set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/spare".
include "theory.ma".
+(*
+inductive pr: Set \def
+| pr_zero: pr
+| pr_succ: pr
+| pr_proj: nat \to pr
+| pr_comp: ((nat \to pr)) \to (pr \to pr)
+| pr_prec: pr \to (pr \to pr).
+definition pr_type:
+ Set
+\def
+ ((nat \to nat)) \to nat.
+
+definition prec_appl:
+ pr_type \to (pr_type \to (nat \to pr_type))
+\def
+ let rec prec_appl (f: pr_type) (g: pr_type) (n: nat) on n: pr_type \def
+(match n with [O \Rightarrow f | (S m) \Rightarrow (\lambda (ns: ((nat \to
+nat))).(g (\lambda (i: nat).(match i with [O \Rightarrow (prec_appl f g m ns)
+| (S n0) \Rightarrow (match n0 with [O \Rightarrow m | (S j) \Rightarrow (ns
+j)])]))))]) in prec_appl.
+
+definition pr_appl:
+ pr \to pr_type
+\def
+ let rec pr_appl (h: pr) on h: pr_type \def (match h with [pr_zero
+\Rightarrow (\lambda (_: ((nat \to nat))).O) | pr_succ \Rightarrow (\lambda
+(ns: ((nat \to nat))).(S (ns O))) | (pr_proj i) \Rightarrow (\lambda (ns:
+((nat \to nat))).(ns i)) | (pr_comp fs g) \Rightarrow (\lambda (ns: ((nat \to
+nat))).(pr_appl g (\lambda (i: nat).(pr_appl (fs i) ns)))) | (pr_prec f g)
+\Rightarrow (\lambda (ns: ((nat \to nat))).(prec_appl (pr_appl f) (pr_appl g)
+(ns O) (\lambda (i: nat).(ns (S i)))))]) in pr_appl.
+
+inductive pr_arity: pr \to (nat \to Prop) \def
+| pr_arity_zero: \forall (n: nat).(pr_arity pr_zero n)
+| pr_arity_succ: \forall (n: nat).((lt O n) \to (pr_arity pr_succ n))
+| pr_arity_proj: \forall (n: nat).(\forall (i: nat).((lt i n) \to (pr_arity
+(pr_proj i) n)))
+| pr_arity_comp: \forall (n: nat).(\forall (m: nat).(\forall (fs: ((nat \to
+pr))).(\forall (g: pr).((pr_arity g m) \to (((\forall (i: nat).((lt i m) \to
+(pr_arity (fs i) n)))) \to (pr_arity (pr_comp fs g) n))))))
+| pr_arity_prec: \forall (n: nat).(\forall (f: pr).(\forall (g: pr).((lt O n)
+\to ((pr_arity f (pred n)) \to ((pr_arity g (S n)) \to (pr_arity (pr_prec f
+g) n)))))).
+
+theorem pr_arity_le:
+ \forall (h: pr).(\forall (m: nat).((pr_arity h m) \to (\forall (n: nat).((le
+m n) \to (pr_arity h n)))))
+\def
+ \lambda (h: pr).(\lambda (m: nat).(\lambda (H: (pr_arity h m)).(pr_arity_ind
+(\lambda (p: pr).(\lambda (n: nat).(\forall (n0: nat).((le n n0) \to
+(pr_arity p n0))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: (le n
+n0)).(pr_arity_zero n0)))) (\lambda (n: nat).(\lambda (H0: (lt O n)).(\lambda
+(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_succ n0 (lt_le_trans O n n0 H0
+H1)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (H0: (lt i n)).(\lambda
+(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_proj n0 i (lt_le_trans i n n0 H0
+H1))))))) (\lambda (n: nat).(\lambda (m0: nat).(\lambda (fs: ((nat \to
+pr))).(\lambda (g: pr).(\lambda (H0: (pr_arity g m0)).(\lambda (_: ((\forall
+(n0: nat).((le m0 n0) \to (pr_arity g n0))))).(\lambda (_: ((\forall (i:
+nat).((lt i m0) \to (pr_arity (fs i) n))))).(\lambda (H3: ((\forall (i:
+nat).((lt i m0) \to (\forall (n0: nat).((le n n0) \to (pr_arity (fs i)
+n0))))))).(\lambda (n0: nat).(\lambda (H4: (le n n0)).(pr_arity_comp n0 m0 fs
+g H0 (\lambda (i: nat).(\lambda (H5: (lt i m0)).(H3 i H5 n0 H4))))))))))))))
+(\lambda (n: nat).(\lambda (f: pr).(\lambda (g: pr).(\lambda (H0: (lt O
+n)).(\lambda (_: (pr_arity f (pred n))).(\lambda (H2: ((\forall (n0:
+nat).((le (pred n) n0) \to (pr_arity f n0))))).(\lambda (_: (pr_arity g (S
+n))).(\lambda (H4: ((\forall (n0: nat).((le (S n) n0) \to (pr_arity g
+n0))))).(\lambda (n0: nat).(\lambda (H5: (le n n0)).(pr_arity_prec n0 f g
+(lt_le_trans O n n0 H0 H5) (H2 (pred n0) (le_n_pred n n0 H5)) (H4 (S n0)
+(le_n_S n n0 H5))))))))))))) h m H))).
+
+theorem pr_arity_appl:
+ \forall (h: pr).(\forall (n: nat).((pr_arity h n) \to (\forall (ns: ((nat
+\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt i n) \to
+(eq nat (ns i) (ms i))))) \to (eq nat (pr_appl h ns) (pr_appl h ms)))))))
+\def
+ \lambda (h: pr).(\lambda (n: nat).(\lambda (H: (pr_arity h n)).(pr_arity_ind
+(\lambda (p: pr).(\lambda (n0: nat).(\forall (ns: ((nat \to nat))).(\forall
+(ms: ((nat \to nat))).(((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
+i))))) \to (eq nat (pr_appl p ns) (pr_appl p ms))))))) (\lambda (n0:
+nat).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
+(_: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(refl_equal
+nat O))))) (\lambda (n0: nat).(\lambda (H0: (lt O n0)).(\lambda (ns: ((nat
+\to nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H1: ((\forall (i:
+nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(f_equal nat nat S (ns O) (ms
+O) (H1 O H0))))))) (\lambda (n0: nat).(\lambda (i: nat).(\lambda (H0: (lt i
+n0)).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
+(H1: ((\forall (i0: nat).((lt i0 n0) \to (eq nat (ns i0) (ms i0)))))).(H1 i
+H0))))))) (\lambda (n0: nat).(\lambda (m: nat).(\lambda (fs: ((nat \to
+pr))).(\lambda (g: pr).(\lambda (_: (pr_arity g m)).(\lambda (H1: ((\forall
+(ns: ((nat \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt
+i m) \to (eq nat (ns i) (ms i))))) \to (eq nat (pr_appl g ns) (pr_appl g
+ms))))))).(\lambda (_: ((\forall (i: nat).((lt i m) \to (pr_arity (fs i)
+n0))))).(\lambda (H3: ((\forall (i: nat).((lt i m) \to (\forall (ns: ((nat
+\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i0: nat).((lt i0 n0)
+\to (eq nat (ns i0) (ms i0))))) \to (eq nat (pr_appl (fs i) ns) (pr_appl (fs
+i) ms))))))))).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to
+nat))).(\lambda (H4: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
+i)))))).(H1 (\lambda (i: nat).(pr_appl (fs i) ns)) (\lambda (i: nat).(pr_appl
+(fs i) ms)) (\lambda (i: nat).(\lambda (H5: (lt i m)).(H3 i H5 ns ms
+H4))))))))))))))) (\lambda (n0: nat).(\lambda (f: pr).(\lambda (g:
+pr).(\lambda (H0: (lt O n0)).(\lambda (_: (pr_arity f (pred n0))).(\lambda
+(H2: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
+nat))).(((\forall (i: nat).((lt i (pred n0)) \to (eq nat (ns i) (ms i)))))
+\to (eq nat (pr_appl f ns) (pr_appl f ms))))))).(\lambda (_: (pr_arity g (S
+n0))).(\lambda (H4: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
+nat))).(((\forall (i: nat).((lt i (S n0)) \to (eq nat (ns i) (ms i))))) \to
+(eq nat (pr_appl g ns) (pr_appl g ms))))))).(\lambda (ns: ((nat \to
+nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H5: ((\forall (i: nat).((lt i
+n0) \to (eq nat (ns i) (ms i)))))).(eq_ind nat (ns O) (\lambda (n1: nat).(eq
+nat (prec_appl (pr_appl f) (pr_appl g) (ns O) (\lambda (i: nat).(ns (S i))))
+(prec_appl (pr_appl f) (pr_appl g) n1 (\lambda (i: nat).(ms (S i)))))) (let
+n1 \def (ns O) in (nat_ind (\lambda (n2: nat).(eq nat (prec_appl (pr_appl f)
+(pr_appl g) n2 (\lambda (i: nat).(ns (S i)))) (prec_appl (pr_appl f) (pr_appl
+g) n2 (\lambda (i: nat).(ms (S i)))))) (H2 (\lambda (i: nat).(ns (S i)))
+(\lambda (i: nat).(ms (S i))) (\lambda (i: nat).(\lambda (H6: (lt i (pred
+n0))).(H5 (S i) (lt_x_pred_y i n0 H6))))) (\lambda (n2: nat).(\lambda (IHn0:
+(eq nat (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ns (S i))))
+(prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ms (S i)))))).(H4
+(\lambda (i: nat).(match i with [O \Rightarrow (prec_appl (pr_appl f)
+(pr_appl g) n2 (\lambda (i0: nat).(ns (S i0)))) | (S n3) \Rightarrow (match
+n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))])])) (\lambda (i:
+nat).(match i with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
+(\lambda (i0: nat).(ms (S i0)))) | (S n3) \Rightarrow (match n3 with [O
+\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])])) (\lambda (i: nat).(\lambda
+(H6: (lt i (S n0))).(nat_ind (\lambda (n3: nat).((lt n3 (S n0)) \to (eq nat
+(match n3 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda
+(i0: nat).(ns (S i0)))) | (S n4) \Rightarrow (match n4 with [O \Rightarrow n2
+| (S j) \Rightarrow (ns (S j))])]) (match n3 with [O \Rightarrow (prec_appl
+(pr_appl f) (pr_appl g) n2 (\lambda (i0: nat).(ms (S i0)))) | (S n4)
+\Rightarrow (match n4 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S
+j))])])))) (\lambda (_: (lt O (S n0))).IHn0) (\lambda (i0: nat).(\lambda (_:
+(((lt i0 (S n0)) \to (eq nat (match i0 with [O \Rightarrow (prec_appl
+(pr_appl f) (pr_appl g) n2 (\lambda (i1: nat).(ns (S i1)))) | (S n3)
+\Rightarrow (match n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S
+j))])]) (match i0 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
+(\lambda (i1: nat).(ms (S i1)))) | (S n3) \Rightarrow (match n3 with [O
+\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])]))))).(\lambda (H7: (lt (S
+i0) (S n0))).(let H_y \def (H5 i0 (lt_S_n i0 n0 H7)) in (nat_ind (\lambda
+(n3: nat).((eq nat (ns n3) (ms n3)) \to (eq nat (match n3 with [O \Rightarrow
+n2 | (S j) \Rightarrow (ns (S j))]) (match n3 with [O \Rightarrow n2 | (S j)
+\Rightarrow (ms (S j))])))) (\lambda (_: (eq nat (ns O) (ms O))).(refl_equal
+nat n2)) (\lambda (i1: nat).(\lambda (_: (((eq nat (ns i1) (ms i1)) \to (eq
+nat (match i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))]) (match
+i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S j))]))))).(\lambda (H8:
+(eq nat (ns (S i1)) (ms (S i1)))).H8))) i0 H_y))))) i H6)))))) n1)) (ms O)
+(H5 O H0))))))))))))) h n H))).
+
+theorem pr_arity_comp_proj_zero:
+ \forall (n: nat).(pr_arity (pr_comp pr_proj pr_zero) n)
+\def
+ \lambda (n: nat).(pr_arity_comp n n pr_proj pr_zero (pr_arity_zero n)
+(\lambda (i: nat).(\lambda (H: (lt i n)).(pr_arity_proj n i H)))).
+
+*)
projects / helm.git / blobdiff