1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/nf2/defs.ma".
19 include "LambdaDelta-1/pr3/pr3.ma".
21 theorem nf2_pr3_unfold:
22 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c
23 t1) \to (eq T t1 t2)))))
25 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
26 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t
27 t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t
28 (pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
29 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0)
30 \to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def
31 (eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def
32 (eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T
33 t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
35 theorem nf2_pr3_confluence:
36 \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2)
37 \to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
39 \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2:
40 T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t
41 t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0))
42 (\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3:
43 (pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1
44 x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1
45 H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y)
46 in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2
47 (\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0:
48 T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2