include "basic_1/wcpr0/getl.ma".
-theorem pc3_wcpr0__pc3_wcpr0_t_aux:
+fact pc3_wcpr0__pc3_wcpr0_t_aux:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall
(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3
(CHead c2 k u) t1 t2))))))))
\def
\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k:
K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
-(CHead c1 k u) t1 t2)).(let TMP_1 \def (CHead c1 k u) in (let TMP_3 \def
-(\lambda (t: T).(\lambda (t0: T).(let TMP_2 \def (CHead c2 k u) in (pc3 TMP_2
-t t0)))) in (let TMP_5 \def (\lambda (t: T).(let TMP_4 \def (CHead c2 k u) in
-(pc3_refl TMP_4 t))) in (let TMP_52 \def (\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H1: (pr2 (CHead c1 k u) t4 t3)).(\lambda (t5: T).(\lambda (_:
-(pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: (pc3 (CHead c2 k u) t3 t5)).(let
-TMP_6 \def (CHead c2 k u) in (let TMP_7 \def (CHead c1 k u) in (let TMP_8
-\def (\lambda (c: C).(pr2 c t4 t3)) in (let TMP_10 \def (\lambda (_: C).(let
-TMP_9 \def (CHead c2 k u) in (pc3 TMP_9 t4 t3))) in (let TMP_50 \def (\lambda
-(y: C).(\lambda (H4: (pr2 y t4 t3)).(let TMP_12 \def (\lambda (c: C).(\lambda
-(t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (let TMP_11 \def (CHead
-c2 k u) in (pc3 TMP_11 t t0)))))) in (let TMP_16 \def (\lambda (c:
-C).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H5: (pr0 t6 t0)).(\lambda (_:
-(eq C c (CHead c1 k u))).(let TMP_13 \def (CHead c2 k u) in (let TMP_14 \def
-(CHead c2 k u) in (let TMP_15 \def (pr2_free TMP_14 t6 t0 H5) in (pc3_pr2_r
-TMP_13 t6 t0 TMP_15))))))))) in (let TMP_49 \def (\lambda (c: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d (Bind
-Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6
+(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0:
+T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3:
+(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead
+c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4
+t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead
+c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda
+(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2
+k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d
+(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6
t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c
-(CHead c1 k u))).(let TMP_19 \def (\lambda (c0: C).(let TMP_17 \def (Bind
-Abbr) in (let TMP_18 \def (CHead d TMP_17 u0) in (getl i c0 TMP_18)))) in
-(let TMP_20 \def (CHead c1 k u) in (let H9 \def (eq_ind C c TMP_19 H5 TMP_20
-H8) in (let TMP_24 \def (\lambda (e2: C).(\lambda (u2: T).(let TMP_21 \def
-(CHead c2 k u) in (let TMP_22 \def (Bind Abbr) in (let TMP_23 \def (CHead e2
-TMP_22 u2) in (getl i TMP_21 TMP_23)))))) in (let TMP_25 \def (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 d e2))) in (let TMP_26 \def (\lambda (_:
-C).(\lambda (u2: T).(pr0 u0 u2))) in (let TMP_27 \def (CHead c2 k u) in (let
-TMP_28 \def (pc3 TMP_27 t6 t) in (let TMP_42 \def (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(let TMP_29 \def
-(\lambda (t7: T).(subst0 i x1 t0 t7)) in (let TMP_30 \def (\lambda (t7:
-T).(pr0 t t7)) in (let TMP_31 \def (CHead c2 k u) in (let TMP_32 \def (pc3
-TMP_31 t6 t) in (let TMP_40 \def (\lambda (x: T).(\lambda (H13: (subst0 i x1
-t0 x)).(\lambda (H14: (pr0 t x)).(let TMP_33 \def (CHead c2 k u) in (let
-TMP_34 \def (CHead c2 k u) in (let TMP_35 \def (pr2_delta TMP_34 x0 x1 i H10
-t6 t0 H6 x H13) in (let TMP_36 \def (CHead c2 k u) in (let TMP_37 \def (CHead
-c2 k u) in (let TMP_38 \def (pr2_free TMP_37 t x H14) in (let TMP_39 \def
-(pc3_pr2_x TMP_36 x t TMP_38) in (pc3_pr2_u TMP_33 x t6 TMP_35 t
-TMP_39))))))))))) in (let TMP_41 \def (pr0_subst0_fwd u0 t0 t i H7 x1 H12) in
-(ex2_ind T TMP_29 TMP_30 TMP_32 TMP_40 TMP_41)))))))))))) in (let TMP_43 \def
-(CHead c1 k u) in (let TMP_44 \def (CHead c2 k u) in (let TMP_45 \def
-(pr0_refl u) in (let TMP_46 \def (wcpr0_comp c1 c2 H u u TMP_45 k) in (let
-TMP_47 \def (Bind Abbr) in (let TMP_48 \def (wcpr0_getl TMP_43 TMP_44 TMP_46
-i d u0 TMP_47 H9) in (ex3_2_ind C T TMP_24 TMP_25 TMP_26 TMP_28 TMP_42
-TMP_48))))))))))))))))))))))))))) in (pr2_ind TMP_12 TMP_16 TMP_49 y t4 t3
-H4)))))) in (let TMP_51 \def (insert_eq C TMP_7 TMP_8 TMP_10 TMP_50 H1) in
-(pc3_t t3 TMP_6 t4 TMP_51 t5 H3))))))))))))) in (pr3_ind TMP_1 TMP_3 TMP_5
-TMP_52 t1 t2 H0)))))))))))).
+(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead
+d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2:
+C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
+(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7:
+T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6
+t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t
+x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6
+t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x
+H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k
+u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr)
+H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
-theorem pc3_wcpr0_t:
+lemma pc3_wcpr0_t:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(let TMP_1 \def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1
-t2) \to (pc3 c0 t1 t2)))))) in (let TMP_2 \def (\lambda (c: C).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0)))))
-in (let TMP_16 \def (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
-c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3
-c3 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr3
-(CHead c0 k u1) t1 t2)).(let TMP_3 \def (pr2_free c0 u1 u2 H2) in (let H4
-\def (pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 TMP_3) in (let TMP_5 \def (\lambda
-(t: T).(let TMP_4 \def (CHead c0 k u2) in (pr3 TMP_4 t1 t))) in (let TMP_7
-\def (\lambda (t: T).(let TMP_6 \def (CHead c0 k u2) in (pr3 TMP_6 t2 t))) in
-(let TMP_8 \def (CHead c3 k u2) in (let TMP_9 \def (pc3 TMP_8 t1 t2) in (let
-TMP_15 \def (\lambda (x: T).(\lambda (H5: (pr3 (CHead c0 k u2) t1
-x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(let TMP_10 \def (CHead c3 k
-u2) in (let TMP_11 \def (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) in
-(let TMP_12 \def (CHead c3 k u2) in (let TMP_13 \def
-(pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x H6) in (let TMP_14 \def (pc3_s
-TMP_12 x t2 TMP_13) in (pc3_t x TMP_10 t1 TMP_11 t2 TMP_14))))))))) in
-(ex2_ind T TMP_5 TMP_7 TMP_9 TMP_15 H4))))))))))))))))))) in (wcpr0_ind TMP_1
-TMP_2 TMP_16 c1 c2 H)))))).
+t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1:
+T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def
+(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k
+u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3
+(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x
+(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2
+(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x
+H6)))))) H4))))))))))))) c1 c2 H))).
-theorem pc3_wcpr0:
+lemma pc3_wcpr0:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
\def
\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (let
-TMP_1 \def (\lambda (t: T).(pr3 c1 t1 t)) in (let TMP_2 \def (\lambda (t:
-T).(pr3 c1 t2 t)) in (let TMP_3 \def (pc3 c2 t1 t2) in (let TMP_7 \def
-(\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2
-x)).(let TMP_4 \def (pc3_wcpr0_t c1 c2 H t1 x H2) in (let TMP_5 \def
-(pc3_wcpr0_t c1 c2 H t2 x H3) in (let TMP_6 \def (pc3_s c2 x t2 TMP_5) in
-(pc3_t x c2 t1 TMP_4 t2 TMP_6))))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_7
-H1))))))))))).
+T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind
+T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2
+x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2
+(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
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