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/asucc/defs.ma".
19 include "basic_1/A/fwd.ma".
21 theorem asucc_gen_sort:
22 \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
23 (ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
24 nat).(eq A a (ASort h0 n0)))))))))
26 \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(let
27 TMP_3 \def (\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (let TMP_2
28 \def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_1 \def (ASort h0 n0) in
29 (eq A a0 TMP_1)))) in (ex_2 nat nat TMP_2)))) in (let TMP_13 \def (\lambda
30 (n0: nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort
31 n0 n1)))).(let TMP_4 \def (\lambda (e: A).e) in (let TMP_5 \def (ASort h n)
32 in (let TMP_7 \def (match n0 with [O \Rightarrow (let TMP_6 \def (next g n1)
33 in (ASort O TMP_6)) | (S h0) \Rightarrow (ASort h0 n1)]) in (let H0 \def
34 (f_equal A A TMP_4 TMP_5 TMP_7 H) in (let TMP_10 \def (\lambda (h0:
35 nat).(\lambda (n2: nat).(let TMP_8 \def (ASort n0 n1) in (let TMP_9 \def
36 (ASort h0 n2) in (eq A TMP_8 TMP_9))))) in (let TMP_11 \def (ASort n0 n1) in
37 (let TMP_12 \def (refl_equal A TMP_11) in (ex_2_intro nat nat TMP_10 n0 n1
38 TMP_12))))))))))) in (let TMP_22 \def (\lambda (a0: A).(\lambda (_: (((eq A
39 (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
40 nat).(eq A a0 (ASort h0 n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A
41 (ASort h n) (asucc g a1)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
42 nat).(eq A a1 (ASort h0 n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g
43 (AHead a0 a1)))).(let TMP_14 \def (ASort h n) in (let TMP_15 \def (\lambda
44 (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _)
45 \Rightarrow False])) in (let TMP_16 \def (AHead a0 a1) in (let TMP_17 \def
46 (asucc g TMP_16) in (let H2 \def (eq_ind A TMP_14 TMP_15 I TMP_17 H1) in (let
47 TMP_20 \def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_18 \def (AHead a0
48 a1) in (let TMP_19 \def (ASort h0 n0) in (eq A TMP_18 TMP_19))))) in (let
49 TMP_21 \def (ex_2 nat nat TMP_20) in (False_ind TMP_21 H2))))))))))))) in
50 (A_ind TMP_3 TMP_13 TMP_22 a))))))).
52 theorem asucc_gen_head:
53 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A
54 (AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1
55 a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
57 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(let TMP_5
58 \def (\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (let TMP_2 \def
59 (\lambda (a3: A).(let TMP_1 \def (AHead a1 a3) in (eq A a0 TMP_1))) in (let
60 TMP_4 \def (\lambda (a3: A).(let TMP_3 \def (asucc g a3) in (eq A a2 TMP_3)))
61 in (ex2 A TMP_2 TMP_4))))) in (let TMP_34 \def (\lambda (n: nat).(\lambda
62 (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc g (ASort n n0)))).(let
63 TMP_11 \def (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g (ASort n1 n0)))
64 \to (let TMP_8 \def (\lambda (a0: A).(let TMP_6 \def (ASort n1 n0) in (let
65 TMP_7 \def (AHead a1 a0) in (eq A TMP_6 TMP_7)))) in (let TMP_10 \def
66 (\lambda (a0: A).(let TMP_9 \def (asucc g a0) in (eq A a2 TMP_9))) in (ex2 A
67 TMP_8 TMP_10))))) in (let TMP_22 \def (\lambda (H0: (eq A (AHead a1 a2)
68 (asucc g (ASort O n0)))).(let TMP_12 \def (AHead a1 a2) in (let TMP_13 \def
69 (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
70 \Rightarrow True])) in (let TMP_14 \def (next g n0) in (let TMP_15 \def
71 (ASort O TMP_14) in (let H1 \def (eq_ind A TMP_12 TMP_13 I TMP_15 H0) in (let
72 TMP_18 \def (\lambda (a0: A).(let TMP_16 \def (ASort O n0) in (let TMP_17
73 \def (AHead a1 a0) in (eq A TMP_16 TMP_17)))) in (let TMP_20 \def (\lambda
74 (a0: A).(let TMP_19 \def (asucc g a0) in (eq A a2 TMP_19))) in (let TMP_21
75 \def (ex2 A TMP_18 TMP_20) in (False_ind TMP_21 H1)))))))))) in (let TMP_33
76 \def (\lambda (n1: nat).(\lambda (_: (((eq A (AHead a1 a2) (asucc g (ASort n1
77 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 a0)))
78 (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H0: (eq A (AHead a1
79 a2) (asucc g (ASort (S n1) n0)))).(let TMP_23 \def (AHead a1 a2) in (let
80 TMP_24 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False |
81 (AHead _ _) \Rightarrow True])) in (let TMP_25 \def (ASort n1 n0) in (let H1
82 \def (eq_ind A TMP_23 TMP_24 I TMP_25 H0) in (let TMP_29 \def (\lambda (a0:
83 A).(let TMP_26 \def (S n1) in (let TMP_27 \def (ASort TMP_26 n0) in (let
84 TMP_28 \def (AHead a1 a0) in (eq A TMP_27 TMP_28))))) in (let TMP_31 \def
85 (\lambda (a0: A).(let TMP_30 \def (asucc g a0) in (eq A a2 TMP_30))) in (let
86 TMP_32 \def (ex2 A TMP_29 TMP_31) in (False_ind TMP_32 H1))))))))))) in
87 (nat_ind TMP_11 TMP_22 TMP_33 n H))))))) in (let TMP_86 \def (\lambda (a0:
88 A).(\lambda (H: (((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
89 A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g
90 a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A (AHead a1 a2) (asucc g a3))
91 \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 a4))) (\lambda (a4: A).(eq A
92 a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead a1 a2) (asucc g (AHead a0
93 a3)))).(let TMP_35 \def (\lambda (e: A).(match e with [(ASort _ _)
94 \Rightarrow a1 | (AHead a4 _) \Rightarrow a4])) in (let TMP_36 \def (AHead a1
95 a2) in (let TMP_37 \def (asucc g a3) in (let TMP_38 \def (AHead a0 TMP_37) in
96 (let H2 \def (f_equal A A TMP_35 TMP_36 TMP_38 H1) in (let TMP_39 \def
97 (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a2 | (AHead _ a4)
98 \Rightarrow a4])) in (let TMP_40 \def (AHead a1 a2) in (let TMP_41 \def
99 (asucc g a3) in (let TMP_42 \def (AHead a0 TMP_41) in (let H3 \def (f_equal A
100 A TMP_39 TMP_40 TMP_42 H1) in (let TMP_85 \def (\lambda (H4: (eq A a1
101 a0)).(let TMP_47 \def (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) \to
102 (let TMP_44 \def (\lambda (a5: A).(let TMP_43 \def (AHead a1 a5) in (eq A a4
103 TMP_43))) in (let TMP_46 \def (\lambda (a5: A).(let TMP_45 \def (asucc g a5)
104 in (eq A a2 TMP_45))) in (ex2 A TMP_44 TMP_46))))) in (let H5 \def (eq_ind_r
105 A a0 TMP_47 H a1 H4) in (let TMP_53 \def (\lambda (a4: A).(let TMP_50 \def
106 (\lambda (a5: A).(let TMP_48 \def (AHead a4 a3) in (let TMP_49 \def (AHead a1
107 a5) in (eq A TMP_48 TMP_49)))) in (let TMP_52 \def (\lambda (a5: A).(let
108 TMP_51 \def (asucc g a5) in (eq A a2 TMP_51))) in (ex2 A TMP_50 TMP_52)))) in
109 (let TMP_58 \def (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a3)) \to (let
110 TMP_55 \def (\lambda (a5: A).(let TMP_54 \def (AHead a1 a5) in (eq A a3
111 TMP_54))) in (let TMP_57 \def (\lambda (a5: A).(let TMP_56 \def (asucc g a5)
112 in (eq A a4 TMP_56))) in (ex2 A TMP_55 TMP_57))))) in (let TMP_59 \def (asucc
113 g a3) in (let H6 \def (eq_ind A a2 TMP_58 H0 TMP_59 H3) in (let TMP_64 \def
114 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to (let TMP_61 \def
115 (\lambda (a5: A).(let TMP_60 \def (AHead a1 a5) in (eq A a1 TMP_60))) in (let
116 TMP_63 \def (\lambda (a5: A).(let TMP_62 \def (asucc g a5) in (eq A a4
117 TMP_62))) in (ex2 A TMP_61 TMP_63))))) in (let TMP_65 \def (asucc g a3) in
118 (let H7 \def (eq_ind A a2 TMP_64 H5 TMP_65 H3) in (let TMP_66 \def (asucc g
119 a3) in (let TMP_72 \def (\lambda (a4: A).(let TMP_69 \def (\lambda (a5:
120 A).(let TMP_67 \def (AHead a1 a3) in (let TMP_68 \def (AHead a1 a5) in (eq A
121 TMP_67 TMP_68)))) in (let TMP_71 \def (\lambda (a5: A).(let TMP_70 \def
122 (asucc g a5) in (eq A a4 TMP_70))) in (ex2 A TMP_69 TMP_71)))) in (let TMP_75
123 \def (\lambda (a4: A).(let TMP_73 \def (AHead a1 a3) in (let TMP_74 \def
124 (AHead a1 a4) in (eq A TMP_73 TMP_74)))) in (let TMP_78 \def (\lambda (a4:
125 A).(let TMP_76 \def (asucc g a3) in (let TMP_77 \def (asucc g a4) in (eq A
126 TMP_76 TMP_77)))) in (let TMP_79 \def (AHead a1 a3) in (let TMP_80 \def
127 (refl_equal A TMP_79) in (let TMP_81 \def (asucc g a3) in (let TMP_82 \def
128 (refl_equal A TMP_81) in (let TMP_83 \def (ex_intro2 A TMP_75 TMP_78 a3
129 TMP_80 TMP_82) in (let TMP_84 \def (eq_ind_r A TMP_66 TMP_72 TMP_83 a2 H3) in
130 (eq_ind A a1 TMP_53 TMP_84 a0 H4))))))))))))))))))))) in (TMP_85
131 H2))))))))))))))))) in (A_ind TMP_5 TMP_34 TMP_86 a))))))).