X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Faplus%2Fprops.ma;h=7edaaa65663e475008093ee6f226e1fd6eeebc2a;hb=02bd27d53c28099b9fc92917cf34ccf3bc72d696;hp=7907d8b967ad4b8dcf5d56f6687482b7c0766998;hpb=bfb39a9bcb10b87ab7d6e09928fb82d340d8feca;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
index 7907d8b96..7edaaa656 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
@@ -29,12 +29,12 @@ theorem aplus_reg_r:
 (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: 
 nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 
 (plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n 
-h1)) (aplus g a2 (plus n h2)))).(sym_equal A (asucc g (aplus g a2 (plus n 
-h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1 
-(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus 
-g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g 
-g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq 
-A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+h1)) (aplus g a2 (plus n h2)))).(sym_eq A (asucc g (aplus g a2 (plus n h2))) 
+(asucc g (aplus g a1 (plus n h1))) (sym_eq A (asucc g (aplus g a1 (plus n 
+h1))) (asucc g (aplus g a2 (plus n h2))) (sym_eq A (asucc g (aplus g a2 (plus 
+n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g g (aplus g 
+a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq A (aplus g 
+a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
 
 theorem aplus_assoc:
  \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
@@ -51,10 +51,10 @@ g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
 n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) 
 n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda 
 (n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g 
-(aplus g a n1)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) 
-(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g 
+(aplus g a n1)))) (sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) 
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_eq A (asucc g (aplus g 
 (asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0)))) 
-(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g 
+(sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g 
 (asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a 
 (plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A 
 (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))) 
@@ -178,53 +178,56 @@ h) a) \to (\forall (P: Prop).P))))
  \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: 
 nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) 
 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A 
-(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in 
-nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow 
-(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) 
-\to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g 
-n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0)) 
-h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g 
-(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2 
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
-with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus 
-(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S 
-i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0) 
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) 
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
+(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O 
+(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) 
+(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 
+\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 
+(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) 
+(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat 
+(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) 
+\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: 
+nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) 
+\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) 
 (minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) 
-(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n: 
-nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in 
+(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: 
+nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in 
 (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g 
 n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n 
-(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1) 
-\Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) 
-n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A 
-a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) 
+(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda 
+(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort 
+O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to 
+P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let 
+H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort 
+(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) 
 (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n 
-| (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def 
-(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m 
-with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1 
-h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) 
-H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def 
-(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n 
-i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g 
-n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) 
-(S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) 
-(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda 
-(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: 
-Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus 
-g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: 
-nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 
-a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g 
-a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc 
-g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A 
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with 
-[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g 
-(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g 
-a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow 
+n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) 
+\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow 
+(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in 
+minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S 
+n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A 
+return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) 
+\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def 
+(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 
+n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus 
+g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 
+h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 
+n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: 
+A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to 
+(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: 
+nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: 
+Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc 
+g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g 
+(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 
+(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) 
+in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
+(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: 
+nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc 
+g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow 
+a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 
+P)))))))))) a)).
 
 theorem aplus_inj:
  \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A