X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FBase-1%2Fspare.ma;h=e19f961cc07e606a47e0c7f9f13070d611c4e820;hb=1776f357e1a69fa1133956660b65d7bafdfe5c25;hp=f66934f78e96c7a2b7175d0cb7d44c397f7f1609;hpb=5bdc90c40499df8b7aa022bbb061c685ac88fb26;p=helm.git diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/spare.ma b/matita/contribs/LAMBDA-TYPES/Base-1/spare.ma index f66934f78..e19f961cc 100644 --- a/matita/contribs/LAMBDA-TYPES/Base-1/spare.ma +++ b/matita/contribs/LAMBDA-TYPES/Base-1/spare.ma @@ -17,4 +17,158 @@ 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)))). + +*)