include "basic_1/drop1/defs.ma".
-let rec drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: (\forall (c:
-C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds:
-PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons h d hds) c1
-c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P p c c0 \def
-match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1 c2 h d0 d1
-c3 hds d2) \Rightarrow (let TMP_1 \def ((drop1_ind P f f0) hds c2 c3 d2) in
-(f0 c1 c2 h d0 d1 c3 hds d2 TMP_1))].
+implied rec lemma drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f:
+(\forall (c: C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3:
+C).(\forall (hds: PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons
+h d hds) c1 c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P
+p c c0 \def match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1
+c2 h d0 d1 c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f
+f0) hds c2 c3 d2))].
-theorem drop1_gen_pnil:
+lemma drop1_gen_pnil:
\forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(let TMP_1
-\def (\lambda (p: PList).(drop1 p c1 c2)) in (let TMP_2 \def (\lambda (_:
-PList).(eq C c1 c2)) in (let TMP_9 \def (\lambda (y: PList).(\lambda (H0:
-(drop1 y c1 c2)).(let TMP_3 \def (\lambda (p: PList).(\lambda (c: C).(\lambda
-(c0: C).((eq PList p PNil) \to (eq C c c0))))) in (let TMP_4 \def (\lambda
-(c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) in (let TMP_8
-\def (\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d:
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq
+PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1
+c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda
+(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c
+c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c)))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d:
nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds:
PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to
-(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let TMP_5
-\def (PCons h d hds) in (let TMP_6 \def (\lambda (ee: PList).(match ee with
-[PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) in (let H5 \def
-(eq_ind PList TMP_5 TMP_6 I PNil H4) in (let TMP_7 \def (eq C c3 c5) in
-(False_ind TMP_7 H5))))))))))))))) in (drop1_ind TMP_3 TMP_4 TMP_8 y c1 c2
-H0)))))) in (insert_eq PList PNil TMP_1 TMP_2 TMP_9 H)))))).
+(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def
+(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee with [PNil
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H4) in
+(False_ind (eq C c3 c5) H5)))))))))))) y c1 c2 H0))) H))).
-theorem drop1_gen_pcons:
+lemma drop1_gen_pcons:
\forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h:
nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda
(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
\def
\lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(let TMP_1
-\def (PCons h d hds) in (let TMP_2 \def (\lambda (p: PList).(drop1 p c1 c3))
-in (let TMP_5 \def (\lambda (_: PList).(let TMP_3 \def (\lambda (c2: C).(drop
-h d c1 c2)) in (let TMP_4 \def (\lambda (c2: C).(drop1 hds c2 c3)) in (ex2 C
-TMP_3 TMP_4)))) in (let TMP_35 \def (\lambda (y: PList).(\lambda (H0: (drop1
-y c1 c3)).(let TMP_8 \def (\lambda (p: PList).(\lambda (c: C).(\lambda (c0:
-C).((eq PList p (PCons h d hds)) \to (let TMP_6 \def (\lambda (c2: C).(drop h
-d c c2)) in (let TMP_7 \def (\lambda (c2: C).(drop1 hds c2 c0)) in (ex2 C
-TMP_6 TMP_7))))))) in (let TMP_14 \def (\lambda (c: C).(\lambda (H1: (eq
-PList PNil (PCons h d hds))).(let TMP_9 \def (\lambda (ee: PList).(match ee
-with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) in (let
-TMP_10 \def (PCons h d hds) in (let H2 \def (eq_ind PList PNil TMP_9 I TMP_10
-H1) in (let TMP_11 \def (\lambda (c2: C).(drop h d c c2)) in (let TMP_12 \def
-(\lambda (c2: C).(drop1 hds c2 c)) in (let TMP_13 \def (ex2 C TMP_11 TMP_12)
-in (False_ind TMP_13 H2))))))))) in (let TMP_34 \def (\lambda (c2:
-C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1: (drop
-h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2: (drop1
-hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2 C
-(\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
-c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let
-TMP_15 \def (\lambda (e: PList).(match e with [PNil \Rightarrow h0 | (PCons n
-_ _) \Rightarrow n])) in (let TMP_16 \def (PCons h0 d0 hds0) in (let TMP_17
-\def (PCons h d hds) in (let H5 \def (f_equal PList nat TMP_15 TMP_16 TMP_17
-H4) in (let TMP_18 \def (\lambda (e: PList).(match e with [PNil \Rightarrow
-d0 | (PCons _ n _) \Rightarrow n])) in (let TMP_19 \def (PCons h0 d0 hds0) in
-(let TMP_20 \def (PCons h d hds) in (let H6 \def (f_equal PList nat TMP_18
-TMP_19 TMP_20 H4) in (let TMP_21 \def (\lambda (e: PList).(match e with [PNil
-\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) in (let TMP_22 \def (PCons
-h0 d0 hds0) in (let TMP_23 \def (PCons h d hds) in (let H7 \def (f_equal
-PList PList TMP_21 TMP_22 TMP_23 H4) in (let TMP_32 \def (\lambda (H8: (eq
-nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let TMP_26 \def (\lambda (p:
-PList).((eq PList p (PCons h d hds)) \to (let TMP_24 \def (\lambda (c6:
-C).(drop h d c4 c6)) in (let TMP_25 \def (\lambda (c6: C).(drop1 hds c6 c5))
-in (ex2 C TMP_24 TMP_25))))) in (let H10 \def (eq_ind PList hds0 TMP_26 H3
-hds H7) in (let TMP_27 \def (\lambda (p: PList).(drop1 p c4 c5)) in (let H11
-\def (eq_ind PList hds0 TMP_27 H2 hds H7) in (let TMP_28 \def (\lambda (n:
-nat).(drop h0 n c2 c4)) in (let H12 \def (eq_ind nat d0 TMP_28 H1 d H8) in
-(let TMP_29 \def (\lambda (n: nat).(drop n d c2 c4)) in (let H13 \def (eq_ind
-nat h0 TMP_29 H12 h H9) in (let TMP_30 \def (\lambda (c6: C).(drop h d c2
-c6)) in (let TMP_31 \def (\lambda (c6: C).(drop1 hds c6 c5)) in (ex_intro2 C
-TMP_30 TMP_31 c4 H13 H11))))))))))))) in (let TMP_33 \def (TMP_32 H6) in
-(TMP_33 H5))))))))))))))))))))))))) in (drop1_ind TMP_8 TMP_14 TMP_34 y c1 c3
-H0)))))) in (insert_eq PList TMP_1 TMP_2 TMP_5 TMP_35 H)))))))))).
+nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq
+PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_:
+PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds
+c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind
+(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d
+hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1
+hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d
+hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee with
+[PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds)
+H1) in (False_ind (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2:
+C).(drop1 hds c2 c))) H2)))) (\lambda (c2: C).(\lambda (c4: C).(\lambda (h0:
+nat).(\lambda (d0: nat).(\lambda (H1: (drop h0 d0 c2 c4)).(\lambda (c5:
+C).(\lambda (hds0: PList).(\lambda (H2: (drop1 hds0 c4 c5)).(\lambda (H3:
+(((eq PList hds0 (PCons h d hds)) \to (ex2 C (\lambda (c6: C).(drop h d c4
+c6)) (\lambda (c6: C).(drop1 hds c6 c5)))))).(\lambda (H4: (eq PList (PCons
+h0 d0 hds0) (PCons h d hds))).(let H5 \def (f_equal PList nat (\lambda (e:
+PList).(match e with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
+(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat
+(\lambda (e: PList).(match e with [PNil \Rightarrow d0 | (PCons _ n _)
+\Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H7 \def
+(f_equal PList PList (\lambda (e: PList).(match e with [PNil \Rightarrow hds0
+| (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds) H4) in
+(\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let H10 \def
+(eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) \to (ex2
+C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 c5)))))
+H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: PList).(drop1 p
+c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0
+n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda (n: nat).(drop n
+d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h d c2 c6))
+(\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) H5)))))))))))) y
+c1 c3 H0))) H)))))).
projects / helm.git / blobdiff