update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / aprem / fwd.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
index ed48846a8a56b55fd4a170dcb286c132ccc7447c..3f415a4c1793019c81ecc2b64711547539198ff8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
@@ -14,9 +14,17 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/aprem/defs.ma".
+include "basic_1/aprem/defs.ma".
 
-theorem aprem_gen_sort:
+implied rec lemma aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall 
+(a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: 
+A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to 
+(\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) 
+(a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) 
+\Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4 
+((aprem_ind P f f0) i a2 a3 a4) a5)].
+
+lemma aprem_gen_sort:
  \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem 
 i (ASort h n) x) \to False))))
 \def
@@ -26,20 +34,16 @@ nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
 (aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: 
 A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: 
 A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A 
-(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
-(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: 
-A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 
-(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) 
-(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee 
-in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | 
-(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False 
+(AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False 
+| (AHead _ _) \Rightarrow True])) I (ASort h n) H1) in (False_ind False 
+H2))))) (\lambda (a2: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (_: 
+(aprem i0 a2 a)).(\lambda (_: (((eq A a2 (ASort h n)) \to False))).(\lambda 
+(a1: A).(\lambda (H3: (eq A (AHead a1 a2) (ASort h n))).(let H4 \def (eq_ind 
+A (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow 
+False | (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False 
 H4))))))))) i y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 227
-END *)
 
-theorem aprem_gen_head_O:
+lemma aprem_gen_head_O:
  \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) 
 \to (eq A x a1))))
 \def
@@ -51,31 +55,26 @@ A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
 y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq 
 nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: 
 A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 
-a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) 
-\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A 
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) 
-in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: 
-A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i 
-O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda 
-(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let 
-H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) 
-with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 
-a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ 
-a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A 
-a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A 
-a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 
-(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) 
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a 
-a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
-(* COMMENTS
-Initial nodes: 500
-END *)
+a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) 
+(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) 
+(AHead a1 a2) H3) in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: 
+A).(\lambda (a: A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda 
+(H3: (((eq nat i O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda 
+(a3: A).(\lambda (H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) 
+(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0) 
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) 
+(AHead a1 a2) H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 
+(\lambda (a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a 
+a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i a4 
+a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
+(eq A a a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
 
-theorem aprem_gen_head_S:
+lemma aprem_gen_head_S:
  \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem 
 (S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
 \def
@@ -88,33 +87,27 @@ x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
 nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead 
 a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda 
 (H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let 
-H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) 
-with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) 
-(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) 
-\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 
-a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat 
-O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind 
-(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: 
-A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat 
-i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: 
-A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) 
-(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) 
+H4 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow 
+a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 
+\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | 
+(AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: 
+(eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def 
+(eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) 
+\Rightarrow False])) I (S i) H2) in (False_ind (aprem i a2 a1) H7)) a0 H6))) 
+H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (H2: 
+(aprem i0 a0 a)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq A a0 (AHead a1 
+a2)) \to (aprem i a2 a))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S i0) (S 
+i))).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def (f_equal 
+A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) 
 \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A 
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) 
-H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: 
-A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 
-a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 
-H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat 
-return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) 
-(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n 
-(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let 
-H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in 
+A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead _ a4) 
+\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3 
+a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A 
+a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 a2 H7) in (let H10 \def (eq_ind A 
+a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 H7) in (let H11 \def (f_equal nat 
+nat (\lambda (e: nat).(match e with [O \Rightarrow i0 | (S n) \Rightarrow 
+n])) (S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq 
+nat n (S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in 
+(let H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in 
 H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
-(* COMMENTS
-Initial nodes: 631
-END *)