X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2Fproblems-2.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2Fproblems-2.ma;h=a0f6f0275f0e2707293b302d08226c8b97204552;hb=f4a4c8ceb91e62b17de591967f8e6f53cceb0b63;hp=0000000000000000000000000000000000000000;hpb=f97af72aec6f23053fff6f1a1f6b0a923dc584c9;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/problems-2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/problems-2.ma
new file mode 100644
index 000000000..a0f6f0275
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/problems-2.ma
@@ -0,0 +1,158 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl 
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2 
+(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: 
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to 
+(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i 
+v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H 
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to 
+(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with 
+[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: 
+(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C 
+c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) 
+(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2: 
+C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i 
+c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 
+H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: 
+(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5: 
+(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 
+hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))) 
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) 
+(refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 
+c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: 
+nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl 
+(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0 
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq 
+C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true 
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false 
+\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow 
+(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c 
+c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2) 
+in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl 
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false 
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt 
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) 
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 
+H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq 
+PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0 
+c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda 
+(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil 
+\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h 
+d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e 
+in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _ 
+n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9 
+\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda 
+(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) 
+(PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n: 
+nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 
+c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: 
+C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) 
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match 
+(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 
+i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))))) 
+(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0 
+hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 
+hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) 
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 
+i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true 
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false 
+\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0 
+hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1) 
+\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl 
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false 
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt 
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) 
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda 
+(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d 
+c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt 
+(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow 
+(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) 
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4 
+c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c) 
+\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true 
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
+(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift 
+h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans 
+hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1 
+hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex 
+C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | 
+false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
+v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x: 
+bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to 
+(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) 
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt 
+(trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 
+\def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 
+(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 
+(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
+v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind 
+b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d 
+(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d) 
+(S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H 
+\def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1 
+(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda 
+(e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2: 
+C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 
+i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0 
+i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
+v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C 
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d 
+(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda 
+(H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1 
+v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 
+i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl 
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda 
+(x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 
+i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x 
+(Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus 
+(trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d 
+(trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq 
+C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10))) 
+h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) 
+hds).
+