_introl p') \Rightarrow p'
209 |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]))
211 intro; simplify; reflexivity.
212 intro q; elim (q (refl_eq ? x)).
217 theorem a:\forall x.x=x\land True.
222 exact (refl_eq Prop x);