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 "basic_1/nf2/defs.ma".
19 include "basic_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)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t
27 t0)))) in (let TMP_4 \def (\lambda (t: T).(\lambda (H0: (nf2 c t)).(let TMP_2
28 \def (pr0_refl t) in (let TMP_3 \def (pr2_free c t t TMP_2) in (H0 t
29 TMP_3))))) in (let TMP_12 \def (\lambda (t0: T).(\lambda (t3: T).(\lambda
30 (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda
31 (H2: (((nf2 c t0) \to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def
32 H3 in (let TMP_5 \def (\lambda (t: T).(nf2 c t)) in (let TMP_6 \def (H4 t0
33 H0) in (let H5 \def (eq_ind T t3 TMP_5 H3 t0 TMP_6) in (let TMP_7 \def
34 (\lambda (t: T).(pr2 c t t0)) in (let TMP_8 \def (H4 t0 H0) in (let H6 \def
35 (eq_ind T t3 TMP_7 H0 t0 TMP_8) in (let TMP_9 \def (\lambda (t: T).(eq T t
36 t4)) in (let TMP_10 \def (H2 H5) in (let TMP_11 \def (H4 t0 H0) in (eq_ind_r
37 T t0 TMP_9 TMP_10 t3 TMP_11)))))))))))))))))) in (pr3_ind c TMP_1 TMP_4
38 TMP_12 t1 t2 H))))))).
40 theorem nf2_pr3_confluence:
41 \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2)
42 \to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
44 \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2:
45 T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t
46 t1)).(\lambda (H2: (pr3 c t t2)).(let TMP_1 \def (\lambda (t0: T).(pr3 c t2
47 t0)) in (let TMP_2 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let TMP_3 \def
48 (eq T t1 t2) in (let TMP_9 \def (\lambda (x: T).(\lambda (H3: (pr3 c t2
49 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 x H4 H) in
50 (let TMP_4 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let H5 \def (eq_ind_r T x
51 TMP_4 H4 t1 H_y) in (let TMP_5 \def (\lambda (t0: T).(pr3 c t2 t0)) in (let
52 H6 \def (eq_ind_r T x TMP_5 H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c t2
53 t1 H6 H0) in (let TMP_6 \def (\lambda (t0: T).(pr3 c t0 t1)) in (let H7 \def
54 (eq_ind T t2 TMP_6 H6 t1 H_y0) in (let TMP_7 \def (\lambda (t0: T).(eq T t1
55 t0)) in (let TMP_8 \def (refl_equal T t1) in (eq_ind_r T t1 TMP_7 TMP_8 t2
56 H_y0)))))))))))))) in (let TMP_10 \def (pr3_confluence c t t2 H2 t1 H1) in
57 (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_9 TMP_10))))))))))))).