X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fpr2%2Fprops.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fpr2%2Fprops.ma;h=fca780fee854d67eaaf9a538c10b3f0f90846acf;hb=44b11d163869b92e2d2abc695c622e64356fb6fd;hp=0000000000000000000000000000000000000000;hpb=8b9d2632ab7edc7102d16f3857fde02139cef5c2;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma new file mode 100644 index 000000000..fca780fee --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma @@ -0,0 +1,285 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props". + +include "pr2/defs.ma". + +include "pr0/props.ma". + +include "getl/drop.ma". + +include "getl/clear.ma". + +theorem pr2_thin_dx: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u +t2))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 +t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u +(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 +t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i +H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 +t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 +u)))))))))))) c t1 t2 H)))))). + +theorem pr2_head_1: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall +(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 +u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: +T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 +(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 +t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 +t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c +u1 u2 H)))))). + +theorem pr2_head_2: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u +t2))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead +k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind +b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind +b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) +(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow +(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 +t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: +C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind +b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 +(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u +t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 +(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) +u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead +(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T +t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) +H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda +(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda +(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 +t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 +t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) +u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T +t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 +t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind +b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: +T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) +\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u +t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) +u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match +i in nat return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d +(Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) +(THead (Bind b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c +(Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 +t2)).(let H13 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d +(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) +H11))) in ((let H14 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) +(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u +(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H15 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4])) (CHead d +(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) +u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in +(\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18 \def (eq_ind +T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind B Abbr +(\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u t2))) +(pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) (pr0_delta u u +(pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13)))) | (S n) \Rightarrow +(\lambda (H11: (getl (S n) (CHead c (Bind b) u) (CHead d (Bind Abbr) +u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Bind b) +n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H11) (THead (Bind b) u +t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Bind b)) +(THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r (Bind b) n) H12 +u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq +C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C +(CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) (\lambda (f: +F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def (match H in +pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq T t0 +t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with +[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) +u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead +c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 +t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: +(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 +t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u +t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead +(Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 +(Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 +(CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t +H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: +(eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) +(\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d +(Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead +(Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0 +t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat +f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T +t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d +(Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead +(Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c +(Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda +(H10: (subst0 i u0 t3 t2)).((match i in nat return (\lambda (n: nat).((getl n +(CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow +(\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) +u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c +(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind +Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) +(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 +H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) +| (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d +(Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 +(r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) +(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 +H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) +n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 +(sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal +C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))). + +theorem clear_pr2_trans: + \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to +(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) +\def + \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 +t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in +pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 +c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 +t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c +c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 +(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 +t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 +t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind +T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 +t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 +H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t +H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 +t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) +\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) +\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 +t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d +(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 +t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 +(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 +t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: +(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i +(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) +t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 +H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T +t2)))))))). + +theorem pr2_cflat: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f: +F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f: +F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda +(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0 +(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))). + +theorem pr2_ctail: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) +t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: +(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail +Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). + +theorem pr2_lift: + \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift +h d t1) (lift h d t2))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 +t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) +\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with +[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: +(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T +t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d +t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 +t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: +(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d +t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) +(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T +t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 +t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 +t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) +\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) +\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda +(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e +(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c +(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 +(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to +((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: +(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda +(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) +(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d +(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C +(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda +(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear +e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda +(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop +h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr) +u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 +x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def +(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C +(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) +u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) +(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr) +(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x +d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x +(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d +t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d +H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u +(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12) +(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) +(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0 +(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 +(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))). +