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/pr3/props.ma".
19 include "basic_1/pr2/pr2.ma".
22 \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
23 (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
24 T).(pr3 c t2 t))))))))
26 \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
27 t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2
28 c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2
29 \def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let
30 TMP_8 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let
31 TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3:
32 T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_pr2 c t t2 H0) in (let TMP_7 \def
33 (pr3_refl c t2) in (ex_intro2 T TMP_4 TMP_5 t2 TMP_6 TMP_7)))))))) in (let
34 TMP_26 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
35 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H2: ((\forall
36 (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t:
37 T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr2 c t3 t5)).(let
38 TMP_9 \def (\lambda (t: T).(pr2 c t5 t)) in (let TMP_10 \def (\lambda (t:
39 T).(pr2 c t2 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t4 t)) in (let
40 TMP_12 \def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_13 \def (ex2 T TMP_11
41 TMP_12) in (let TMP_24 \def (\lambda (x: T).(\lambda (H4: (pr2 c t5
42 x)).(\lambda (H5: (pr2 c t2 x)).(let TMP_14 \def (\lambda (t: T).(pr3 c t4
43 t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c x t)) in (let TMP_16 \def
44 (\lambda (t: T).(pr3 c t4 t)) in (let TMP_17 \def (\lambda (t: T).(pr3 c t5
45 t)) in (let TMP_18 \def (ex2 T TMP_16 TMP_17) in (let TMP_22 \def (\lambda
46 (x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(let TMP_19
47 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_20 \def (\lambda (t: T).(pr3 c
48 t5 t)) in (let TMP_21 \def (pr3_sing c x t5 H4 x0 H7) in (ex_intro2 T TMP_19
49 TMP_20 x0 H6 TMP_21))))))) in (let TMP_23 \def (H2 x H5) in (ex2_ind T TMP_14
50 TMP_15 TMP_18 TMP_22 TMP_23))))))))))) in (let TMP_25 \def (pr2_confluence c
51 t3 t5 H3 t2 H0) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_24
52 TMP_25)))))))))))))))) in (pr3_ind c TMP_3 TMP_8 TMP_26 t0 t1 H))))))).
54 theorem pr3_confluence:
55 \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
56 (t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
57 T).(pr3 c t2 t))))))))
59 \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
60 t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3
61 c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2
62 \def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let
63 TMP_7 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(let
64 TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3:
65 T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_refl c t2) in (ex_intro2 T TMP_4
66 TMP_5 t2 H0 TMP_6))))))) in (let TMP_25 \def (\lambda (t2: T).(\lambda (t3:
67 T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2
68 t4)).(\lambda (H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t:
69 T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda
70 (H3: (pr3 c t3 t5)).(let TMP_8 \def (\lambda (t: T).(pr3 c t5 t)) in (let
71 TMP_9 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_10 \def (\lambda (t:
72 T).(pr3 c t4 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t5 t)) in (let
73 TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda (x:
74 T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 x)).(let TMP_13 \def
75 (\lambda (t: T).(pr3 c t4 t)) in (let TMP_14 \def (\lambda (t: T).(pr3 c x
76 t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_16 \def
77 (\lambda (t: T).(pr3 c t5 t)) in (let TMP_17 \def (ex2 T TMP_15 TMP_16) in
78 (let TMP_21 \def (\lambda (x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7:
79 (pr3 c x x0)).(let TMP_18 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_19
80 \def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_20 \def (pr3_t x t5 c H4 x0
81 H7) in (ex_intro2 T TMP_18 TMP_19 x0 H6 TMP_20))))))) in (let TMP_22 \def (H2
82 x H5) in (ex2_ind T TMP_13 TMP_14 TMP_17 TMP_21 TMP_22))))))))))) in (let
83 TMP_24 \def (pr3_strip c t3 t5 H3 t2 H0) in (ex2_ind T TMP_8 TMP_9 TMP_12
84 TMP_23 TMP_24)))))))))))))))) in (pr3_ind c TMP_3 TMP_7 TMP_25 t0 t1 H))))))).