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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 set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/spare".
21 inductive pr: Set \def
25 | pr_comp: ((nat \to pr)) \to (pr \to pr)
26 | pr_prec: pr \to (pr \to pr).
31 ((nat \to nat)) \to nat.
34 pr_type \to (pr_type \to (nat \to pr_type))
36 let rec prec_appl (f: pr_type) (g: pr_type) (n: nat) on n: pr_type \def
37 (match n with [O \Rightarrow f | (S m) \Rightarrow (\lambda (ns: ((nat \to
38 nat))).(g (\lambda (i: nat).(match i with [O \Rightarrow (prec_appl f g m ns)
39 | (S n0) \Rightarrow (match n0 with [O \Rightarrow m | (S j) \Rightarrow (ns
40 j)])]))))]) in prec_appl.
45 let rec pr_appl (h: pr) on h: pr_type \def (match h with [pr_zero
46 \Rightarrow (\lambda (_: ((nat \to nat))).O) | pr_succ \Rightarrow (\lambda
47 (ns: ((nat \to nat))).(S (ns O))) | (pr_proj i) \Rightarrow (\lambda (ns:
48 ((nat \to nat))).(ns i)) | (pr_comp fs g) \Rightarrow (\lambda (ns: ((nat \to
49 nat))).(pr_appl g (\lambda (i: nat).(pr_appl (fs i) ns)))) | (pr_prec f g)
50 \Rightarrow (\lambda (ns: ((nat \to nat))).(prec_appl (pr_appl f) (pr_appl g)
51 (ns O) (\lambda (i: nat).(ns (S i)))))]) in pr_appl.
53 inductive pr_arity: pr \to (nat \to Prop) \def
54 | pr_arity_zero: \forall (n: nat).(pr_arity pr_zero n)
55 | pr_arity_succ: \forall (n: nat).((lt O n) \to (pr_arity pr_succ n))
56 | pr_arity_proj: \forall (n: nat).(\forall (i: nat).((lt i n) \to (pr_arity
58 | pr_arity_comp: \forall (n: nat).(\forall (m: nat).(\forall (fs: ((nat \to
59 pr))).(\forall (g: pr).((pr_arity g m) \to (((\forall (i: nat).((lt i m) \to
60 (pr_arity (fs i) n)))) \to (pr_arity (pr_comp fs g) n))))))
61 | pr_arity_prec: \forall (n: nat).(\forall (f: pr).(\forall (g: pr).((lt O n)
62 \to ((pr_arity f (pred n)) \to ((pr_arity g (S n)) \to (pr_arity (pr_prec f
66 \forall (h: pr).(\forall (m: nat).((pr_arity h m) \to (\forall (n: nat).((le
67 m n) \to (pr_arity h n)))))
69 \lambda (h: pr).(\lambda (m: nat).(\lambda (H: (pr_arity h m)).(pr_arity_ind
70 (\lambda (p: pr).(\lambda (n: nat).(\forall (n0: nat).((le n n0) \to
71 (pr_arity p n0))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: (le n
72 n0)).(pr_arity_zero n0)))) (\lambda (n: nat).(\lambda (H0: (lt O n)).(\lambda
73 (n0: nat).(\lambda (H1: (le n n0)).(pr_arity_succ n0 (lt_le_trans O n n0 H0
74 H1)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (H0: (lt i n)).(\lambda
75 (n0: nat).(\lambda (H1: (le n n0)).(pr_arity_proj n0 i (lt_le_trans i n n0 H0
76 H1))))))) (\lambda (n: nat).(\lambda (m0: nat).(\lambda (fs: ((nat \to
77 pr))).(\lambda (g: pr).(\lambda (H0: (pr_arity g m0)).(\lambda (_: ((\forall
78 (n0: nat).((le m0 n0) \to (pr_arity g n0))))).(\lambda (_: ((\forall (i:
79 nat).((lt i m0) \to (pr_arity (fs i) n))))).(\lambda (H3: ((\forall (i:
80 nat).((lt i m0) \to (\forall (n0: nat).((le n n0) \to (pr_arity (fs i)
81 n0))))))).(\lambda (n0: nat).(\lambda (H4: (le n n0)).(pr_arity_comp n0 m0 fs
82 g H0 (\lambda (i: nat).(\lambda (H5: (lt i m0)).(H3 i H5 n0 H4))))))))))))))
83 (\lambda (n: nat).(\lambda (f: pr).(\lambda (g: pr).(\lambda (H0: (lt O
84 n)).(\lambda (_: (pr_arity f (pred n))).(\lambda (H2: ((\forall (n0:
85 nat).((le (pred n) n0) \to (pr_arity f n0))))).(\lambda (_: (pr_arity g (S
86 n))).(\lambda (H4: ((\forall (n0: nat).((le (S n) n0) \to (pr_arity g
87 n0))))).(\lambda (n0: nat).(\lambda (H5: (le n n0)).(pr_arity_prec n0 f g
88 (lt_le_trans O n n0 H0 H5) (H2 (pred n0) (le_n_pred n n0 H5)) (H4 (S n0)
89 (le_n_S n n0 H5))))))))))))) h m H))).
91 theorem pr_arity_appl:
92 \forall (h: pr).(\forall (n: nat).((pr_arity h n) \to (\forall (ns: ((nat
93 \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt i n) \to
94 (eq nat (ns i) (ms i))))) \to (eq nat (pr_appl h ns) (pr_appl h ms)))))))
96 \lambda (h: pr).(\lambda (n: nat).(\lambda (H: (pr_arity h n)).(pr_arity_ind
97 (\lambda (p: pr).(\lambda (n0: nat).(\forall (ns: ((nat \to nat))).(\forall
98 (ms: ((nat \to nat))).(((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
99 i))))) \to (eq nat (pr_appl p ns) (pr_appl p ms))))))) (\lambda (n0:
100 nat).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
101 (_: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(refl_equal
102 nat O))))) (\lambda (n0: nat).(\lambda (H0: (lt O n0)).(\lambda (ns: ((nat
103 \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H1: ((\forall (i:
104 nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(f_equal nat nat S (ns O) (ms
105 O) (H1 O H0))))))) (\lambda (n0: nat).(\lambda (i: nat).(\lambda (H0: (lt i
106 n0)).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
107 (H1: ((\forall (i0: nat).((lt i0 n0) \to (eq nat (ns i0) (ms i0)))))).(H1 i
108 H0))))))) (\lambda (n0: nat).(\lambda (m: nat).(\lambda (fs: ((nat \to
109 pr))).(\lambda (g: pr).(\lambda (_: (pr_arity g m)).(\lambda (H1: ((\forall
110 (ns: ((nat \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt
111 i m) \to (eq nat (ns i) (ms i))))) \to (eq nat (pr_appl g ns) (pr_appl g
112 ms))))))).(\lambda (_: ((\forall (i: nat).((lt i m) \to (pr_arity (fs i)
113 n0))))).(\lambda (H3: ((\forall (i: nat).((lt i m) \to (\forall (ns: ((nat
114 \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i0: nat).((lt i0 n0)
115 \to (eq nat (ns i0) (ms i0))))) \to (eq nat (pr_appl (fs i) ns) (pr_appl (fs
116 i) ms))))))))).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to
117 nat))).(\lambda (H4: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
118 i)))))).(H1 (\lambda (i: nat).(pr_appl (fs i) ns)) (\lambda (i: nat).(pr_appl
119 (fs i) ms)) (\lambda (i: nat).(\lambda (H5: (lt i m)).(H3 i H5 ns ms
120 H4))))))))))))))) (\lambda (n0: nat).(\lambda (f: pr).(\lambda (g:
121 pr).(\lambda (H0: (lt O n0)).(\lambda (_: (pr_arity f (pred n0))).(\lambda
122 (H2: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
123 nat))).(((\forall (i: nat).((lt i (pred n0)) \to (eq nat (ns i) (ms i)))))
124 \to (eq nat (pr_appl f ns) (pr_appl f ms))))))).(\lambda (_: (pr_arity g (S
125 n0))).(\lambda (H4: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
126 nat))).(((\forall (i: nat).((lt i (S n0)) \to (eq nat (ns i) (ms i))))) \to
127 (eq nat (pr_appl g ns) (pr_appl g ms))))))).(\lambda (ns: ((nat \to
128 nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H5: ((\forall (i: nat).((lt i
129 n0) \to (eq nat (ns i) (ms i)))))).(eq_ind nat (ns O) (\lambda (n1: nat).(eq
130 nat (prec_appl (pr_appl f) (pr_appl g) (ns O) (\lambda (i: nat).(ns (S i))))
131 (prec_appl (pr_appl f) (pr_appl g) n1 (\lambda (i: nat).(ms (S i)))))) (let
132 n1 \def (ns O) in (nat_ind (\lambda (n2: nat).(eq nat (prec_appl (pr_appl f)
133 (pr_appl g) n2 (\lambda (i: nat).(ns (S i)))) (prec_appl (pr_appl f) (pr_appl
134 g) n2 (\lambda (i: nat).(ms (S i)))))) (H2 (\lambda (i: nat).(ns (S i)))
135 (\lambda (i: nat).(ms (S i))) (\lambda (i: nat).(\lambda (H6: (lt i (pred
136 n0))).(H5 (S i) (lt_x_pred_y i n0 H6))))) (\lambda (n2: nat).(\lambda (IHn0:
137 (eq nat (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ns (S i))))
138 (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ms (S i)))))).(H4
139 (\lambda (i: nat).(match i with [O \Rightarrow (prec_appl (pr_appl f)
140 (pr_appl g) n2 (\lambda (i0: nat).(ns (S i0)))) | (S n3) \Rightarrow (match
141 n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))])])) (\lambda (i:
142 nat).(match i with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
143 (\lambda (i0: nat).(ms (S i0)))) | (S n3) \Rightarrow (match n3 with [O
144 \Rightarrow n2 | (S j) \Rightarrow (ms (S j))])])) (\lambda (i: nat).(\lambda
145 (H6: (lt i (S n0))).(nat_ind (\lambda (n3: nat).((lt n3 (S n0)) \to (eq nat
146 (match n3 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda
147 (i0: nat).(ns (S i0)))) | (S n4) \Rightarrow (match n4 with [O \Rightarrow n2
148 | (S j) \Rightarrow (ns (S j))])]) (match n3 with [O \Rightarrow (prec_appl
149 (pr_appl f) (pr_appl g) n2 (\lambda (i0: nat).(ms (S i0)))) | (S n4)
150 \Rightarrow (match n4 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S
151 j))])])))) (\lambda (_: (lt O (S n0))).IHn0) (\lambda (i0: nat).(\lambda (_:
152 (((lt i0 (S n0)) \to (eq nat (match i0 with [O \Rightarrow (prec_appl
153 (pr_appl f) (pr_appl g) n2 (\lambda (i1: nat).(ns (S i1)))) | (S n3)
154 \Rightarrow (match n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S
155 j))])]) (match i0 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
156 (\lambda (i1: nat).(ms (S i1)))) | (S n3) \Rightarrow (match n3 with [O
157 \Rightarrow n2 | (S j) \Rightarrow (ms (S j))])]))))).(\lambda (H7: (lt (S
158 i0) (S n0))).(let H_y \def (H5 i0 (lt_S_n i0 n0 H7)) in (nat_ind (\lambda
159 (n3: nat).((eq nat (ns n3) (ms n3)) \to (eq nat (match n3 with [O \Rightarrow
160 n2 | (S j) \Rightarrow (ns (S j))]) (match n3 with [O \Rightarrow n2 | (S j)
161 \Rightarrow (ms (S j))])))) (\lambda (_: (eq nat (ns O) (ms O))).(refl_equal
162 nat n2)) (\lambda (i1: nat).(\lambda (_: (((eq nat (ns i1) (ms i1)) \to (eq
163 nat (match i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))]) (match
164 i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S j))]))))).(\lambda (H8:
165 (eq nat (ns (S i1)) (ms (S i1)))).H8))) i0 H_y))))) i H6)))))) n1)) (ms O)
166 (H5 O H0))))))))))))) h n H))).
168 theorem pr_arity_comp_proj_zero:
169 \forall (n: nat).(pr_arity (pr_comp pr_proj pr_zero) n)
171 \lambda (n: nat).(pr_arity_comp n n pr_proj pr_zero (pr_arity_zero n)
172 (\lambda (i: nat).(\lambda (H: (lt i n)).(pr_arity_proj n i H)))).