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 Prop.
32 P ? (refl_eq ? x) \to \forall y:A. \forall p:x=y. P y p.
35 (match p return \lambda y. \lambda p.P y p with
36 [refl_eq \Rightarrow H]).
39 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
40 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 theorem 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 theorem 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.reflexivity.
68 coercion cic:/matita/logic/equality/sym_eq.con.
69 coercion cic:/matita/logic/equality/eq_f.con.
72 cic:/matita/logic/equality/eq.ind
73 cic:/matita/logic/equality/sym_eq.con
74 cic:/matita/logic/equality/trans_eq.con
75 cic:/matita/logic/equality/eq_ind.con
76 cic:/matita/logic/equality/eq_elim_r.con
77 cic:/matita/logic/equality/eq_f.con
78 cic:/matita/logic/equality/eq_OF_eq.con. (* \x.sym (eq_f x) *)
80 theorem eq_f2: \forall A,B,C:Type.\forall f:A\to B \to C.
81 \forall x1,x2:A. \forall y1,y2:B.
82 x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
83 intros.elim H1.elim H.reflexivity.
91 eq_ind ? ? (\lambda a.a=y') eq2 ? eq1.
97 comp ? ? ? ? u u = refl_eq ? y.
99 apply (eq_ind' ? ? ? ? ? u).
105 \lambda H: \forall x,y:A. decidable (x=y).
106 \lambda x,y. \lambda p:x=y.
108 [ (or_introl p') \Rightarrow p'
109 | (or_intror K) \Rightarrow False_ind ? (K p) ].
113 \forall H: \forall x,y:A. decidable (x=y).
116 nu ? H ? ? u = nu ? H ? ? v.
119 unfold decidable in H.
120 apply (Or_ind' ? ? ? ? ? (H x y)); simplify.
125 definition nu_inv \def
127 \lambda H: \forall x,y:A. decidable (x=y).
130 comp ? ? ? ? (nu ? H ? ? (refl_eq ? x)) v.
134 \forall H: \forall x,y:A. decidable (x=y).
137 nu_inv ? H ? ? (nu ? H ? ? u) = u.
139 apply (eq_ind' ? ? ? ? ? u).
144 theorem eq_to_eq_to_eq_p_q:
145 \forall A. \forall x,y:A.
146 (\forall x,y:A. decidable (x=y)) \to
147 \forall p,q:x=y. p=q.
149 rewrite < (nu_left_inv ? H ? ? p).
150 rewrite < (nu_left_inv ? H ? ? q).
151 elim (nu_constant ? H ? ? q).
155 (*CSC: alternative proof that does not pollute the environment with
156 technical lemmata. Unfortunately, it is a pain to do without proper
158 theorem eq_to_eq_to_eq_p_q:
159 \forall A. \forall x,y:A.
160 (\forall x,y:A. decidable (x=y)) \to
161 \forall p,q:x=y. p=q.
164 (\lambda x,y. \lambda p:x=y.
166 [ (or_introl p') \Rightarrow p'
167 | (or_intror K) \Rightarrow False_ind ? (K p) ]).
170 eq_ind ? ? (\lambda z. z=y) (nu ? ? q) ? (nu ? ? (refl_eq ? x))
175 apply (eq_ind' ? ? ? ? ? q);
176 fold simplify (nu ? ? (refl_eq ? x)).
177 generalize in match (nu ? ? (refl_eq ? x)); intro.
180 (\lambda y. \lambda u.
181 eq_ind A x (\lambda a.a=y) u y u = refl_eq ? y)
185 rewrite < (Hcut p); fold simplify (nu ? ? p).
186 rewrite < (Hcut q); fold simplify (nu ? ? q).
187 apply (Or_ind' (x=x) (x \neq x)
188 (\lambda p:decidable (x=x). eq_ind A x (\lambda z.z=y) (nu x y p) x
189 ([\lambda H1.eq A x x]
191 [(or_introl p') \Rightarrow p'
192 |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]) =
193 eq_ind A x (\lambda z.z=y) (nu x y q) x
194 ([\lambda H1.eq A x x]
196 [(or_introl p') \Rightarrow p'
197 |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]))
199 intro; simplify; reflexivity.
200 intro q; elim (q (refl_eq ? x)).
205 theorem a:\forall x.x=x\land True.
210 exact (refl_eq Prop x);