From: Ferruccio Guidi Date: Tue, 24 Feb 2015 19:27:32 +0000 (+0000) Subject: components: wcpr0 pr1 pr2 X-Git-Tag: make_still_working~737 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=40e0eed5dc1a5f56f089f491841a5077723b891a;p=helm.git components: wcpr0 pr1 pr2 --- diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma index 84db71acb..b466ad925 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr0/defs.ma". +include "basic_1/pr0/defs.ma". inductive pr1: T \to (T \to Prop) \def | pr1_refl: \forall (t: T).(pr1 t t) diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma new file mode 100644 index 000000000..1bdb15327 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *********************) + +include "basic_1/pr1/defs.ma". + +let rec pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0: +(\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pr1 t2 +t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t t0) on p: P +t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | (pr1_sing t2 t1 p0 +t3 p1) \Rightarrow (let TMP_1 \def ((pr1_ind P f f0) t2 t3 p1) in (f0 t2 t1 +p0 t3 p1 TMP_1))]. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma index ec469da50..db9c74c5a 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma @@ -14,57 +14,69 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr1/props.ma". +include "basic_1/pr1/props.ma". -include "Basic-1/pr0/pr0.ma". +include "basic_1/pr0/pr0.ma". theorem pr1_strip: \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) \def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda -(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T -(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: -T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda -(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let -H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: -T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) -(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 -t2 H0)))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 317 -END *) + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(let TMP_3 \def +(\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (let TMP_1 +\def (\lambda (t4: T).(pr1 t2 t4)) in (let TMP_2 \def (\lambda (t4: T).(pr1 +t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let TMP_8 \def (\lambda (t: +T).(\lambda (t2: T).(\lambda (H0: (pr0 t t2)).(let TMP_4 \def (\lambda (t3: +T).(pr1 t t3)) in (let TMP_5 \def (\lambda (t3: T).(pr1 t2 t3)) in (let TMP_6 +\def (pr1_pr0 t t2 H0) in (let TMP_7 \def (pr1_refl t2) in (ex_intro2 T TMP_4 +TMP_5 t2 TMP_6 TMP_7)))))))) in (let TMP_26 \def (\lambda (t2: T).(\lambda +(t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2 +t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T (\lambda (t: +T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda +(H3: (pr0 t3 t5)).(let TMP_9 \def (\lambda (t: T).(pr0 t5 t)) in (let TMP_10 +\def (\lambda (t: T).(pr0 t2 t)) in (let TMP_11 \def (\lambda (t: T).(pr1 t4 +t)) in (let TMP_12 \def (\lambda (t: T).(pr1 t5 t)) in (let TMP_13 \def (ex2 +T TMP_11 TMP_12) in (let TMP_24 \def (\lambda (x: T).(\lambda (H4: (pr0 t5 +x)).(\lambda (H5: (pr0 t2 x)).(let H6 \def (H2 x H5) in (let TMP_14 \def +(\lambda (t: T).(pr1 t4 t)) in (let TMP_15 \def (\lambda (t: T).(pr1 x t)) in +(let TMP_16 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_17 \def (\lambda (t: +T).(pr1 t5 t)) in (let TMP_18 \def (ex2 T TMP_16 TMP_17) in (let TMP_23 \def +(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x x0)).(let +TMP_19 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_20 \def (\lambda (t: +T).(pr1 t5 t)) in (let TMP_21 \def (pr1_pr0 t5 x H4) in (let TMP_22 \def +(pr1_t x t5 TMP_21 x0 H8) in (ex_intro2 T TMP_19 TMP_20 x0 H7 TMP_22)))))))) +in (ex2_ind T TMP_14 TMP_15 TMP_18 TMP_23 H6))))))))))) in (let TMP_25 \def +(pr0_confluence t3 t5 H3 t2 H0) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_24 +TMP_25)))))))))))))))) in (pr1_ind TMP_3 TMP_8 TMP_26 t0 t1 H)))))). theorem pr1_confluence: \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) \def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: -(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda -(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def -H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) -(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: -T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x -H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda -(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 311 -END *) + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(let TMP_3 \def +(\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (let TMP_1 +\def (\lambda (t4: T).(pr1 t2 t4)) in (let TMP_2 \def (\lambda (t4: T).(pr1 +t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let TMP_7 \def (\lambda (t: +T).(\lambda (t2: T).(\lambda (H0: (pr1 t t2)).(let TMP_4 \def (\lambda (t3: +T).(pr1 t t3)) in (let TMP_5 \def (\lambda (t3: T).(pr1 t2 t3)) in (let TMP_6 +\def (pr1_refl t2) in (ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6))))))) in (let +TMP_23 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 +t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: +T).((pr1 t2 t5) \to (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 +t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr1 t3 t5)).(let H_x \def +(pr1_strip t3 t5 H3 t2 H0) in (let H4 \def H_x in (let TMP_8 \def (\lambda +(t: T).(pr1 t5 t)) in (let TMP_9 \def (\lambda (t: T).(pr1 t2 t)) in (let +TMP_10 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_11 \def (\lambda (t: +T).(pr1 t5 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_22 \def +(\lambda (x: T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 +\def (H2 x H6) in (let H7 \def H_x0 in (let TMP_13 \def (\lambda (t: T).(pr1 +t4 t)) in (let TMP_14 \def (\lambda (t: T).(pr1 x t)) in (let TMP_15 \def +(\lambda (t: T).(pr1 t4 t)) in (let TMP_16 \def (\lambda (t: T).(pr1 t5 t)) +in (let TMP_17 \def (ex2 T TMP_15 TMP_16) in (let TMP_21 \def (\lambda (x0: +T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x x0)).(let TMP_18 \def +(\lambda (t: T).(pr1 t4 t)) in (let TMP_19 \def (\lambda (t: T).(pr1 t5 t)) +in (let TMP_20 \def (pr1_t x t5 H5 x0 H9) in (ex_intro2 T TMP_18 TMP_19 x0 H8 +TMP_20))))))) in (ex2_ind T TMP_13 TMP_14 TMP_17 TMP_21 H7)))))))))))) in +(ex2_ind T TMP_8 TMP_9 TMP_12 TMP_22 H4))))))))))))))))) in (pr1_ind TMP_3 +TMP_7 TMP_23 t0 t1 H)))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma index 0615bed96..ab17c0dc0 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma @@ -14,113 +14,151 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr1/defs.ma". +include "basic_1/pr1/fwd.ma". -include "Basic-1/pr0/subst1.ma". +include "basic_1/pr0/subst1.ma". -include "Basic-1/subst1/props.ma". +include "basic_1/subst1/props.ma". -include "Basic-1/T/props.ma". +include "basic_1/T/props.ma". theorem pr1_pr0: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) \def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H -t2 (pr1_refl t2)))). -(* COMMENTS -Initial nodes: 23 -END *) + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_1 \def +(pr1_refl t2) in (pr1_sing t2 t1 H t2 TMP_1)))). theorem pr1_t: \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 t3) \to (pr1 t1 t3))))) \def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) -(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda -(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 -t5 H3)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 103 -END *) + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(let TMP_1 \def +(\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t +t3))))) in (let TMP_2 \def (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: +(pr1 t t3)).H0))) in (let TMP_4 \def (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 +t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 +t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(let TMP_3 \def (H2 t5 +H3) in (pr1_sing t0 t3 H0 t5 TMP_3)))))))))) in (pr1_ind TMP_1 TMP_2 TMP_4 t1 +t2 H)))))). theorem pr1_head_1: \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall (k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) \def \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k -t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda -(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing -(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k -t3 t) H2))))))) u1 u2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) +T).(\lambda (k: K).(let TMP_3 \def (\lambda (t0: T).(\lambda (t1: T).(let +TMP_1 \def (THead k t0 t) in (let TMP_2 \def (THead k t1 t) in (pr1 TMP_1 +TMP_2))))) in (let TMP_5 \def (\lambda (t0: T).(let TMP_4 \def (THead k t0 t) +in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t2: T).(\lambda (t1: +T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (_: (pr1 t2 +t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(let TMP_6 \def +(THead k t2 t) in (let TMP_7 \def (THead k t1 t) in (let TMP_8 \def (pr0_refl +t) in (let TMP_9 \def (pr0_comp t1 t2 H0 t t TMP_8 k) in (let TMP_10 \def +(THead k t3 t) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in +(pr1_ind TMP_3 TMP_5 TMP_11 u1 u2 H)))))))). theorem pr1_head_2: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall (k: K).(pr1 (THead k u t1) (THead k u t2)))))) \def \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u -t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing -(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k -u t4) H2))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) +T).(\lambda (k: K).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: T).(let +TMP_1 \def (THead k u t) in (let TMP_2 \def (THead k u t0) in (pr1 TMP_1 +TMP_2))))) in (let TMP_5 \def (\lambda (t: T).(let TMP_4 \def (THead k u t) +in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 +t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(let TMP_6 \def +(THead k u t0) in (let TMP_7 \def (THead k u t3) in (let TMP_8 \def (pr0_refl +u) in (let TMP_9 \def (pr0_comp u u TMP_8 t3 t0 H0 k) in (let TMP_10 \def +(THead k u t4) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in +(pr1_ind TMP_3 TMP_5 TMP_11 t1 t2 H)))))))). theorem pr1_comp: \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u)))))))) \def - \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall -(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda -(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 -t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 -t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: -T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead -k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t -u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k -t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 -H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 -t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k -t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp -t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v -w H))). -(* COMMENTS -Initial nodes: 273 -END *) + \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(let TMP_3 \def +(\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) +\to (\forall (k: K).(let TMP_1 \def (THead k t t1) in (let TMP_2 \def (THead +k t0 u) in (pr1 TMP_1 TMP_2))))))))) in (let TMP_4 \def (\lambda (t: +T).(\lambda (t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: +K).(pr1_head_2 t0 u H0 t k)))))) in (let TMP_16 \def (\lambda (t2: +T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (H1: +(pr1 t2 t3)).(\lambda (_: ((\forall (t: T).(\forall (u: T).((pr1 t u) \to +(\forall (k: K).(pr1 (THead k t2 t) (THead k t3 u)))))))).(\lambda (t: +T).(\lambda (u: T).(\lambda (H3: (pr1 t u)).(\lambda (k: K).(let TMP_7 \def +(\lambda (t0: T).(\lambda (t4: T).(let TMP_5 \def (THead k t1 t0) in (let +TMP_6 \def (THead k t3 t4) in (pr1 TMP_5 TMP_6))))) in (let TMP_9 \def +(\lambda (t0: T).(let TMP_8 \def (pr1_sing t2 t1 H0 t3 H1) in (pr1_head_1 t1 +t3 TMP_8 t0 k))) in (let TMP_15 \def (\lambda (t0: T).(\lambda (t4: +T).(\lambda (H4: (pr0 t4 t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 +t5)).(\lambda (H6: (pr1 (THead k t1 t0) (THead k t3 t5))).(let TMP_10 \def +(THead k t1 t0) in (let TMP_11 \def (THead k t1 t4) in (let TMP_12 \def +(pr0_refl t1) in (let TMP_13 \def (pr0_comp t1 t1 TMP_12 t4 t0 H4 k) in (let +TMP_14 \def (THead k t3 t5) in (pr1_sing TMP_10 TMP_11 TMP_13 TMP_14 +H6)))))))))))) in (pr1_ind TMP_7 TMP_9 TMP_15 t u H3)))))))))))))) in +(pr1_ind TMP_3 TMP_4 TMP_16 v w H)))))). theorem pr1_eta: \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in (\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))) \def - \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in -(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift -(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef -O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) -u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind -Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) -(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl -(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u -(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr -not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O -(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))). -(* COMMENTS -Initial nodes: 463 -END *) + \lambda (w: T).(\lambda (u: T).(let TMP_1 \def (Bind Abst) in (let t \def +(THead TMP_1 w u) in (\lambda (v: T).(\lambda (H: (pr1 v w)).(let TMP_2 \def +(Bind Abst) in (let TMP_3 \def (S O) in (let TMP_4 \def (lift TMP_3 O w) in +(let TMP_5 \def (S O) in (let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_5 +TMP_6 u) in (let TMP_8 \def (THead TMP_2 TMP_4 TMP_7) in (let TMP_16 \def +(\lambda (t0: T).(let TMP_9 \def (Bind Abst) in (let TMP_10 \def (Flat Appl) +in (let TMP_11 \def (TLRef O) in (let TMP_12 \def (THead TMP_10 TMP_11 t0) in +(let TMP_13 \def (THead TMP_9 v TMP_12) in (let TMP_14 \def (Bind Abst) in +(let TMP_15 \def (THead TMP_14 w u) in (pr1 TMP_13 TMP_15))))))))) in (let +TMP_17 \def (Flat Appl) in (let TMP_18 \def (TLRef O) in (let TMP_19 \def +(Bind Abst) in (let TMP_20 \def (S O) in (let TMP_21 \def (lift TMP_20 O w) +in (let TMP_22 \def (S O) in (let TMP_23 \def (S O) in (let TMP_24 \def (lift +TMP_22 TMP_23 u) in (let TMP_25 \def (THead TMP_19 TMP_21 TMP_24) in (let +TMP_26 \def (THead TMP_17 TMP_18 TMP_25) in (let TMP_27 \def (Bind Abbr) in +(let TMP_28 \def (TLRef O) in (let TMP_29 \def (S O) in (let TMP_30 \def (S +O) in (let TMP_31 \def (lift TMP_29 TMP_30 u) in (let TMP_32 \def (THead +TMP_27 TMP_28 TMP_31) in (let TMP_33 \def (Flat Appl) in (let TMP_34 \def +(TLRef O) in (let TMP_35 \def (Bind Abst) in (let TMP_36 \def (S O) in (let +TMP_37 \def (lift TMP_36 O w) in (let TMP_38 \def (S O) in (let TMP_39 \def +(S O) in (let TMP_40 \def (lift TMP_38 TMP_39 u) in (let TMP_41 \def (THead +TMP_35 TMP_37 TMP_40) in (let TMP_42 \def (THead TMP_33 TMP_34 TMP_41) in +(let TMP_43 \def (S O) in (let TMP_44 \def (lift TMP_43 O w) in (let TMP_45 +\def (TLRef O) in (let TMP_46 \def (TLRef O) in (let TMP_47 \def (TLRef O) in +(let TMP_48 \def (pr0_refl TMP_47) in (let TMP_49 \def (S O) in (let TMP_50 +\def (S O) in (let TMP_51 \def (lift TMP_49 TMP_50 u) in (let TMP_52 \def (S +O) in (let TMP_53 \def (S O) in (let TMP_54 \def (lift TMP_52 TMP_53 u) in +(let TMP_55 \def (S O) in (let TMP_56 \def (S O) in (let TMP_57 \def (lift +TMP_55 TMP_56 u) in (let TMP_58 \def (pr0_refl TMP_57) in (let TMP_59 \def +(pr0_beta TMP_44 TMP_45 TMP_46 TMP_48 TMP_51 TMP_54 TMP_58) in (let TMP_60 +\def (Bind Abbr) in (let TMP_61 \def (TLRef O) in (let TMP_62 \def (S O) in +(let TMP_63 \def (lift TMP_62 O u) in (let TMP_64 \def (THead TMP_60 TMP_61 +TMP_63) in (let TMP_65 \def (Bind Abbr) in (let TMP_66 \def (TLRef O) in (let +TMP_67 \def (S O) in (let TMP_68 \def (S O) in (let TMP_69 \def (lift TMP_67 +TMP_68 u) in (let TMP_70 \def (THead TMP_65 TMP_66 TMP_69) in (let TMP_71 +\def (TLRef O) in (let TMP_72 \def (TLRef O) in (let TMP_73 \def (TLRef O) in +(let TMP_74 \def (pr0_refl TMP_73) in (let TMP_75 \def (S O) in (let TMP_76 +\def (S O) in (let TMP_77 \def (lift TMP_75 TMP_76 u) in (let TMP_78 \def (S +O) in (let TMP_79 \def (S O) in (let TMP_80 \def (lift TMP_78 TMP_79 u) in +(let TMP_81 \def (S O) in (let TMP_82 \def (S O) in (let TMP_83 \def (lift +TMP_81 TMP_82 u) in (let TMP_84 \def (pr0_refl TMP_83) in (let TMP_85 \def (S +O) in (let TMP_86 \def (lift TMP_85 O u) in (let TMP_87 \def (le_O_n O) in +(let TMP_88 \def (subst1_lift_S u O O TMP_87) in (let TMP_89 \def (pr0_delta1 +TMP_71 TMP_72 TMP_74 TMP_77 TMP_80 TMP_84 TMP_86 TMP_88) in (let TMP_90 \def +(Bind Abbr) in (let TMP_91 \def (TLRef O) in (let TMP_92 \def (S O) in (let +TMP_93 \def (lift TMP_92 O u) in (let TMP_94 \def (THead TMP_90 TMP_91 +TMP_93) in (let TMP_95 \def (pr0_refl u) in (let TMP_96 \def (TLRef O) in +(let TMP_97 \def (pr0_zeta Abbr not_abbr_abst u u TMP_95 TMP_96) in (let +TMP_98 \def (pr1_pr0 TMP_94 u TMP_97) in (let TMP_99 \def (pr1_sing TMP_64 +TMP_70 TMP_89 u TMP_98) in (let TMP_100 \def (pr1_sing TMP_32 TMP_42 TMP_59 u +TMP_99) in (let TMP_101 \def (Bind Abst) in (let TMP_102 \def (pr1_comp v w H +TMP_26 u TMP_100 TMP_101) in (let TMP_103 \def (S O) in (let TMP_104 \def +(Bind Abst) in (let TMP_105 \def (THead TMP_104 w u) in (let TMP_106 \def +(lift TMP_103 O TMP_105) in (let TMP_107 \def (S O) in (let TMP_108 \def +(lift_bind Abst w u TMP_107 O) in (eq_ind_r T TMP_8 TMP_16 TMP_102 TMP_106 +TMP_108))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) +))))))))))))))))))))))))))))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma index df9ea1fd9..0b256622f 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/props.ma". +include "basic_1/pr2/props.ma". -include "Basic-1/clen/getl.ma". +include "basic_1/clen/getl.ma". theorem pr2_gen_ctail: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall @@ -25,59 +25,103 @@ T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen c) u t t2))))))))) \def \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T -(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda -(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind -Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 -t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) -(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) -(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 -t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda -(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let -H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind -(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c -(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) -(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d -(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) -u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: -C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k -u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c -t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 -t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 -H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen -c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq -nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: -nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 -T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) -(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: -(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T -u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda -(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 -(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind -Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda -(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 -(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 -t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1161 -END *) +(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(let TMP_1 \def (CTail k u c) +in (let TMP_2 \def (\lambda (c0: C).(pr2 c0 t1 t2)) in (let TMP_10 \def +(\lambda (_: C).(let TMP_3 \def (pr2 c t1 t2) in (let TMP_5 \def (\lambda (_: +T).(let TMP_4 \def (Bind Abbr) in (eq K k TMP_4))) in (let TMP_6 \def +(\lambda (t: T).(pr0 t1 t)) in (let TMP_8 \def (\lambda (t: T).(let TMP_7 +\def (clen c) in (subst0 TMP_7 u t t2))) in (let TMP_9 \def (ex3 T TMP_5 +TMP_6 TMP_8) in (or TMP_3 TMP_9))))))) in (let TMP_126 \def (\lambda (y: +C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_18 \def (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (let TMP_11 \def (pr2 c +t t0) in (let TMP_13 \def (\lambda (_: T).(let TMP_12 \def (Bind Abbr) in (eq +K k TMP_12))) in (let TMP_14 \def (\lambda (t3: T).(pr0 t t3)) in (let TMP_16 +\def (\lambda (t3: T).(let TMP_15 \def (clen c) in (subst0 TMP_15 u t3 t0))) +in (let TMP_17 \def (ex3 T TMP_13 TMP_14 TMP_16) in (or TMP_11 +TMP_17)))))))))) in (let TMP_27 \def (\lambda (c0: C).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k +u c))).(let TMP_19 \def (pr2 c t3 t4) in (let TMP_21 \def (\lambda (_: +T).(let TMP_20 \def (Bind Abbr) in (eq K k TMP_20))) in (let TMP_22 \def +(\lambda (t: T).(pr0 t3 t)) in (let TMP_24 \def (\lambda (t: T).(let TMP_23 +\def (clen c) in (subst0 TMP_23 u t t4))) in (let TMP_25 \def (ex3 T TMP_21 +TMP_22 TMP_24) in (let TMP_26 \def (pr2_free c t3 t4 H1) in (or_introl TMP_19 +TMP_25 TMP_26)))))))))))) in (let TMP_125 \def (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind +Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0 +(CTail k u c))).(let TMP_30 \def (\lambda (c1: C).(let TMP_28 \def (Bind +Abbr) in (let TMP_29 \def (CHead d TMP_28 u0) in (getl i c1 TMP_29)))) in +(let TMP_31 \def (CTail k u c) in (let H5 \def (eq_ind C c0 TMP_30 H1 TMP_31 +H4) in (let H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x +in (let TMP_33 \def (\lambda (e: C).(let TMP_32 \def (CTail k u e) in (eq C d +TMP_32))) in (let TMP_36 \def (\lambda (e: C).(let TMP_34 \def (Bind Abbr) in +(let TMP_35 \def (CHead e TMP_34 u0) in (getl i c TMP_35)))) in (let TMP_37 +\def (ex2 C TMP_33 TMP_36) in (let TMP_39 \def (\lambda (_: nat).(let TMP_38 +\def (clen c) in (eq nat i TMP_38))) in (let TMP_41 \def (\lambda (_: +nat).(let TMP_40 \def (Bind Abbr) in (eq K k TMP_40))) in (let TMP_42 \def +(\lambda (_: nat).(eq T u u0)) in (let TMP_44 \def (\lambda (n: nat).(let +TMP_43 \def (CSort n) in (eq C d TMP_43))) in (let TMP_45 \def (ex4 nat +TMP_39 TMP_41 TMP_42 TMP_44) in (let TMP_46 \def (pr2 c t3 t) in (let TMP_48 +\def (\lambda (_: T).(let TMP_47 \def (Bind Abbr) in (eq K k TMP_47))) in +(let TMP_49 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_51 \def (\lambda +(t0: T).(let TMP_50 \def (clen c) in (subst0 TMP_50 u t0 t))) in (let TMP_52 +\def (ex3 T TMP_48 TMP_49 TMP_51) in (let TMP_53 \def (or TMP_46 TMP_52) in +(let TMP_76 \def (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) +(\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))))).(let TMP_55 \def +(\lambda (e: C).(let TMP_54 \def (CTail k u e) in (eq C d TMP_54))) in (let +TMP_58 \def (\lambda (e: C).(let TMP_56 \def (Bind Abbr) in (let TMP_57 \def +(CHead e TMP_56 u0) in (getl i c TMP_57)))) in (let TMP_59 \def (pr2 c t3 t) +in (let TMP_61 \def (\lambda (_: T).(let TMP_60 \def (Bind Abbr) in (eq K k +TMP_60))) in (let TMP_62 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_64 +\def (\lambda (t0: T).(let TMP_63 \def (clen c) in (subst0 TMP_63 u t0 t))) +in (let TMP_65 \def (ex3 T TMP_61 TMP_62 TMP_64) in (let TMP_66 \def (or +TMP_59 TMP_65) in (let TMP_75 \def (\lambda (x: C).(\lambda (_: (eq C d +(CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(let +TMP_67 \def (pr2 c t3 t) in (let TMP_69 \def (\lambda (_: T).(let TMP_68 \def +(Bind Abbr) in (eq K k TMP_68))) in (let TMP_70 \def (\lambda (t0: T).(pr0 t3 +t0)) in (let TMP_72 \def (\lambda (t0: T).(let TMP_71 \def (clen c) in +(subst0 TMP_71 u t0 t))) in (let TMP_73 \def (ex3 T TMP_69 TMP_70 TMP_72) in +(let TMP_74 \def (pr2_delta c x u0 i H9 t3 t4 H2 t H3) in (or_introl TMP_67 +TMP_73 TMP_74)))))))))) in (ex2_ind C TMP_55 TMP_58 TMP_66 TMP_75 +H7))))))))))) in (let TMP_124 \def (\lambda (H7: (ex4 nat (\lambda (_: +nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda +(_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))))).(let TMP_78 +\def (\lambda (_: nat).(let TMP_77 \def (clen c) in (eq nat i TMP_77))) in +(let TMP_80 \def (\lambda (_: nat).(let TMP_79 \def (Bind Abbr) in (eq K k +TMP_79))) in (let TMP_81 \def (\lambda (_: nat).(eq T u u0)) in (let TMP_83 +\def (\lambda (n: nat).(let TMP_82 \def (CSort n) in (eq C d TMP_82))) in +(let TMP_84 \def (pr2 c t3 t) in (let TMP_86 \def (\lambda (_: T).(let TMP_85 +\def (Bind Abbr) in (eq K k TMP_85))) in (let TMP_87 \def (\lambda (t0: +T).(pr0 t3 t0)) in (let TMP_89 \def (\lambda (t0: T).(let TMP_88 \def (clen +c) in (subst0 TMP_88 u t0 t))) in (let TMP_90 \def (ex3 T TMP_86 TMP_87 +TMP_89) in (let TMP_91 \def (or TMP_84 TMP_90) in (let TMP_123 \def (\lambda +(x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda (H9: (eq K k (Bind +Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d (CSort x0))).(let +TMP_92 \def (\lambda (n: nat).(subst0 n u0 t4 t)) in (let TMP_93 \def (clen +c) in (let H12 \def (eq_ind nat i TMP_92 H3 TMP_93 H8) in (let TMP_95 \def +(\lambda (t0: T).(let TMP_94 \def (clen c) in (subst0 TMP_94 t0 t4 t))) in +(let H13 \def (eq_ind_r T u0 TMP_95 H12 u H10) in (let TMP_96 \def (Bind +Abbr) in (let TMP_104 \def (\lambda (k0: K).(let TMP_97 \def (pr2 c t3 t) in +(let TMP_99 \def (\lambda (_: T).(let TMP_98 \def (Bind Abbr) in (eq K k0 +TMP_98))) in (let TMP_100 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_102 +\def (\lambda (t0: T).(let TMP_101 \def (clen c) in (subst0 TMP_101 u t0 t))) +in (let TMP_103 \def (ex3 T TMP_99 TMP_100 TMP_102) in (or TMP_97 +TMP_103))))))) in (let TMP_105 \def (pr2 c t3 t) in (let TMP_108 \def +(\lambda (_: T).(let TMP_106 \def (Bind Abbr) in (let TMP_107 \def (Bind +Abbr) in (eq K TMP_106 TMP_107)))) in (let TMP_109 \def (\lambda (t0: T).(pr0 +t3 t0)) in (let TMP_111 \def (\lambda (t0: T).(let TMP_110 \def (clen c) in +(subst0 TMP_110 u t0 t))) in (let TMP_112 \def (ex3 T TMP_108 TMP_109 +TMP_111) in (let TMP_115 \def (\lambda (_: T).(let TMP_113 \def (Bind Abbr) +in (let TMP_114 \def (Bind Abbr) in (eq K TMP_113 TMP_114)))) in (let TMP_116 +\def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_118 \def (\lambda (t0: T).(let +TMP_117 \def (clen c) in (subst0 TMP_117 u t0 t))) in (let TMP_119 \def (Bind +Abbr) in (let TMP_120 \def (refl_equal K TMP_119) in (let TMP_121 \def +(ex3_intro T TMP_115 TMP_116 TMP_118 t4 TMP_120 H2 H13) in (let TMP_122 \def +(or_intror TMP_105 TMP_112 TMP_121) in (eq_ind_r K TMP_96 TMP_104 TMP_122 k +H9))))))))))))))))))))))))) in (ex4_ind nat TMP_78 TMP_80 TMP_81 TMP_83 +TMP_91 TMP_123 H7))))))))))))) in (or_ind TMP_37 TMP_45 TMP_53 TMP_76 TMP_124 +H6))))))))))))))))))))))))))))))))) in (pr2_ind TMP_18 TMP_27 TMP_125 y t1 t2 +H0)))))) in (insert_eq C TMP_1 TMP_2 TMP_10 TMP_126 H)))))))))). theorem pr2_gen_cbind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall @@ -85,77 +129,119 @@ theorem pr2_gen_cbind: (THead (Bind b) v t2))))))) \def \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c -(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead -(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y -t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v -t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead -(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 -(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) -v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d -(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S -j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead -(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) -(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead -d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \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) u) (CHead c (Bind b) -v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) -H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def -(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def -(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr -(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) -(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v -(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: -(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c -(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) -(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) -v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S -x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal -nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i -(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t -(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead -c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t -H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1085 -END *) +(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let TMP_1 \def (Bind +b) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0: +C).(pr2 c0 t1 t2)) in (let TMP_8 \def (\lambda (_: C).(let TMP_4 \def (Bind +b) in (let TMP_5 \def (THead TMP_4 v t1) in (let TMP_6 \def (Bind b) in (let +TMP_7 \def (THead TMP_6 v t2) in (pr2 c TMP_5 TMP_7)))))) in (let TMP_119 +\def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_13 \def (\lambda +(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v)) \to +(let TMP_9 \def (Bind b) in (let TMP_10 \def (THead TMP_9 v t) in (let TMP_11 +\def (Bind b) in (let TMP_12 \def (THead TMP_11 v t0) in (pr2 c TMP_10 +TMP_12))))))))) in (let TMP_21 \def (\lambda (c0: C).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c +(Bind b) v))).(let TMP_14 \def (Bind b) in (let TMP_15 \def (THead TMP_14 v +t3) in (let TMP_16 \def (Bind b) in (let TMP_17 \def (THead TMP_16 v t4) in +(let TMP_18 \def (pr0_refl v) in (let TMP_19 \def (Bind b) in (let TMP_20 +\def (pr0_comp v v TMP_18 t3 t4 H1 TMP_19) in (pr2_free c TMP_15 TMP_17 +TMP_20))))))))))))) in (let TMP_118 \def (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4: (eq C c0 +(CHead c (Bind b) v))).(let TMP_24 \def (\lambda (c1: C).(let TMP_22 \def +(Bind Abbr) in (let TMP_23 \def (CHead d TMP_22 u) in (getl i c1 TMP_23)))) +in (let TMP_25 \def (Bind b) in (let TMP_26 \def (CHead c TMP_25 v) in (let +H5 \def (eq_ind C c0 TMP_24 H1 TMP_26 H4) in (let TMP_27 \def (Bind Abbr) in +(let TMP_28 \def (CHead d TMP_27 u) in (let H_x \def (getl_gen_bind b c +TMP_28 v i H5) in (let H6 \def H_x in (let TMP_29 \def (eq nat i O) in (let +TMP_30 \def (Bind Abbr) in (let TMP_31 \def (CHead d TMP_30 u) in (let TMP_32 +\def (Bind b) in (let TMP_33 \def (CHead c TMP_32 v) in (let TMP_34 \def (eq +C TMP_31 TMP_33) in (let TMP_35 \def (land TMP_29 TMP_34) in (let TMP_37 \def +(\lambda (j: nat).(let TMP_36 \def (S j) in (eq nat i TMP_36))) in (let +TMP_40 \def (\lambda (j: nat).(let TMP_38 \def (Bind Abbr) in (let TMP_39 +\def (CHead d TMP_38 u) in (getl j c TMP_39)))) in (let TMP_41 \def (ex2 nat +TMP_37 TMP_40) in (let TMP_42 \def (Bind b) in (let TMP_43 \def (THead TMP_42 +v t3) in (let TMP_44 \def (Bind b) in (let TMP_45 \def (THead TMP_44 v t) in +(let TMP_46 \def (pr2 c TMP_43 TMP_45) in (let TMP_90 \def (\lambda (H7: +(land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(let +TMP_47 \def (eq nat i O) in (let TMP_48 \def (Bind Abbr) in (let TMP_49 \def +(CHead d TMP_48 u) in (let TMP_50 \def (Bind b) in (let TMP_51 \def (CHead c +TMP_50 v) in (let TMP_52 \def (eq C TMP_49 TMP_51) in (let TMP_53 \def (Bind +b) in (let TMP_54 \def (THead TMP_53 v t3) in (let TMP_55 \def (Bind b) in +(let TMP_56 \def (THead TMP_55 v t) in (let TMP_57 \def (pr2 c TMP_54 TMP_56) +in (let TMP_89 \def (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead d +(Bind Abbr) u) (CHead c (Bind b) v))).(let TMP_58 \def (\lambda (e: C).(match +e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) in (let +TMP_59 \def (Bind Abbr) in (let TMP_60 \def (CHead d TMP_59 u) in (let TMP_61 +\def (Bind b) in (let TMP_62 \def (CHead c TMP_61 v) in (let H10 \def +(f_equal C C TMP_58 TMP_60 TMP_62 H9) in (let TMP_63 \def (\lambda (e: +C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow +(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) in +(let TMP_64 \def (Bind Abbr) in (let TMP_65 \def (CHead d TMP_64 u) in (let +TMP_66 \def (Bind b) in (let TMP_67 \def (CHead c TMP_66 v) in (let H11 \def +(f_equal C B TMP_63 TMP_65 TMP_67 H9) in (let TMP_68 \def (\lambda (e: +C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +in (let TMP_69 \def (Bind Abbr) in (let TMP_70 \def (CHead d TMP_69 u) in +(let TMP_71 \def (Bind b) in (let TMP_72 \def (CHead c TMP_71 v) in (let H12 +\def (f_equal C T TMP_68 TMP_70 TMP_72 H9) in (let TMP_87 \def (\lambda (H13: +(eq B Abbr b)).(\lambda (_: (eq C d c)).(let TMP_73 \def (\lambda (n: +nat).(subst0 n u t4 t)) in (let H15 \def (eq_ind nat i TMP_73 H3 O H8) in +(let TMP_74 \def (\lambda (t0: T).(subst0 O t0 t4 t)) in (let H16 \def +(eq_ind T u TMP_74 H15 v H12) in (let TMP_79 \def (\lambda (b0: B).(let +TMP_75 \def (Bind b0) in (let TMP_76 \def (THead TMP_75 v t3) in (let TMP_77 +\def (Bind b0) in (let TMP_78 \def (THead TMP_77 v t) in (pr2 c TMP_76 +TMP_78)))))) in (let TMP_80 \def (Bind Abbr) in (let TMP_81 \def (THead +TMP_80 v t3) in (let TMP_82 \def (Bind Abbr) in (let TMP_83 \def (THead +TMP_82 v t) in (let TMP_84 \def (pr0_refl v) in (let TMP_85 \def (pr0_delta v +v TMP_84 t3 t4 H2 t H16) in (let TMP_86 \def (pr2_free c TMP_81 TMP_83 +TMP_85) in (eq_ind B Abbr TMP_79 TMP_86 b H13))))))))))))))) in (let TMP_88 +\def (TMP_87 H11) in (TMP_88 H10))))))))))))))))))))))) in (land_ind TMP_47 +TMP_52 TMP_57 TMP_89 H7)))))))))))))) in (let TMP_117 \def (\lambda (H7: (ex2 +nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d +(Bind Abbr) u))))).(let TMP_92 \def (\lambda (j: nat).(let TMP_91 \def (S j) +in (eq nat i TMP_91))) in (let TMP_95 \def (\lambda (j: nat).(let TMP_93 \def +(Bind Abbr) in (let TMP_94 \def (CHead d TMP_93 u) in (getl j c TMP_94)))) in +(let TMP_96 \def (Bind b) in (let TMP_97 \def (THead TMP_96 v t3) in (let +TMP_98 \def (Bind b) in (let TMP_99 \def (THead TMP_98 v t) in (let TMP_100 +\def (pr2 c TMP_97 TMP_99) in (let TMP_116 \def (\lambda (x: nat).(\lambda +(H8: (eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let +TMP_101 \def (\lambda (e: nat).e) in (let TMP_102 \def (S x) in (let H10 \def +(f_equal nat nat TMP_101 i TMP_102 H8) in (let TMP_103 \def (\lambda (n: +nat).(subst0 n u t4 t)) in (let TMP_104 \def (S x) in (let H11 \def (eq_ind +nat i TMP_103 H3 TMP_104 H10) in (let TMP_105 \def (Bind b) in (let TMP_106 +\def (Bind b) in (let TMP_107 \def (CHead c TMP_106 v) in (let TMP_108 \def +(S x) in (let TMP_109 \def (Bind b) in (let TMP_110 \def (CHead c TMP_109 v) +in (let TMP_111 \def (clear_bind b c v) in (let TMP_112 \def (Bind Abbr) in +(let TMP_113 \def (CHead d TMP_112 u) in (let TMP_114 \def (getl_clear_bind b +TMP_110 c v TMP_111 TMP_113 x H9) in (let TMP_115 \def (pr2_delta TMP_107 d u +TMP_108 TMP_114 t3 t4 H2 t H11) in (pr2_head_2 c v t3 t TMP_105 +TMP_115))))))))))))))))))))) in (ex2_ind nat TMP_92 TMP_95 TMP_100 TMP_116 +H7)))))))))) in (or_ind TMP_35 TMP_41 TMP_46 TMP_90 TMP_117 +H6))))))))))))))))))))))))))))))))))))) in (pr2_ind TMP_13 TMP_21 TMP_118 y +t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3 TMP_8 TMP_119 H))))))))))). theorem pr2_gen_cflat: \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) \def \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c -(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) -(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 -c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 -H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) -v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) -H)))))). -(* COMMENTS -Initial nodes: 293 -END *) +(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let TMP_1 \def (Flat +f) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0: +C).(pr2 c0 t1 t2)) in (let TMP_4 \def (\lambda (_: C).(pr2 c t1 t2)) in (let +TMP_15 \def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_5 \def +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) +v)) \to (pr2 c t t0))))) in (let TMP_6 \def (\lambda (c0: C).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c +(Flat f) v))).(pr2_free c t3 t4 H1)))))) in (let TMP_14 \def (\lambda (c0: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 +(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: +(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4: +(eq C c0 (CHead c (Flat f) v))).(let TMP_9 \def (\lambda (c1: C).(let TMP_7 +\def (Bind Abbr) in (let TMP_8 \def (CHead d TMP_7 u) in (getl i c1 TMP_8)))) +in (let TMP_10 \def (Flat f) in (let TMP_11 \def (CHead c TMP_10 v) in (let +H5 \def (eq_ind C c0 TMP_9 H1 TMP_11 H4) in (let TMP_12 \def (Bind Abbr) in +(let TMP_13 \def (CHead d TMP_12 u) in (let H_y \def (getl_gen_flat f c +TMP_13 v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3))))))))))))))))))) in +(pr2_ind TMP_5 TMP_6 TMP_14 y t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3 +TMP_4 TMP_15 H))))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma index e21568232..1bcb0b108 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr0/defs.ma". +include "basic_1/pr0/defs.ma". -include "Basic-1/getl/defs.ma". +include "basic_1/getl/defs.ma". inductive pr2: C \to (T \to (T \to Prop)) \def | pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma index 1a18f7dbc..7694da8fb 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma @@ -14,13 +14,31 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/defs.ma". +include "basic_1/pr2/defs.ma". -include "Basic-1/pr0/fwd.ma". +include "basic_1/pr0/fwd.ma". -include "Basic-1/getl/drop.ma". +include "basic_1/getl/clear.ma". -include "Basic-1/getl/clear.ma". +include "basic_1/getl/drop.ma". + +theorem pr2_ind: + \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c: +C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d +(Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to +(\forall (t: T).((subst0 i u t2 t) \to (P c t1 t)))))))))))) \to (\forall (c: +C).(\forall (t: T).(\forall (t0: T).((pr2 c t t0) \to (P c t t0))))))) +\def + \lambda (P: ((C \to (T \to (T \to Prop))))).(\lambda (f: ((\forall (c: +C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 +t2))))))).(\lambda (f0: ((\forall (c: C).(\forall (d: C).(\forall (u: +T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to +(P c t1 t))))))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: +T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2) +\Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8) +\Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))). theorem pr2_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to @@ -43,9 +61,6 @@ t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) (let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) (pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 347 -END *) theorem pr2_gen_lref: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to @@ -96,9 +111,6 @@ u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) (subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 1003 -END *) theorem pr2_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c @@ -221,9 +233,6 @@ H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) (pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2383 -END *) theorem pr2_gen_cast: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c @@ -356,9 +365,6 @@ t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) (pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2659 -END *) theorem pr2_gen_csort: \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) @@ -376,9 +382,6 @@ t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c (CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d (Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))). -(* COMMENTS -Initial nodes: 221 -END *) theorem pr2_gen_appl: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c @@ -1747,17 +1750,16 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 -(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u -x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind -x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b @@ -1771,131 +1773,179 @@ T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O -x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22 -\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u -x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +y2) z1 z2))))))))) (\lambda (H20: (lt (s (Bind x0) i) (S +O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind +x0) i)) H20 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) +(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) +(S O) O H19 Hle) in (let H20 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 +(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) -O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H21: (eq T x7 (lift (S O) O +x8))).(\lambda (H22: (subst0 (minus i O) u x3 x8)).(eq_ind_r T (lift (S O) O +x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H23 \def (eq_ind_r +nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x8)) H22 i (minus_n_O i)) in +(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift +(S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) +(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat +Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) +y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) +(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))) +(pr2_delta c0 d u i H1 u1 x3 H10 x8 H23) (pr2_free c0 x1 x4 H11) (pr2_free +(CHead c0 (Bind x0) x4) x2 x5 H12)))) x7 H21)))) H20))))) x6 H18)))) H17)) +(\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S +O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O +x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 +x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4 -H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20))))) -(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n -i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T +x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: (subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O +x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: +T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda +(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda -(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: -(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat -Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: 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x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead -(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat +Appl) (lift (S O) O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: @@ -1981,9 +2031,32 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) +(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S +O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind +x0) i)) H21 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: @@ -2005,115 +2078,139 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda -(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) -i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) -(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T -(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) -O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) +(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) +(S O) O H19 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 +(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4 -H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead -c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 -x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S -O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat -Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) -(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) -(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t -(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: -(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x7 (lift (S O) O +x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O +x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r +nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in +(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift +(S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) +(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat +Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) +(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8))) +(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_free c0 x1 x4 H11) (pr2_delta +(CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) +c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 +H20)))) x7 H22)))) H21))))) x6 H18)))))) H17)) (subst0_gen_head (Flat Appl) u +(lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) (\lambda (H14: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind x0) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) +O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat +Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) +x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t +(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: +(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: @@ -2204,17 +2301,16 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 -(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u -x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind -x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b @@ -2228,131 +2324,180 @@ T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O -x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23 -\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u -x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S +O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind +x0) i)) H21 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) +(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) +(S O) O H20 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 +(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift -(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u -i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 -H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i -(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T -x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 -(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x8 (lift (S O) O +x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O +x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) +(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r +nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in +(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift +(S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift -(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 -x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) -t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) +(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat +Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) +(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))) +(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_delta c0 d u i H1 x1 x4 H11 x6 +H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12)))) x8 H22)))) H21))))) x7 +H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat +Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s +(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat +Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s +(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat -Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda +(H19: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x8))).(\lambda (H20: +(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat +Appl) (lift (S O) O x3) x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead +(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: @@ -2438,9 +2583,32 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) +(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (H22: (lt (s (Bind x0) i) (S +O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind +x0) i)) H22 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: @@ -2462,37 +2630,62 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda -(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind -x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S -O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in -(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) +(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) +(S O) O H20 Hle) in (let H22 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 +(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind +x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (x10: T).(\lambda (H23: (eq T x8 (lift (S O) O +x10))).(\lambda (H24: (subst0 (minus i O) u x3 x10)).(eq_ind_r T (lift (S O) +O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq +T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H25 \def (eq_ind_r +nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x10)) H24 i (minus_n_O i)) +in (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) @@ -2526,746 +2719,14 @@ T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) -(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta +(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H25) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) -(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n -i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 -x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 -(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) +(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21)))) x8 H23)))) H22))))) x7 +H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s +(Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 (THead +(Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) (pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 38859 -END *) - -theorem pr2_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T -T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) -(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 -t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) -u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind -Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 -(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) -(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O -t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in -(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead -c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 -t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) -u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr) -x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 -x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda -(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 -T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c0 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(pr2_delta c0 d u i H1 u1 x0 H8 x3 -H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) -t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) -(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O -(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O -H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda -(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 -O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 -(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) -u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl -Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S -i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl -x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda -(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 -x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 -(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 -u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O -(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) -u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 -x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) -H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) -H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) -H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) -(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 11671 -END *) - -theorem pr2_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift -(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 -t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind -Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) -t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) -(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 -x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 -H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 -(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t -(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind -Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 -x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 -x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda -(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: -T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 -(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t -(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: -(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) -(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S -O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 -t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3467 -END *) theorem pr2_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall @@ -3337,7 +2798,4 @@ t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 t2 h d H6)))))))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1579 -END *) diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma index f8df4e9e2..406d8bcd2 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma @@ -14,11 +14,11 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/defs.ma". +include "basic_1/pr2/defs.ma". -include "Basic-1/pr0/pr0.ma". +include "basic_1/pr0/pr0.ma". -include "Basic-1/getl/props.ma". +include "basic_1/getl/fwd.ma". theorem pr2_confluence__pr2_free_free: \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 @@ -26,15 +26,16 @@ t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))) \def \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 -t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 -x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) -(pr0_confluence t0 t2 H0 t1 H))))))). -(* COMMENTS -Initial nodes: 135 -END *) +(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(let TMP_1 \def (\lambda (t: +T).(pr0 t2 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3 +\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c +t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_10 \def (\lambda +(x: T).(\lambda (H1: (pr0 t2 x)).(\lambda (H2: (pr0 t1 x)).(let TMP_6 \def +(\lambda (t: T).(pr2 c t1 t)) in (let TMP_7 \def (\lambda (t: T).(pr2 c t2 +t)) in (let TMP_8 \def (pr2_free c t1 x H2) in (let TMP_9 \def (pr2_free c t2 +x H1) in (ex_intro2 T TMP_6 TMP_7 x TMP_8 TMP_9)))))))) in (let TMP_11 \def +(pr0_confluence t0 t2 H0 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_10 +TMP_11))))))))))))). theorem pr2_confluence__pr2_free_delta: \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall @@ -46,24 +47,33 @@ t)))))))))))))) \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 -t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 -t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: -(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 -c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 -x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: -T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: -(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 -H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) -(pr0_confluence t0 t4 H1 t1 H))))))))))))). -(* COMMENTS -Initial nodes: 403 -END *) +t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(let TMP_1 \def (\lambda (t: +T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3 +\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c +t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_31 \def (\lambda +(x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: (pr0 t1 x)).(let TMP_6 \def +(pr0 t2 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_8 +\def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T TMP_7 +TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11 \def +(\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in +(let TMP_17 \def (\lambda (H5: (pr0 t2 x)).(let TMP_13 \def (\lambda (t: +T).(pr2 c t1 t)) in (let TMP_14 \def (\lambda (t: T).(pr2 c t2 t)) in (let +TMP_15 \def (pr2_free c t1 x H4) in (let TMP_16 \def (pr2_free c t2 x H5) in +(ex_intro2 T TMP_13 TMP_14 x TMP_15 TMP_16)))))) in (let TMP_28 \def (\lambda +(H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i u x +w2)))).(let TMP_18 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_19 \def +(\lambda (w2: T).(subst0 i u x w2)) in (let TMP_20 \def (\lambda (t: T).(pr2 +c t1 t)) in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22 +\def (ex2 T TMP_20 TMP_21) in (let TMP_27 \def (\lambda (x0: T).(\lambda (H6: +(pr0 t2 x0)).(\lambda (H7: (subst0 i u x x0)).(let TMP_23 \def (\lambda (t: +T).(pr2 c t1 t)) in (let TMP_24 \def (\lambda (t: T).(pr2 c t2 t)) in (let +TMP_25 \def (pr2_delta c d u i H0 t1 x H4 x0 H7) in (let TMP_26 \def +(pr2_free c t2 x0 H6) in (ex_intro2 T TMP_23 TMP_24 x0 TMP_25 TMP_26)))))))) +in (ex2_ind T TMP_18 TMP_19 TMP_22 TMP_27 H5)))))))) in (let TMP_29 \def +(pr0_refl u) in (let TMP_30 \def (pr0_subst0 t4 x H3 u t2 i H2 u TMP_29) in +(or_ind TMP_6 TMP_9 TMP_12 TMP_17 TMP_28 TMP_30))))))))))))))) in (let TMP_32 +\def (pr0_confluence t0 t4 H1 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_31 +TMP_32))))))))))))))))))). theorem pr2_confluence__pr2_delta_delta: \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall @@ -79,84 +89,143 @@ t)))))))))))))))))))) T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda -(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: -T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 -x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 -x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda -(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 -u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x -x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) -x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T -(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: -T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 -t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H -t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) -(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 -w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 -x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: -T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i -i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 -i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: -(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d -u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 -(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def -(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 -\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) -H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: -C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 -(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda -(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x -x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c -(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 -(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind -(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: -T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda -(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 -x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in -(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 -(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T -(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i -u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(let TMP_1 \def +(\lambda (t: T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t3 t)) in +(let TMP_3 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: +T).(pr2 c t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_165 \def +(\lambda (x: T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (pr0 t3 x)).(let +TMP_6 \def (pr0 t1 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t1 w2)) in +(let TMP_8 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T +TMP_7 TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11 +\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) +in (let TMP_38 \def (\lambda (H7: (pr0 t1 x)).(let TMP_13 \def (pr0 t2 x) in +(let TMP_14 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_15 \def (\lambda +(w2: T).(subst0 i0 u0 x w2)) in (let TMP_16 \def (ex2 T TMP_14 TMP_15) in +(let TMP_17 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_18 \def (\lambda +(t: T).(pr2 c t2 t)) in (let TMP_19 \def (ex2 T TMP_17 TMP_18) in (let TMP_24 +\def (\lambda (H8: (pr0 t2 x)).(let TMP_20 \def (\lambda (t: T).(pr2 c t1 t)) +in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22 \def +(pr2_free c t1 x H7) in (let TMP_23 \def (pr2_free c t2 x H8) in (ex_intro2 T +TMP_20 TMP_21 x TMP_22 TMP_23)))))) in (let TMP_35 \def (\lambda (H8: (ex2 T +(\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let +TMP_25 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_26 \def (\lambda (w2: +T).(subst0 i0 u0 x w2)) in (let TMP_27 \def (\lambda (t: T).(pr2 c t1 t)) in +(let TMP_28 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_29 \def (ex2 T +TMP_27 TMP_28) in (let TMP_34 \def (\lambda (x0: T).(\lambda (H9: (pr0 t2 +x0)).(\lambda (H10: (subst0 i0 u0 x x0)).(let TMP_30 \def (\lambda (t: +T).(pr2 c t1 t)) in (let TMP_31 \def (\lambda (t: T).(pr2 c t2 t)) in (let +TMP_32 \def (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) in (let TMP_33 \def +(pr2_free c t2 x0 H9) in (ex_intro2 T TMP_30 TMP_31 x0 TMP_32 TMP_33)))))))) +in (ex2_ind T TMP_25 TMP_26 TMP_29 TMP_34 H8)))))))) in (let TMP_36 \def +(pr0_refl u0) in (let TMP_37 \def (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 TMP_36) +in (or_ind TMP_13 TMP_16 TMP_19 TMP_24 TMP_35 TMP_37))))))))))))) in (let +TMP_162 \def (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: +T).(subst0 i u x w2)))).(let TMP_39 \def (\lambda (w2: T).(pr0 t1 w2)) in +(let TMP_40 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_41 \def +(\lambda (t: T).(pr2 c t1 t)) in (let TMP_42 \def (\lambda (t: T).(pr2 c t2 +t)) in (let TMP_43 \def (ex2 T TMP_41 TMP_42) in (let TMP_161 \def (\lambda +(x0: T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(let +TMP_44 \def (pr0 t2 x) in (let TMP_45 \def (\lambda (w2: T).(pr0 t2 w2)) in +(let TMP_46 \def (\lambda (w2: T).(subst0 i0 u0 x w2)) in (let TMP_47 \def +(ex2 T TMP_45 TMP_46) in (let TMP_48 \def (\lambda (t: T).(pr2 c t1 t)) in +(let TMP_49 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_50 \def (ex2 T +TMP_48 TMP_49) in (let TMP_55 \def (\lambda (H10: (pr0 t2 x)).(let TMP_51 +\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_52 \def (\lambda (t: T).(pr2 c +t2 t)) in (let TMP_53 \def (pr2_free c t1 x0 H8) in (let TMP_54 \def +(pr2_delta c d u i H t2 x H10 x0 H9) in (ex_intro2 T TMP_51 TMP_52 x0 TMP_53 +TMP_54)))))) in (let TMP_158 \def (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 +t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let TMP_56 \def (\lambda +(w2: T).(pr0 t2 w2)) in (let TMP_57 \def (\lambda (w2: T).(subst0 i0 u0 x +w2)) in (let TMP_58 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_59 \def +(\lambda (t: T).(pr2 c t2 t)) in (let TMP_60 \def (ex2 T TMP_58 TMP_59) in +(let TMP_157 \def (\lambda (x1: T).(\lambda (H11: (pr0 t2 x1)).(\lambda (H12: +(subst0 i0 u0 x x1)).(let TMP_61 \def (\lambda (t: T).(pr2 c t1 t)) in (let +TMP_62 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_63 \def (ex2 T TMP_61 +TMP_62) in (let TMP_76 \def (\lambda (H13: (not (eq nat i i0))).(let TMP_64 +\def (\lambda (t: T).(subst0 i u x1 t)) in (let TMP_65 \def (\lambda (t: +T).(subst0 i0 u0 x0 t)) in (let TMP_66 \def (\lambda (t: T).(pr2 c t1 t)) in +(let TMP_67 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_68 \def (ex2 T +TMP_66 TMP_67) in (let TMP_73 \def (\lambda (x2: T).(\lambda (H14: (subst0 i +u x1 x2)).(\lambda (H15: (subst0 i0 u0 x0 x2)).(let TMP_69 \def (\lambda (t: +T).(pr2 c t1 t)) in (let TMP_70 \def (\lambda (t: T).(pr2 c t2 t)) in (let +TMP_71 \def (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) in (let TMP_72 \def +(pr2_delta c d u i H t2 x1 H11 x2 H14) in (ex_intro2 T TMP_69 TMP_70 x2 +TMP_71 TMP_72)))))))) in (let TMP_74 \def (sym_not_eq nat i i0 H13) in (let +TMP_75 \def (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 TMP_74) in +(ex2_ind T TMP_64 TMP_65 TMP_68 TMP_73 TMP_75)))))))))) in (let TMP_156 \def +(\lambda (H13: (eq nat i i0)).(let TMP_77 \def (\lambda (n: nat).(subst0 n u0 +x x1)) in (let H14 \def (eq_ind_r nat i0 TMP_77 H12 i H13) in (let TMP_80 +\def (\lambda (n: nat).(let TMP_78 \def (Bind Abbr) in (let TMP_79 \def +(CHead d0 TMP_78 u0) in (getl n c TMP_79)))) in (let H15 \def (eq_ind_r nat +i0 TMP_80 H2 i H13) in (let TMP_81 \def (Bind Abbr) in (let TMP_82 \def +(CHead d TMP_81 u) in (let TMP_83 \def (\lambda (c0: C).(getl i c c0)) in +(let TMP_84 \def (Bind Abbr) in (let TMP_85 \def (CHead d0 TMP_84 u0) in (let +TMP_86 \def (Bind Abbr) in (let TMP_87 \def (CHead d TMP_86 u) in (let TMP_88 +\def (Bind Abbr) in (let TMP_89 \def (CHead d0 TMP_88 u0) in (let TMP_90 \def +(getl_mono c TMP_87 i H TMP_89 H15) in (let H16 \def (eq_ind C TMP_82 TMP_83 +H TMP_85 TMP_90) in (let TMP_91 \def (\lambda (e: C).(match e with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) in (let TMP_92 \def (Bind +Abbr) in (let TMP_93 \def (CHead d TMP_92 u) in (let TMP_94 \def (Bind Abbr) +in (let TMP_95 \def (CHead d0 TMP_94 u0) in (let TMP_96 \def (Bind Abbr) in +(let TMP_97 \def (CHead d TMP_96 u) in (let TMP_98 \def (Bind Abbr) in (let +TMP_99 \def (CHead d0 TMP_98 u0) in (let TMP_100 \def (getl_mono c TMP_97 i H +TMP_99 H15) in (let H17 \def (f_equal C C TMP_91 TMP_93 TMP_95 TMP_100) in +(let TMP_101 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) in (let TMP_102 \def (Bind Abbr) in (let +TMP_103 \def (CHead d TMP_102 u) in (let TMP_104 \def (Bind Abbr) in (let +TMP_105 \def (CHead d0 TMP_104 u0) in (let TMP_106 \def (Bind Abbr) in (let +TMP_107 \def (CHead d TMP_106 u) in (let TMP_108 \def (Bind Abbr) in (let +TMP_109 \def (CHead d0 TMP_108 u0) in (let TMP_110 \def (getl_mono c TMP_107 +i H TMP_109 H15) in (let H18 \def (f_equal C T TMP_101 TMP_103 TMP_105 +TMP_110) in (let TMP_155 \def (\lambda (H19: (eq C d d0)).(let TMP_111 \def +(\lambda (t: T).(subst0 i t x x1)) in (let H20 \def (eq_ind_r T u0 TMP_111 +H14 u H18) in (let TMP_114 \def (\lambda (t: T).(let TMP_112 \def (Bind Abbr) +in (let TMP_113 \def (CHead d0 TMP_112 t) in (getl i c TMP_113)))) in (let +H21 \def (eq_ind_r T u0 TMP_114 H16 u H18) in (let TMP_117 \def (\lambda (c0: +C).(let TMP_115 \def (Bind Abbr) in (let TMP_116 \def (CHead c0 TMP_115 u) in +(getl i c TMP_116)))) in (let H22 \def (eq_ind_r C d0 TMP_117 H21 d H19) in +(let TMP_118 \def (eq T x1 x0) in (let TMP_119 \def (\lambda (t: T).(subst0 i +u x1 t)) in (let TMP_120 \def (\lambda (t: T).(subst0 i u x0 t)) in (let +TMP_121 \def (ex2 T TMP_119 TMP_120) in (let TMP_122 \def (subst0 i u x1 x0) +in (let TMP_123 \def (subst0 i u x0 x1) in (let TMP_124 \def (\lambda (t: +T).(pr2 c t1 t)) in (let TMP_125 \def (\lambda (t: T).(pr2 c t2 t)) in (let +TMP_126 \def (ex2 T TMP_124 TMP_125) in (let TMP_132 \def (\lambda (H23: (eq +T x1 x0)).(let TMP_127 \def (\lambda (t: T).(pr0 t2 t)) in (let H24 \def +(eq_ind T x1 TMP_127 H11 x0 H23) in (let TMP_128 \def (\lambda (t: T).(pr2 c +t1 t)) in (let TMP_129 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_130 +\def (pr2_free c t1 x0 H8) in (let TMP_131 \def (pr2_free c t2 x0 H24) in +(ex_intro2 T TMP_128 TMP_129 x0 TMP_130 TMP_131)))))))) in (let TMP_143 \def +(\lambda (H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: +T).(subst0 i u x0 t)))).(let TMP_133 \def (\lambda (t: T).(subst0 i u x1 t)) +in (let TMP_134 \def (\lambda (t: T).(subst0 i u x0 t)) in (let TMP_135 \def +(\lambda (t: T).(pr2 c t1 t)) in (let TMP_136 \def (\lambda (t: T).(pr2 c t2 +t)) in (let TMP_137 \def (ex2 T TMP_135 TMP_136) in (let TMP_142 \def (\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i -u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c -t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 -x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 -x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u -x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) -(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x -H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 -H0))))))))))))))))))). -(* COMMENTS -Initial nodes: 1901 -END *) +u x0 x2)).(let TMP_138 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_139 +\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_140 \def (pr2_delta c d u i +H22 t1 x0 H8 x2 H25) in (let TMP_141 \def (pr2_delta c d u i H22 t2 x1 H11 x2 +H24) in (ex_intro2 T TMP_138 TMP_139 x2 TMP_140 TMP_141)))))))) in (ex2_ind T +TMP_133 TMP_134 TMP_137 TMP_142 H23)))))))) in (let TMP_148 \def (\lambda +(H23: (subst0 i u x1 x0)).(let TMP_144 \def (\lambda (t: T).(pr2 c t1 t)) in +(let TMP_145 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_146 \def +(pr2_free c t1 x0 H8) in (let TMP_147 \def (pr2_delta c d u i H22 t2 x1 H11 +x0 H23) in (ex_intro2 T TMP_144 TMP_145 x0 TMP_146 TMP_147)))))) in (let +TMP_153 \def (\lambda (H23: (subst0 i u x0 x1)).(let TMP_149 \def (\lambda +(t: T).(pr2 c t1 t)) in (let TMP_150 \def (\lambda (t: T).(pr2 c t2 t)) in +(let TMP_151 \def (pr2_delta c d u i H22 t1 x0 H8 x1 H23) in (let TMP_152 +\def (pr2_free c t2 x1 H11) in (ex_intro2 T TMP_149 TMP_150 x1 TMP_151 +TMP_152)))))) in (let TMP_154 \def (subst0_confluence_eq x x1 u i H20 x0 H9) +in (or4_ind TMP_118 TMP_121 TMP_122 TMP_123 TMP_126 TMP_132 TMP_143 TMP_148 +TMP_153 TMP_154)))))))))))))))))))))) in (TMP_155 +H17)))))))))))))))))))))))))))))))))))))))) in (neq_eq_e i i0 TMP_63 TMP_76 +TMP_156))))))))) in (ex2_ind T TMP_56 TMP_57 TMP_60 TMP_157 H10)))))))) in +(let TMP_159 \def (pr0_refl u0) in (let TMP_160 \def (pr0_subst0 t4 x H5 u0 +t2 i0 H4 u0 TMP_159) in (or_ind TMP_44 TMP_47 TMP_50 TMP_55 TMP_158 +TMP_160))))))))))))))) in (ex2_ind T TMP_39 TMP_40 TMP_43 TMP_161 H7)))))))) +in (let TMP_163 \def (pr0_refl u) in (let TMP_164 \def (pr0_subst0 t3 x H6 u +t1 i H1 u TMP_163) in (or_ind TMP_6 TMP_9 TMP_12 TMP_38 TMP_162 +TMP_164))))))))))))))) in (let TMP_166 \def (pr0_confluence t0 t4 H3 t3 H0) +in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_165 TMP_166))))))))))))))))))))))))). theorem pr2_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall @@ -164,95 +233,120 @@ theorem pr2_confluence: T).(pr2 c t2 t)))))))) \def \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in -pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: -(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T -(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with +t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H with [(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: -(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T -t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c -t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind -T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 -t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 -t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: -T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) -\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: -T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda -(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 -t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 -t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: -T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 -t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 -H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | -(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 -c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c -(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d -(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda -(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: -(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c -(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T -(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) -(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda -(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: -(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda -(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i -H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 -(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T -t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 -H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 -t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) -\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) -\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda -(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 -(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to -((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 -t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind -T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) -\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda -(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 -\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: -T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 -t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 -t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C -c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c -(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda -(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 -t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 -t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) -(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 -(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 -H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow -(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T -t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to -((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 -t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T -t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to -((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 -(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to -((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) -u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 +(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(let TMP_54 \def (\lambda (_: +C).((eq T t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (let TMP_52 \def +(\lambda (t: T).(pr2 c t1 t)) in (let TMP_53 \def (\lambda (t: T).(pr2 c t2 +t)) in (ex2 T TMP_52 TMP_53))))))) in (let TMP_98 \def (\lambda (H5: (eq T t3 +t0)).(let TMP_57 \def (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (let +TMP_55 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_56 \def (\lambda (t5: +T).(pr2 c t2 t5)) in (ex2 T TMP_55 TMP_56)))))) in (let TMP_96 \def (\lambda +(H6: (eq T t4 t1)).(let TMP_60 \def (\lambda (t: T).((pr0 t0 t) \to (let +TMP_58 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_59 \def (\lambda (t5: +T).(pr2 c t2 t5)) in (ex2 T TMP_58 TMP_59))))) in (let TMP_94 \def (\lambda +(H7: (pr0 t0 t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8) +\Rightarrow (\lambda (H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda +(H11: (eq T t6 t2)).(let TMP_78 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T +t6 t2) \to ((pr0 t5 t6) \to (let TMP_76 \def (\lambda (t: T).(pr2 c t1 t)) in +(let TMP_77 \def (\lambda (t: T).(pr2 c t2 t)) in (ex2 T TMP_76 TMP_77))))))) +in (let TMP_89 \def (\lambda (H12: (eq T t5 t0)).(let TMP_81 \def (\lambda +(t: T).((eq T t6 t2) \to ((pr0 t t6) \to (let TMP_79 \def (\lambda (t7: +T).(pr2 c t1 t7)) in (let TMP_80 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 +T TMP_79 TMP_80)))))) in (let TMP_87 \def (\lambda (H13: (eq T t6 t2)).(let +TMP_84 \def (\lambda (t: T).((pr0 t0 t) \to (let TMP_82 \def (\lambda (t7: +T).(pr2 c t1 t7)) in (let TMP_83 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 +T TMP_82 TMP_83))))) in (let TMP_85 \def (\lambda (H14: (pr0 t0 +t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) in (let TMP_86 \def +(sym_eq T t6 t2 H13) in (eq_ind T t2 TMP_84 TMP_85 t6 TMP_86))))) in (let +TMP_88 \def (sym_eq T t5 t0 H12) in (eq_ind T t0 TMP_81 TMP_87 t5 TMP_88))))) +in (let TMP_90 \def (sym_eq C c1 c H9) in (eq_ind C c TMP_78 TMP_89 c1 TMP_90 +H10 H11 H8))))))) | (pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow +(\lambda (H11: (eq C c1 c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T +t t2)).(let TMP_63 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to +((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) +\to (let TMP_61 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let TMP_62 \def +(\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_61 TMP_62))))))))) in (let +TMP_74 \def (\lambda (H14: (eq T t5 t0)).(let TMP_66 \def (\lambda (t7: +T).((eq T t t2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to +((subst0 i u t6 t) \to (let TMP_64 \def (\lambda (t8: T).(pr2 c t1 t8)) in +(let TMP_65 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_64 +TMP_65)))))))) in (let TMP_72 \def (\lambda (H15: (eq T t t2)).(let TMP_69 +\def (\lambda (t7: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) +\to ((subst0 i u t6 t7) \to (let TMP_67 \def (\lambda (t8: T).(pr2 c t1 t8)) +in (let TMP_68 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_67 +TMP_68))))))) in (let TMP_70 \def (\lambda (H16: (getl i c (CHead d (Bind +Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda (H18: (subst0 i u t6 +t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i H7 H16 H17 H18)))) +in (let TMP_71 \def (sym_eq T t t2 H15) in (eq_ind T t2 TMP_69 TMP_70 t +TMP_71))))) in (let TMP_73 \def (sym_eq T t5 t0 H14) in (eq_ind T t0 TMP_66 +TMP_72 t5 TMP_73))))) in (let TMP_75 \def (sym_eq C c1 c H11) in (eq_ind C c +TMP_63 TMP_74 c1 TMP_75 H12 H13 H8 H9 H10)))))))]) in (let TMP_91 \def +(refl_equal C c) in (let TMP_92 \def (refl_equal T t0) in (let TMP_93 \def +(refl_equal T t2) in (H8 TMP_91 TMP_92 TMP_93)))))) in (let TMP_95 \def +(sym_eq T t4 t1 H6) in (eq_ind T t1 TMP_60 TMP_94 t4 TMP_95))))) in (let +TMP_97 \def (sym_eq T t3 t0 H5) in (eq_ind T t0 TMP_57 TMP_96 t3 TMP_97))))) +in (let TMP_99 \def (sym_eq C c0 c H2) in (eq_ind C c TMP_54 TMP_98 c0 TMP_99 +H3 H4 H1))))))) | (pr2_delta c0 d u i H1 t3 t4 H2 t H3) \Rightarrow (\lambda +(H4: (eq C c0 c)).(\lambda (H5: (eq T t3 t0)).(\lambda (H6: (eq T t t1)).(let +TMP_3 \def (\lambda (c1: C).((eq T t3 t0) \to ((eq T t t1) \to ((getl i c1 +(CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) \to ((subst0 i u t4 t) \to (let +TMP_1 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_2 \def (\lambda (t5: +T).(pr2 c t2 t5)) in (ex2 T TMP_1 TMP_2))))))))) in (let TMP_50 \def (\lambda +(H7: (eq T t3 t0)).(let TMP_6 \def (\lambda (t5: T).((eq T t t1) \to ((getl i +c (CHead d (Bind Abbr) u)) \to ((pr0 t5 t4) \to ((subst0 i u t4 t) \to (let +TMP_4 \def (\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_5 \def (\lambda (t6: +T).(pr2 c t2 t6)) in (ex2 T TMP_4 TMP_5)))))))) in (let TMP_48 \def (\lambda +(H8: (eq T t t1)).(let TMP_9 \def (\lambda (t5: T).((getl i c (CHead d (Bind +Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t5) \to (let TMP_7 \def +(\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_8 \def (\lambda (t6: T).(pr2 c t2 +t6)) in (ex2 T TMP_7 TMP_8))))))) in (let TMP_46 \def (\lambda (H9: (getl i c +(CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 +i u t4 t1)).(let H12 \def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow +(\lambda (H13: (eq C c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T +t6 t2)).(let TMP_27 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to +((pr0 t5 t6) \to (let TMP_25 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let +TMP_26 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_25 TMP_26))))))) in +(let TMP_41 \def (\lambda (H16: (eq T t5 t0)).(let TMP_30 \def (\lambda (t7: +T).((eq T t6 t2) \to ((pr0 t7 t6) \to (let TMP_28 \def (\lambda (t8: T).(pr2 +c t1 t8)) in (let TMP_29 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T +TMP_28 TMP_29)))))) in (let TMP_39 \def (\lambda (H17: (eq T t6 t2)).(let +TMP_33 \def (\lambda (t7: T).((pr0 t0 t7) \to (let TMP_31 \def (\lambda (t8: +T).(pr2 c t1 t8)) in (let TMP_32 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 +T TMP_31 TMP_32))))) in (let TMP_37 \def (\lambda (H18: (pr0 t0 t2)).(let +TMP_34 \def (pr2 c t2) in (let TMP_35 \def (pr2 c t1) in (let TMP_36 \def +(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11) in +(ex2_sym T TMP_34 TMP_35 TMP_36))))) in (let TMP_38 \def (sym_eq T t6 t2 H17) +in (eq_ind T t2 TMP_33 TMP_37 t6 TMP_38))))) in (let TMP_40 \def (sym_eq T t5 +t0 H16) in (eq_ind T t0 TMP_30 TMP_39 t5 TMP_40))))) in (let TMP_42 \def +(sym_eq C c1 c H13) in (eq_ind C c TMP_27 TMP_41 c1 TMP_42 H14 H15 H12))))))) +| (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow (\lambda (H15: (eq +C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T t7 t2)).(let +TMP_12 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl i0 c2 +(CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) \to +(let TMP_10 \def (\lambda (t8: T).(pr2 c t1 t8)) in (let TMP_11 \def (\lambda +(t8: T).(pr2 c t2 t8)) in (ex2 T TMP_10 TMP_11))))))))) in (let TMP_23 \def +(\lambda (H18: (eq T t5 t0)).(let TMP_15 \def (\lambda (t8: T).((eq T t7 t2) +\to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to ((subst0 i0 +u0 t6 t7) \to (let TMP_13 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_14 +\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_13 TMP_14)))))))) in (let +TMP_21 \def (\lambda (H19: (eq T t7 t2)).(let TMP_18 \def (\lambda (t8: +T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 +t6 t8) \to (let TMP_16 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_17 +\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_16 TMP_17))))))) in (let +TMP_19 \def (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda +(H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 -H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 -(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T -t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C -c) (refl_equal T t0) (refl_equal T t1)))))))). -(* COMMENTS -Initial nodes: 2087 -END *) +H11 H20 H21 H22)))) in (let TMP_20 \def (sym_eq T t7 t2 H19) in (eq_ind T t2 +TMP_18 TMP_19 t7 TMP_20))))) in (let TMP_22 \def (sym_eq T t5 t0 H18) in +(eq_ind T t0 TMP_15 TMP_21 t5 TMP_22))))) in (let TMP_24 \def (sym_eq C c1 c +H15) in (eq_ind C c TMP_12 TMP_23 c1 TMP_24 H16 H17 H12 H13 H14)))))))]) in +(let TMP_43 \def (refl_equal C c) in (let TMP_44 \def (refl_equal T t0) in +(let TMP_45 \def (refl_equal T t2) in (H12 TMP_43 TMP_44 TMP_45)))))))) in +(let TMP_47 \def (sym_eq T t t1 H8) in (eq_ind T t1 TMP_9 TMP_46 t +TMP_47))))) in (let TMP_49 \def (sym_eq T t3 t0 H7) in (eq_ind T t0 TMP_6 +TMP_48 t3 TMP_49))))) in (let TMP_51 \def (sym_eq C c0 c H4) in (eq_ind C c +TMP_3 TMP_50 c0 TMP_51 H5 H6 H1 H2 H3)))))))]) in (let TMP_100 \def +(refl_equal C c) in (let TMP_101 \def (refl_equal T t0) in (let TMP_102 \def +(refl_equal T t1) in (H1 TMP_100 TMP_101 TMP_102)))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma index 2faeb6ebd..19cb5d831 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma @@ -14,13 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/defs.ma". +include "basic_1/pr2/fwd.ma". -include "Basic-1/pr0/props.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". +include "basic_1/pr0/subst0.ma". theorem pr2_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall @@ -39,9 +35,6 @@ 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)))))). -(* COMMENTS -Initial nodes: 239 -END *) theorem pr2_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall @@ -58,9 +51,6 @@ 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)))))). -(* COMMENTS -Initial nodes: 219 -END *) theorem pr2_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall @@ -88,73 +78,68 @@ C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) -u) H4) in (let H6 \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) H5))) in ((let H7 \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) +u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e 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) H5))) in ((let H8 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (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) H5))) in -(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T -u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda -(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) -t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u -t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) -(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) -(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead -(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) -i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind -Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall -(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c -(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 -(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: -(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: -C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in -(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) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) -(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) -(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u -t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) -(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead -(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) -i)))))) k) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1947 -END *) +(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H7 +\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | +(CHead _ k0 _) \Rightarrow (match k0 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) H5))) in ((let H8 \def (f_equal C T (\lambda (e: +C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) +(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) +H5))) in (\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def +(eq_ind T u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr +(\lambda (b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) +(pr2_free c (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u +(pr0_refl u) t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: +nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: +T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to +((eq C c0 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind +b) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) +u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda +(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c +(Bind b) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 +(CHead d (Bind Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def +(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to +(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 +n u0 t6 t0) \to ((eq C c1 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u +t5) (THead (Bind b) u t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta +c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) +(THead (Bind b) u t3) (THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 +H3 (Bind b)) (THead (Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) +H4 u))))))))))))) i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 +(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) +\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) +\to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: +(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 +t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 +(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) +u) H4) in (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) H5))) (THead (Flat f) u t3) +(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead +(Flat f) u t) (subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: +nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: +T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to +((eq C c0 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat +f) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) +u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda +(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c +(Flat f) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 +(CHead d (Bind Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def +(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to +(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 +n u0 t6 t0) \to ((eq C c1 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u +t5) (THead (Flat f) u t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta +c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) +(THead (Flat f) u t3) (THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 +H3 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) +H4 u))))))))))))) i)))))) k) y t1 t2 H0))) H)))))). theorem clear_pr2_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to @@ -170,9 +155,6 @@ Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c (CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))). -(* COMMENTS -Initial nodes: 171 -END *) theorem pr2_cflat: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall @@ -188,9 +170,6 @@ u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 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)))))). -(* COMMENTS -Initial nodes: 175 -END *) theorem pr2_ctail: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall @@ -205,9 +184,6 @@ 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)))))). -(* COMMENTS -Initial nodes: 171 -END *) theorem pr2_change: \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: @@ -232,35 +208,30 @@ C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \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) u) (CHead c (Bind b) -v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind -b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 +C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d +(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) +H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) +\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) +(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) -u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in -(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind -T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def -(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B -Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match -(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c -(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda -(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda +\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | +(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) +v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind +b) v1) (CHead d (Bind Abbr) u) H7))) in (\lambda (H12: (eq B Abbr +b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind T u (\lambda (t0: +T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def (eq_ind_r B b (\lambda +(b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B Abbr (\lambda (b0: +B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match (H15 (refl_equal B +Abbr)) in False with []) in H16) b H12)))))) H10)) H9)))) (\lambda (i0: +nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) +\to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 913 -END *) theorem pr2_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h @@ -281,27 +252,747 @@ T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c (lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def -(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 -(Bind Abbr) u) H6) 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 t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: -(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: -(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) -(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i -H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u -H11) 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 t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (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 H9 H14) (lift h d t3) (lift h d +(drop_getl_trans_le i d (le_S_n i d (le_S_n (S i) (S d) (le_S (S (S i)) (S d) +(le_n_S (S i) d H7)))) c e h H (CHead d0 (Bind Abbr) u) H6) 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 t3) (lift h d t)) (\lambda +(x0: C).(\lambda (x1: C).(\lambda (H9: (drop i O c x0)).(\lambda (H10: (drop +h (minus d i) x0 x1)).(\lambda (H11: (clear x1 (CHead d0 (Bind Abbr) +u))).(let H12 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 +x1)) H10 (S (minus d (S i))) (minus_x_Sy d i H7)) in (let H13 \def +(drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u H11) 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 t3) +(lift h d t)) (\lambda (x: C).(\lambda (H14: (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 H9 H14) (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 h))))) H13)))))))) H8))) (\lambda (H7: (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) H6 H7) (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 849 -END *) + +theorem pr2_gen_abbr: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T +T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T +(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 +t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) +(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind +Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 +(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O +t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: +(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def +(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in +(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind +b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead +c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 +t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T +(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 +t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) +u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr) +x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 +x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda +(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 +T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1 +x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind +Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 +(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O +t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 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T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda +(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 +O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead 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T).(pr2 (CHead +c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 +(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda +(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: +T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) +u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl +Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S +i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl +x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda +(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 +(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t +(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: +(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 +x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 +x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda +(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead +c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 +(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 +u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T +(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O +(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) +u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 +x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) +H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) +H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) +(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) +H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) +(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). + +theorem pr2_gen_void: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) +(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift +(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 +t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind +Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind +b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) +t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead +c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 +x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) +(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead +c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 +x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 +H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 +(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 +t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T +t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 +x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T +(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 +i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t +(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind +Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 +x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 +x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead +(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda +(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: +T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 +(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t +(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: +(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head +(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) +(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 +t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: +T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head +(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S +O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 +t2 H5)))))))))))))) c y x H0))) H))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma index 47704dd52..c27e3c809 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma @@ -14,19 +14,13 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/defs.ma". +include "basic_1/pr2/fwd.ma". -include "Basic-1/pr0/subst1.ma". +include "basic_1/pr0/subst1.ma". -include "Basic-1/pr0/fwd.ma". +include "basic_1/csubst1/getl.ma". -include "Basic-1/csubst1/getl.ma". - -include "Basic-1/csubst1/fwd.ma". - -include "Basic-1/subst1/subst1.ma". - -include "Basic-1/getl/drop.ma". +include "basic_1/subst1/subst1.ma". theorem pr2_delta1: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c @@ -39,9 +33,6 @@ T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) (\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 H0 t0 H2))) t H1)))))))))). -(* COMMENTS -Initial nodes: 111 -END *) theorem pr2_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c @@ -90,13 +81,12 @@ H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal -C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) -(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d -(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ -t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) -(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in +C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) +\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono +c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in ((let H15 \def +(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow v | +(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) +u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u (\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C @@ -109,9 +99,6 @@ T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) (subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 H5))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1311 -END *) theorem pr2_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall @@ -229,13 +216,12 @@ T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ -_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in -((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d +e0 with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d +(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) +i H0 (CHead e (Bind Abbr) u0) H16)) in ((let H19 \def (f_equal C T (\lambda +(e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow +t0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d +(Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r @@ -266,16 +252,14 @@ x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 (lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u (minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 -(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d -(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: -nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 -H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S -O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S -O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift -(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) -(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 -x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 3757 -END *) +(csubst1_getl_ge d0 i (le_S_n d0 i (le_S_n (S d0) (S i) (le_S (S (S d0)) (S +i) (le_n_S (S d0) i H12)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) +d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 +(S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 H16)))))) (subst1_gen_lift_ge u x0 +x2 i (S O) d0 H14 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) +H12 (plus d0 (S O)) (plus_sym d0 (S O)))))))) (subst1_confluence_neq t4 t u i +(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i +(lt_neq d0 i H12)))))))))) (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 +t4 H1 u0 (lift (S O) d0 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 +t2 H)))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma index cdc9bf3a4..feef3e87d 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr0/defs.ma". +include "basic_1/pr0/defs.ma". -include "Basic-1/C/defs.ma". +include "basic_1/C/defs.ma". inductive wcpr0: C \to (C \to Prop) \def | wcpr0_refl: \forall (c: C).(wcpr0 c c) diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma index 2b0531a8a..deda207ba 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma @@ -14,29 +14,40 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/wcpr0/defs.ma". +include "basic_1/wcpr0/defs.ma". + +let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c c))) +(f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to +(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P (CHead +c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on w: P c +c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp c1 c2 +w0 u1 u2 p k) \Rightarrow (let TMP_1 \def ((wcpr0_ind P f f0) c1 c2 w0) in +(f0 c1 c2 w0 TMP_1 u1 u2 p k))]. theorem wcpr0_gen_sort: \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort n)))) \def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) -x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C -x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: -C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) -(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda -(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C -(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) -H5))))))))))) y x H0))) H))). -(* COMMENTS -Initial nodes: 249 -END *) + \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let +TMP_1 \def (CSort n) in (let TMP_2 \def (\lambda (c: C).(wcpr0 c x)) in (let +TMP_3 \def (\lambda (c: C).(eq C x c)) in (let TMP_19 \def (\lambda (y: +C).(\lambda (H0: (wcpr0 y x)).(let TMP_4 \def (\lambda (c: C).(\lambda (c0: +C).((eq C c (CSort n)) \to (eq C c0 c)))) in (let TMP_11 \def (\lambda (c: +C).(\lambda (H1: (eq C c (CSort n))).(let TMP_5 \def (\lambda (e: C).e) in +(let TMP_6 \def (CSort n) in (let H2 \def (f_equal C C TMP_5 c TMP_6 H1) in +(let TMP_7 \def (CSort n) in (let TMP_8 \def (\lambda (c0: C).(eq C c0 c0)) +in (let TMP_9 \def (CSort n) in (let TMP_10 \def (refl_equal C TMP_9) in +(eq_ind_r C TMP_7 TMP_8 TMP_10 c H2)))))))))) in (let TMP_18 \def (\lambda +(c1: C).(\lambda (c2: C).(\lambda (_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 +(CSort n)) \to (eq C c2 c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: +(pr0 u1 u2)).(\lambda (k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort +n))).(let TMP_12 \def (CHead c1 k u1) in (let TMP_13 \def (\lambda (ee: +C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow +True])) in (let TMP_14 \def (CSort n) in (let H5 \def (eq_ind C TMP_12 TMP_13 +I TMP_14 H4) in (let TMP_15 \def (CHead c2 k u2) in (let TMP_16 \def (CHead +c1 k u1) in (let TMP_17 \def (eq C TMP_15 TMP_16) in (False_ind TMP_17 +H5))))))))))))))))) in (wcpr0_ind TMP_4 TMP_11 TMP_18 y x H0)))))) in +(insert_eq C TMP_1 TMP_2 TMP_3 TMP_19 H))))))). theorem wcpr0_gen_head: \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 @@ -45,61 +56,99 @@ C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) \def \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: -C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda -(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) -(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal -C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) -(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 -c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C -(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) -(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: +(H: (wcpr0 (CHead c1 k u1) x)).(let TMP_1 \def (CHead c1 k u1) in (let TMP_2 +\def (\lambda (c: C).(wcpr0 c x)) in (let TMP_9 \def (\lambda (c: C).(let +TMP_3 \def (eq C x c) in (let TMP_5 \def (\lambda (c2: C).(\lambda (u2: +T).(let TMP_4 \def (CHead c2 k u2) in (eq C x TMP_4)))) in (let TMP_6 \def +(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_7 \def (\lambda +(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_8 \def (ex3_2 C T TMP_5 +TMP_6 TMP_7) in (or TMP_3 TMP_8))))))) in (let TMP_111 \def (\lambda (y: +C).(\lambda (H0: (wcpr0 y x)).(let TMP_16 \def (\lambda (c: C).(\lambda (c0: +C).((eq C c (CHead c1 k u1)) \to (let TMP_10 \def (eq C c0 c) in (let TMP_12 +\def (\lambda (c2: C).(\lambda (u2: T).(let TMP_11 \def (CHead c2 k u2) in +(eq C c0 TMP_11)))) in (let TMP_13 \def (\lambda (c2: C).(\lambda (_: +T).(wcpr0 c1 c2))) in (let TMP_14 \def (\lambda (_: C).(\lambda (u2: T).(pr0 +u1 u2))) in (let TMP_15 \def (ex3_2 C T TMP_12 TMP_13 TMP_14) in (or TMP_10 +TMP_15))))))))) in (let TMP_39 \def (\lambda (c: C).(\lambda (H1: (eq C c +(CHead c1 k u1))).(let TMP_17 \def (\lambda (e: C).e) in (let TMP_18 \def +(CHead c1 k u1) in (let H2 \def (f_equal C C TMP_17 c TMP_18 H1) in (let +TMP_19 \def (CHead c1 k u1) in (let TMP_26 \def (\lambda (c0: C).(let TMP_20 +\def (eq C c0 c0) in (let TMP_22 \def (\lambda (c2: C).(\lambda (u2: T).(let +TMP_21 \def (CHead c2 k u2) in (eq C c0 TMP_21)))) in (let TMP_23 \def +(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_24 \def (\lambda +(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_25 \def (ex3_2 C T TMP_22 +TMP_23 TMP_24) in (or TMP_20 TMP_25))))))) in (let TMP_27 \def (CHead c1 k +u1) in (let TMP_28 \def (CHead c1 k u1) in (let TMP_29 \def (eq C TMP_27 +TMP_28) in (let TMP_32 \def (\lambda (c2: C).(\lambda (u2: T).(let TMP_30 +\def (CHead c1 k u1) in (let TMP_31 \def (CHead c2 k u2) in (eq C TMP_30 +TMP_31))))) in (let TMP_33 \def (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 +c2))) in (let TMP_34 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in +(let TMP_35 \def (ex3_2 C T TMP_32 TMP_33 TMP_34) in (let TMP_36 \def (CHead +c1 k u1) in (let TMP_37 \def (refl_equal C TMP_36) in (let TMP_38 \def +(or_introl TMP_29 TMP_35 TMP_37) in (eq_ind_r C TMP_19 TMP_26 TMP_38 c +H2)))))))))))))))))) in (let TMP_110 \def (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 -k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K -k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C -(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: -T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: -T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 -\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 -(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k -u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C -c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def -(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 -(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T -(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) -(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq -C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 -c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C -(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x -H0))) H))))). -(* COMMENTS -Initial nodes: 1133 -END *) +k0 u0) (CHead c1 k u1))).(let TMP_40 \def (\lambda (e: C).(match e with +[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) in (let TMP_41 +\def (CHead c0 k0 u0) in (let TMP_42 \def (CHead c1 k u1) in (let H5 \def +(f_equal C C TMP_40 TMP_41 TMP_42 H4) in (let TMP_43 \def (\lambda (e: +C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) +in (let TMP_44 \def (CHead c0 k0 u0) in (let TMP_45 \def (CHead c1 k u1) in +(let H6 \def (f_equal C K TMP_43 TMP_44 TMP_45 H4) in (let TMP_46 \def +(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) +\Rightarrow t])) in (let TMP_47 \def (CHead c0 k0 u0) in (let TMP_48 \def +(CHead c1 k u1) in (let H7 \def (f_equal C T TMP_46 TMP_47 TMP_48 H4) in (let +TMP_108 \def (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let +TMP_58 \def (\lambda (k1: K).(let TMP_49 \def (CHead c2 k1 u2) in (let TMP_50 +\def (CHead c0 k1 u0) in (let TMP_51 \def (eq C TMP_49 TMP_50) in (let TMP_54 +\def (\lambda (c3: C).(\lambda (u3: T).(let TMP_52 \def (CHead c2 k1 u2) in +(let TMP_53 \def (CHead c3 k u3) in (eq C TMP_52 TMP_53))))) in (let TMP_55 +\def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_56 \def +(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_57 \def (ex3_2 C T +TMP_54 TMP_55 TMP_56) in (or TMP_51 TMP_57))))))))) in (let TMP_59 \def +(\lambda (t: T).(pr0 t u2)) in (let H10 \def (eq_ind T u0 TMP_59 H3 u1 H7) in +(let TMP_69 \def (\lambda (t: T).(let TMP_60 \def (CHead c2 k u2) in (let +TMP_61 \def (CHead c0 k t) in (let TMP_62 \def (eq C TMP_60 TMP_61) in (let +TMP_65 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_63 \def (CHead c2 k +u2) in (let TMP_64 \def (CHead c3 k u3) in (eq C TMP_63 TMP_64))))) in (let +TMP_66 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_67 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_68 \def +(ex3_2 C T TMP_65 TMP_66 TMP_67) in (or TMP_62 TMP_68))))))))) in (let TMP_76 +\def (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (let TMP_70 \def (eq C c2 +c) in (let TMP_72 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_71 \def +(CHead c3 k u3) in (eq C c2 TMP_71)))) in (let TMP_73 \def (\lambda (c3: +C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_74 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_75 \def (ex3_2 C T TMP_72 +TMP_73 TMP_74) in (or TMP_70 TMP_75)))))))) in (let H11 \def (eq_ind C c0 +TMP_76 H2 c1 H9) in (let TMP_77 \def (\lambda (c: C).(wcpr0 c c2)) in (let +H12 \def (eq_ind C c0 TMP_77 H1 c1 H9) in (let TMP_87 \def (\lambda (c: +C).(let TMP_78 \def (CHead c2 k u2) in (let TMP_79 \def (CHead c k u1) in +(let TMP_80 \def (eq C TMP_78 TMP_79) in (let TMP_83 \def (\lambda (c3: +C).(\lambda (u3: T).(let TMP_81 \def (CHead c2 k u2) in (let TMP_82 \def +(CHead c3 k u3) in (eq C TMP_81 TMP_82))))) in (let TMP_84 \def (\lambda (c3: +C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_85 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_86 \def (ex3_2 C T TMP_83 +TMP_84 TMP_85) in (or TMP_80 TMP_86))))))))) in (let TMP_88 \def (CHead c2 k +u2) in (let TMP_89 \def (CHead c1 k u1) in (let TMP_90 \def (eq C TMP_88 +TMP_89) in (let TMP_93 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_91 +\def (CHead c2 k u2) in (let TMP_92 \def (CHead c3 k u3) in (eq C TMP_91 +TMP_92))))) in (let TMP_94 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 +c3))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in +(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_99 \def +(\lambda (c3: C).(\lambda (u3: T).(let TMP_97 \def (CHead c2 k u2) in (let +TMP_98 \def (CHead c3 k u3) in (eq C TMP_97 TMP_98))))) in (let TMP_100 \def +(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_101 \def +(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_102 \def (CHead c2 +k u2) in (let TMP_103 \def (refl_equal C TMP_102) in (let TMP_104 \def +(ex3_2_intro C T TMP_99 TMP_100 TMP_101 c2 u2 TMP_103 H12 H10) in (let +TMP_105 \def (or_intror TMP_90 TMP_96 TMP_104) in (let TMP_106 \def (eq_ind_r +C c1 TMP_87 TMP_105 c0 H9) in (let TMP_107 \def (eq_ind_r T u1 TMP_69 TMP_106 +u0 H7) in (eq_ind_r K k TMP_58 TMP_107 k0 H8)))))))))))))))))))))))))))) in +(let TMP_109 \def (TMP_108 H6) in (TMP_109 H5)))))))))))))))))))))))) in +(wcpr0_ind TMP_16 TMP_39 TMP_110 y x H0)))))) in (insert_eq C TMP_1 TMP_2 +TMP_9 TMP_111 H))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma index d3a109e0d..4eb36afbe 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/wcpr0/defs.ma". +include "basic_1/wcpr0/fwd.ma". -include "Basic-1/getl/props.ma". +include "basic_1/getl/props.ma". theorem wcpr0_drop: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: @@ -25,100 +25,146 @@ e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) \def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) -\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k -u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) -(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1) -(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k -u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 -u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: +(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (let TMP_2 \def +(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (drop h +O c0 TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in +(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c: +C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: +K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2: +C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (drop h O c TMP_6)))) +in (let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let +TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_10 \def +(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7 +TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_132 \def (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall +(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c3 +(CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O +c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: +nat).(let TMP_18 \def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: +T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead e1 k0 u3)) \to (let +TMP_15 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c4 k +u2) in (let TMP_14 \def (CHead e2 k0 u4) in (drop n O TMP_13 TMP_14))))) in +(let TMP_16 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let +TMP_17 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T +TMP_15 TMP_16 TMP_17))))))))) in (let TMP_67 \def (\lambda (e1: C).(\lambda +(u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) (CHead e1 k0 +u0))).(let TMP_19 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow +c3 | (CHead c _ _) \Rightarrow c])) in (let TMP_20 \def (CHead c3 k u1) in +(let TMP_21 \def (CHead e1 k0 u0) in (let TMP_22 \def (CHead c3 k u1) in (let +TMP_23 \def (CHead e1 k0 u0) in (let TMP_24 \def (drop_gen_refl TMP_22 TMP_23 +H3) in (let H4 \def (f_equal C C TMP_19 TMP_20 TMP_21 TMP_24) in (let TMP_25 +\def (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k1 _) +\Rightarrow k1])) in (let TMP_26 \def (CHead c3 k u1) in (let TMP_27 \def +(CHead e1 k0 u0) in (let TMP_28 \def (CHead c3 k u1) in (let TMP_29 \def +(CHead e1 k0 u0) in (let TMP_30 \def (drop_gen_refl TMP_28 TMP_29 H3) in (let +H5 \def (f_equal C K TMP_25 TMP_26 TMP_27 TMP_30) in (let TMP_31 \def +(\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead _ _ t) +\Rightarrow t])) in (let TMP_32 \def (CHead c3 k u1) in (let TMP_33 \def +(CHead e1 k0 u0) in (let TMP_34 \def (CHead c3 k u1) in (let TMP_35 \def +(CHead e1 k0 u0) in (let TMP_36 \def (drop_gen_refl TMP_34 TMP_35 H3) in (let +H6 \def (f_equal C T TMP_31 TMP_32 TMP_33 TMP_36) in (let TMP_65 \def +(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(let TMP_42 \def +(\lambda (k1: K).(let TMP_39 \def (\lambda (e2: C).(\lambda (u3: T).(let +TMP_37 \def (CHead c4 k u2) in (let TMP_38 \def (CHead e2 k1 u3) in (drop O O +TMP_37 TMP_38))))) in (let TMP_40 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) in (let TMP_41 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u0 u3))) in (ex3_2 C T TMP_39 TMP_40 TMP_41))))) in (let TMP_48 \def (\lambda +(t: T).(let TMP_45 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_43 \def +(CHead c4 k u2) in (let TMP_44 \def (CHead e2 k u3) in (drop O O TMP_43 +TMP_44))))) in (let TMP_46 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) in (let TMP_47 \def (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))) in +(ex3_2 C T TMP_45 TMP_46 TMP_47))))) in (let TMP_54 \def (\lambda (c: C).(let +TMP_51 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_49 \def (CHead c4 k +u2) in (let TMP_50 \def (CHead e2 k u3) in (drop O O TMP_49 TMP_50))))) in +(let TMP_52 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) in (let +TMP_53 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (ex3_2 C T +TMP_51 TMP_52 TMP_53))))) in (let TMP_57 \def (\lambda (e2: C).(\lambda (u3: +T).(let TMP_55 \def (CHead c4 k u2) in (let TMP_56 \def (CHead e2 k u3) in +(drop O O TMP_55 TMP_56))))) in (let TMP_58 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 c3 e2))) in (let TMP_59 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u1 u3))) in (let TMP_60 \def (CHead c4 k u2) in (let TMP_61 \def +(drop_refl TMP_60) in (let TMP_62 \def (ex3_2_intro C T TMP_57 TMP_58 TMP_59 +c4 u2 TMP_61 H0 H2) in (let TMP_63 \def (eq_ind C c3 TMP_54 TMP_62 e1 H8) in +(let TMP_64 \def (eq_ind T u1 TMP_48 TMP_63 u0 H6) in (eq_ind K k TMP_42 +TMP_64 k0 H7)))))))))))))) in (let TMP_66 \def (TMP_65 H5) in (TMP_66 +H4)))))))))))))))))))))))))))) in (let TMP_74 \def (\lambda (k0: K).(\forall +(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O +(CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda +(u4: T).(drop n O (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 +u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S +n) O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (let TMP_71 \def (\lambda (e2: +C).(\lambda (u4: T).(let TMP_68 \def (S n) in (let TMP_69 \def (CHead c4 k0 +u2) in (let TMP_70 \def (CHead e2 k1 u4) in (drop TMP_68 O TMP_69 +TMP_70)))))) in (let TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) in (let TMP_73 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))) in +(ex3_2 C T TMP_71 TMP_72 TMP_73))))))))))) in (let TMP_101 \def (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 -k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 -x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop -(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead -e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 -u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 -(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) -h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1755 -END *) +e1 k0 u0))).(let TMP_75 \def (Bind b) in (let TMP_76 \def (CHead e1 k0 u0) in +(let TMP_77 \def (drop_gen_drop TMP_75 c3 TMP_76 u1 n H4) in (let H5 \def (H1 +n e1 u0 k0 TMP_77) in (let TMP_79 \def (\lambda (e2: C).(\lambda (u3: T).(let +TMP_78 \def (CHead e2 k0 u3) in (drop n O c4 TMP_78)))) in (let TMP_80 \def +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_81 \def (\lambda +(_: C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_86 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_82 \def (S n) in (let TMP_83 \def (Bind b) in +(let TMP_84 \def (CHead c4 TMP_83 u2) in (let TMP_85 \def (CHead e2 k0 u3) in +(drop TMP_82 O TMP_84 TMP_85))))))) in (let TMP_87 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_88 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_89 \def (ex3_2 C T TMP_86 +TMP_87 TMP_88) in (let TMP_100 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (drop n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 +x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_94 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_90 \def (S n) in (let TMP_91 \def (Bind b) in (let TMP_92 +\def (CHead c4 TMP_91 u2) in (let TMP_93 \def (CHead e2 k0 u3) in (drop +TMP_90 O TMP_92 TMP_93))))))) in (let TMP_95 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e1 e2))) in (let TMP_96 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u0 u3))) in (let TMP_97 \def (Bind b) in (let TMP_98 \def (CHead x0 +k0 x1) in (let TMP_99 \def (drop_drop TMP_97 n c4 TMP_98 H6 u2) in +(ex3_2_intro C T TMP_94 TMP_95 TMP_96 x0 x1 TMP_99 H7 H8)))))))))))) in +(ex3_2_ind C T TMP_79 TMP_80 TMP_81 TMP_89 TMP_100 H5)))))))))))))))))))) in +(let TMP_130 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 (Flat f) u1) +(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O +(CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 +u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(drop (S n) O (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_102 \def (S +n) in (let TMP_103 \def (Flat f) in (let TMP_104 \def (CHead e1 k0 u0) in +(let TMP_105 \def (drop_gen_drop TMP_103 c3 TMP_104 u1 n H4) in (let H5 \def +(H1 TMP_102 e1 u0 k0 TMP_105) in (let TMP_108 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_106 \def (S n) in (let TMP_107 \def (CHead e2 k0 u3) in +(drop TMP_106 O c4 TMP_107))))) in (let TMP_109 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_110 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_115 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_111 \def (S n) in (let TMP_112 \def (Flat f) in +(let TMP_113 \def (CHead c4 TMP_112 u2) in (let TMP_114 \def (CHead e2 k0 u3) +in (drop TMP_111 O TMP_113 TMP_114))))))) in (let TMP_116 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_117 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_118 \def (ex3_2 C T TMP_115 +TMP_116 TMP_117) in (let TMP_129 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (drop (S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 +x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_123 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_119 \def (S n) in (let TMP_120 \def (Flat f) in (let TMP_121 +\def (CHead c4 TMP_120 u2) in (let TMP_122 \def (CHead e2 k0 u3) in (drop +TMP_119 O TMP_121 TMP_122))))))) in (let TMP_124 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_125 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_126 \def (Flat f) in (let +TMP_127 \def (CHead x0 k0 x1) in (let TMP_128 \def (drop_drop TMP_126 n c4 +TMP_127 H6 u2) in (ex3_2_intro C T TMP_123 TMP_124 TMP_125 x0 x1 TMP_128 H7 +H8)))))))))))) in (ex3_2_ind C T TMP_108 TMP_109 TMP_110 TMP_118 TMP_129 +H5))))))))))))))))))))) in (let TMP_131 \def (K_ind TMP_74 TMP_101 TMP_130 k) +in (nat_ind TMP_18 TMP_67 TMP_131 h)))))))))))))))) in (wcpr0_ind TMP_5 +TMP_12 TMP_132 c1 c2 H)))))). theorem wcpr0_drop_back: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: @@ -127,100 +173,146 @@ e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) \def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 -u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) -\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k -u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) -(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) -(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k -u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 -u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: +(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (let TMP_2 \def +(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (drop h +O c TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in +(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c: +C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: +K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2: +C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (drop h O c TMP_6)))) +in (let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let +TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in (let TMP_10 \def +(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7 +TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_132 \def (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall +(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c4 +(CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O +c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: +nat).(let TMP_18 \def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: +T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead e1 k0 u3)) \to (let +TMP_15 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c3 k +u1) in (let TMP_14 \def (CHead e2 k0 u4) in (drop n O TMP_13 TMP_14))))) in +(let TMP_16 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let +TMP_17 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T +TMP_15 TMP_16 TMP_17))))))))) in (let TMP_67 \def (\lambda (e1: C).(\lambda +(u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) (CHead e1 k0 +u0))).(let TMP_19 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow +c4 | (CHead c _ _) \Rightarrow c])) in (let TMP_20 \def (CHead c4 k u2) in +(let TMP_21 \def (CHead e1 k0 u0) in (let TMP_22 \def (CHead c4 k u2) in (let +TMP_23 \def (CHead e1 k0 u0) in (let TMP_24 \def (drop_gen_refl TMP_22 TMP_23 +H3) in (let H4 \def (f_equal C C TMP_19 TMP_20 TMP_21 TMP_24) in (let TMP_25 +\def (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k1 _) +\Rightarrow k1])) in (let TMP_26 \def (CHead c4 k u2) in (let TMP_27 \def +(CHead e1 k0 u0) in (let TMP_28 \def (CHead c4 k u2) in (let TMP_29 \def +(CHead e1 k0 u0) in (let TMP_30 \def (drop_gen_refl TMP_28 TMP_29 H3) in (let +H5 \def (f_equal C K TMP_25 TMP_26 TMP_27 TMP_30) in (let TMP_31 \def +(\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead _ _ t) +\Rightarrow t])) in (let TMP_32 \def (CHead c4 k u2) in (let TMP_33 \def +(CHead e1 k0 u0) in (let TMP_34 \def (CHead c4 k u2) in (let TMP_35 \def +(CHead e1 k0 u0) in (let TMP_36 \def (drop_gen_refl TMP_34 TMP_35 H3) in (let +H6 \def (f_equal C T TMP_31 TMP_32 TMP_33 TMP_36) in (let TMP_65 \def +(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(let TMP_42 \def +(\lambda (k1: K).(let TMP_39 \def (\lambda (e2: C).(\lambda (u3: T).(let +TMP_37 \def (CHead c3 k u1) in (let TMP_38 \def (CHead e2 k1 u3) in (drop O O +TMP_37 TMP_38))))) in (let TMP_40 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) in (let TMP_41 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u0))) in (ex3_2 C T TMP_39 TMP_40 TMP_41))))) in (let TMP_48 \def (\lambda +(t: T).(let TMP_45 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_43 \def +(CHead c3 k u1) in (let TMP_44 \def (CHead e2 k u3) in (drop O O TMP_43 +TMP_44))))) in (let TMP_46 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) in (let TMP_47 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))) in +(ex3_2 C T TMP_45 TMP_46 TMP_47))))) in (let TMP_54 \def (\lambda (c: C).(let +TMP_51 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_49 \def (CHead c3 k +u1) in (let TMP_50 \def (CHead e2 k u3) in (drop O O TMP_49 TMP_50))))) in +(let TMP_52 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) in (let +TMP_53 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) in (ex3_2 C T +TMP_51 TMP_52 TMP_53))))) in (let TMP_57 \def (\lambda (e2: C).(\lambda (u3: +T).(let TMP_55 \def (CHead c3 k u1) in (let TMP_56 \def (CHead e2 k u3) in +(drop O O TMP_55 TMP_56))))) in (let TMP_58 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 c4))) in (let TMP_59 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u3 u2))) in (let TMP_60 \def (CHead c3 k u1) in (let TMP_61 \def +(drop_refl TMP_60) in (let TMP_62 \def (ex3_2_intro C T TMP_57 TMP_58 TMP_59 +c3 u1 TMP_61 H0 H2) in (let TMP_63 \def (eq_ind C c4 TMP_54 TMP_62 e1 H8) in +(let TMP_64 \def (eq_ind T u2 TMP_48 TMP_63 u0 H6) in (eq_ind K k TMP_42 +TMP_64 k0 H7)))))))))))))) in (let TMP_66 \def (TMP_65 H5) in (TMP_66 +H4)))))))))))))))))))))))))))) in (let TMP_74 \def (\lambda (k0: K).(\forall +(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O +(CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda +(u4: T).(drop n O (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 +u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S +n) O (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (let TMP_71 \def (\lambda (e2: +C).(\lambda (u4: T).(let TMP_68 \def (S n) in (let TMP_69 \def (CHead c3 k0 +u1) in (let TMP_70 \def (CHead e2 k1 u4) in (drop TMP_68 O TMP_69 +TMP_70)))))) in (let TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) in (let TMP_73 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))) in +(ex3_2 C T TMP_71 TMP_72 TMP_73))))))))))) in (let TMP_101 \def (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 -k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 -u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop -(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead -e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 -x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 -(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) -h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 1755 -END *) +e1 k0 u0))).(let TMP_75 \def (Bind b) in (let TMP_76 \def (CHead e1 k0 u0) in +(let TMP_77 \def (drop_gen_drop TMP_75 c4 TMP_76 u2 n H4) in (let H5 \def (H1 +n e1 u0 k0 TMP_77) in (let TMP_79 \def (\lambda (e2: C).(\lambda (u3: T).(let +TMP_78 \def (CHead e2 k0 u3) in (drop n O c3 TMP_78)))) in (let TMP_80 \def +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_81 \def (\lambda +(_: C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_86 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_82 \def (S n) in (let TMP_83 \def (Bind b) in +(let TMP_84 \def (CHead c3 TMP_83 u1) in (let TMP_85 \def (CHead e2 k0 u3) in +(drop TMP_82 O TMP_84 TMP_85))))))) in (let TMP_87 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_88 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_89 \def (ex3_2 C T TMP_86 +TMP_87 TMP_88) in (let TMP_100 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (drop n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 +e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_94 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_90 \def (S n) in (let TMP_91 \def (Bind b) in (let TMP_92 +\def (CHead c3 TMP_91 u1) in (let TMP_93 \def (CHead e2 k0 u3) in (drop +TMP_90 O TMP_92 TMP_93))))))) in (let TMP_95 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 e1))) in (let TMP_96 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u3 u0))) in (let TMP_97 \def (Bind b) in (let TMP_98 \def (CHead x0 +k0 x1) in (let TMP_99 \def (drop_drop TMP_97 n c3 TMP_98 H6 u1) in +(ex3_2_intro C T TMP_94 TMP_95 TMP_96 x0 x1 TMP_99 H7 H8)))))))))))) in +(ex3_2_ind C T TMP_79 TMP_80 TMP_81 TMP_89 TMP_100 H5)))))))))))))))))))) in +(let TMP_130 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 (Flat f) u2) +(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O +(CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 +u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(drop (S n) O (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_102 \def (S +n) in (let TMP_103 \def (Flat f) in (let TMP_104 \def (CHead e1 k0 u0) in +(let TMP_105 \def (drop_gen_drop TMP_103 c4 TMP_104 u2 n H4) in (let H5 \def +(H1 TMP_102 e1 u0 k0 TMP_105) in (let TMP_108 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_106 \def (S n) in (let TMP_107 \def (CHead e2 k0 u3) in +(drop TMP_106 O c3 TMP_107))))) in (let TMP_109 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_110 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_115 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_111 \def (S n) in (let TMP_112 \def (Flat f) in +(let TMP_113 \def (CHead c3 TMP_112 u1) in (let TMP_114 \def (CHead e2 k0 u3) +in (drop TMP_111 O TMP_113 TMP_114))))))) in (let TMP_116 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_117 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_118 \def (ex3_2 C T TMP_115 +TMP_116 TMP_117) in (let TMP_129 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (drop (S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 +e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_123 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_119 \def (S n) in (let TMP_120 \def (Flat f) in (let TMP_121 +\def (CHead c3 TMP_120 u1) in (let TMP_122 \def (CHead e2 k0 u3) in (drop +TMP_119 O TMP_121 TMP_122))))))) in (let TMP_124 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_125 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_126 \def (Flat f) in (let +TMP_127 \def (CHead x0 k0 x1) in (let TMP_128 \def (drop_drop TMP_126 n c3 +TMP_127 H6 u1) in (ex3_2_intro C T TMP_123 TMP_124 TMP_125 x0 x1 TMP_128 H7 +H8)))))))))))) in (ex3_2_ind C T TMP_108 TMP_109 TMP_110 TMP_118 TMP_129 +H5))))))))))))))))))))) in (let TMP_131 \def (K_ind TMP_74 TMP_101 TMP_130 k) +in (nat_ind TMP_18 TMP_67 TMP_131 h)))))))))))))))) in (wcpr0_ind TMP_5 +TMP_12 TMP_132 c2 c1 H)))))). theorem wcpr0_getl: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: @@ -229,118 +321,180 @@ k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) \def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k -u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 -(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r -T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) -(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 -u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 -(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 -(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) -(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 -x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) +(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (let TMP_2 \def +(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (getl h +c0 TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in +(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c: +C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: +K).(\lambda (H0: (getl h c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2: +C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (getl h c TMP_6)))) in +(let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let +TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_10 \def +(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7 +TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_175 \def (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall +(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c3 (CHead +e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c4 (CHead +e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(let TMP_18 +\def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: +K).((getl n (CHead c3 k u1) (CHead e1 k0 u3)) \to (let TMP_15 \def (\lambda +(e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c4 k u2) in (let TMP_14 \def +(CHead e2 k0 u4) in (getl n TMP_13 TMP_14))))) in (let TMP_16 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_17 \def (\lambda (_: +C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T TMP_15 TMP_16 TMP_17))))))))) +in (let TMP_110 \def (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: +K).(\lambda (H3: (getl O (CHead c3 k u1) (CHead e1 k0 u0))).(let TMP_24 \def +(\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 k0 u0)) \to (let TMP_21 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_19 \def (CHead c4 k1 u2) in +(let TMP_20 \def (CHead e2 k0 u3) in (getl O TMP_19 TMP_20))))) in (let +TMP_22 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_23 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in (ex3_2 C T TMP_21 +TMP_22 TMP_23)))))) in (let TMP_80 \def (\lambda (b: B).(\lambda (H4: (clear +(CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let TMP_25 \def (\lambda (e: +C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) +in (let TMP_26 \def (CHead e1 k0 u0) in (let TMP_27 \def (Bind b) in (let +TMP_28 \def (CHead c3 TMP_27 u1) in (let TMP_29 \def (CHead e1 k0 u0) in (let +TMP_30 \def (clear_gen_bind b c3 TMP_29 u1 H4) in (let H5 \def (f_equal C C +TMP_25 TMP_26 TMP_28 TMP_30) in (let TMP_31 \def (\lambda (e: C).(match e +with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) in (let +TMP_32 \def (CHead e1 k0 u0) in (let TMP_33 \def (Bind b) in (let TMP_34 \def +(CHead c3 TMP_33 u1) in (let TMP_35 \def (CHead e1 k0 u0) in (let TMP_36 \def +(clear_gen_bind b c3 TMP_35 u1 H4) in (let H6 \def (f_equal C K TMP_31 TMP_32 +TMP_34 TMP_36) in (let TMP_37 \def (\lambda (e: C).(match e with [(CSort _) +\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) in (let TMP_38 \def (CHead e1 +k0 u0) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead c3 TMP_39 u1) +in (let TMP_41 \def (CHead e1 k0 u0) in (let TMP_42 \def (clear_gen_bind b c3 +TMP_41 u1 H4) in (let H7 \def (f_equal C T TMP_37 TMP_38 TMP_40 TMP_42) in +(let TMP_78 \def (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 +c3)).(let TMP_43 \def (Bind b) in (let TMP_50 \def (\lambda (k1: K).(let +TMP_47 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_44 \def (Bind b) in +(let TMP_45 \def (CHead c4 TMP_44 u2) in (let TMP_46 \def (CHead e2 k1 u3) in +(getl O TMP_45 TMP_46)))))) in (let TMP_48 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) in (let TMP_49 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u0 u3))) in (ex3_2 C T TMP_47 TMP_48 TMP_49))))) in (let TMP_58 \def (\lambda +(t: T).(let TMP_55 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_51 \def +(Bind b) in (let TMP_52 \def (CHead c4 TMP_51 u2) in (let TMP_53 \def (Bind +b) in (let TMP_54 \def (CHead e2 TMP_53 u3) in (getl O TMP_52 TMP_54))))))) +in (let TMP_56 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let +TMP_57 \def (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))) in (ex3_2 C T +TMP_55 TMP_56 TMP_57))))) in (let TMP_66 \def (\lambda (c: C).(let TMP_63 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_59 \def (Bind b) in (let +TMP_60 \def (CHead c4 TMP_59 u2) in (let TMP_61 \def (Bind b) in (let TMP_62 +\def (CHead e2 TMP_61 u3) in (getl O TMP_60 TMP_62))))))) in (let TMP_64 \def +(\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) in (let TMP_65 \def (\lambda +(_: C).(\lambda (u3: T).(pr0 u1 u3))) in (ex3_2 C T TMP_63 TMP_64 TMP_65))))) +in (let TMP_71 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_67 \def (Bind +b) in (let TMP_68 \def (CHead c4 TMP_67 u2) in (let TMP_69 \def (Bind b) in +(let TMP_70 \def (CHead e2 TMP_69 u3) in (getl O TMP_68 TMP_70))))))) in (let +TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) in (let TMP_73 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_74 \def +(getl_refl b c4 u2) in (let TMP_75 \def (ex3_2_intro C T TMP_71 TMP_72 TMP_73 +c4 u2 TMP_74 H0 H2) in (let TMP_76 \def (eq_ind_r C c3 TMP_66 TMP_75 e1 H9) +in (let TMP_77 \def (eq_ind_r T u1 TMP_58 TMP_76 u0 H7) in (eq_ind_r K TMP_43 +TMP_50 TMP_77 k0 H8)))))))))))))) in (let TMP_79 \def (TMP_78 H6) in (TMP_79 +H5)))))))))))))))))))))))))) in (let TMP_106 \def (\lambda (f: F).(\lambda +(H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_81 \def (CHead +e1 k0 u0) in (let TMP_82 \def (drop_refl c3) in (let TMP_83 \def (CHead e1 k0 +u0) in (let TMP_84 \def (clear_gen_flat f c3 TMP_83 u1 H4) in (let TMP_85 +\def (getl_intro O c3 TMP_81 c3 TMP_82 TMP_84) in (let H5 \def (H1 O e1 u0 k0 +TMP_85) in (let TMP_87 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_86 +\def (CHead e2 k0 u3) in (getl O c4 TMP_86)))) in (let TMP_88 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_89 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_93 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_90 \def (Flat f) in (let TMP_91 \def (CHead c4 +TMP_90 u2) in (let TMP_92 \def (CHead e2 k0 u3) in (getl O TMP_91 +TMP_92)))))) in (let TMP_94 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in +(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_105 \def +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 +x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_100 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_97 \def (Flat f) in (let +TMP_98 \def (CHead c4 TMP_97 u2) in (let TMP_99 \def (CHead e2 k0 u3) in +(getl O TMP_98 TMP_99)))))) in (let TMP_101 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e1 e2))) in (let TMP_102 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u0 u3))) in (let TMP_103 \def (CHead x0 k0 x1) in (let TMP_104 \def +(getl_flat c4 TMP_103 O H6 f u2) in (ex3_2_intro C T TMP_100 TMP_101 TMP_102 +x0 x1 TMP_104 H7 H8))))))))))) in (ex3_2_ind C T TMP_87 TMP_88 TMP_89 TMP_96 +TMP_105 H5))))))))))))))))) in (let TMP_107 \def (CHead c3 k u1) in (let +TMP_108 \def (CHead e1 k0 u0) in (let TMP_109 \def (getl_gen_O TMP_107 +TMP_108 H3) in (K_ind TMP_24 TMP_80 TMP_106 k TMP_109))))))))))) in (let +TMP_117 \def (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall +(u3: T).(\forall (k1: K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to +(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead +e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: +T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (let +TMP_114 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_111 \def (S n) in +(let TMP_112 \def (CHead c4 k0 u2) in (let TMP_113 \def (CHead e2 k1 u4) in +(getl TMP_111 TMP_112 TMP_113)))))) in (let TMP_115 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_116 \def (\lambda (_: +C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T TMP_114 TMP_115 +TMP_116))))))))))) in (let TMP_144 \def (\lambda (b: B).(\lambda (n: +nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl +n (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl n (CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda +(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: +K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let +TMP_118 \def (Bind b) in (let TMP_119 \def (CHead e1 k0 u0) in (let TMP_120 +\def (getl_gen_S TMP_118 c3 TMP_119 u1 n H4) in (let H5 \def (H1 n e1 u0 k0 +TMP_120) in (let TMP_122 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_121 +\def (CHead e2 k0 u3) in (getl n c4 TMP_121)))) in (let TMP_123 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_124 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_129 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_125 \def (S n) in (let TMP_126 \def (Bind b) in +(let TMP_127 \def (CHead c4 TMP_126 u2) in (let TMP_128 \def (CHead e2 k0 u3) +in (getl TMP_125 TMP_127 TMP_128))))))) in (let TMP_130 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_131 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_132 \def (ex3_2 C T TMP_129 +TMP_130 TMP_131) in (let TMP_143 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 +x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_137 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_133 \def (S n) in (let TMP_134 \def (Bind b) in (let TMP_135 +\def (CHead c4 TMP_134 u2) in (let TMP_136 \def (CHead e2 k0 u3) in (getl +TMP_133 TMP_135 TMP_136))))))) in (let TMP_138 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e1 e2))) in (let TMP_139 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u0 u3))) in (let TMP_140 \def (Bind b) in (let TMP_141 \def (CHead x0 +k0 x1) in (let TMP_142 \def (getl_head TMP_140 n c4 TMP_141 H6 u2) in +(ex3_2_intro C T TMP_137 TMP_138 TMP_139 x0 x1 TMP_142 H7 H8)))))))))))) in +(ex3_2_ind C T TMP_122 TMP_123 TMP_124 TMP_132 TMP_143 H5)))))))))))))))))))) +in (let TMP_173 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +(CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat -f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 2103 -END *) +(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_145 \def (S n) +in (let TMP_146 \def (Flat f) in (let TMP_147 \def (CHead e1 k0 u0) in (let +TMP_148 \def (getl_gen_S TMP_146 c3 TMP_147 u1 n H4) in (let H5 \def (H1 +TMP_145 e1 u0 k0 TMP_148) in (let TMP_151 \def (\lambda (e2: C).(\lambda (u3: +T).(let TMP_149 \def (S n) in (let TMP_150 \def (CHead e2 k0 u3) in (getl +TMP_149 c4 TMP_150))))) in (let TMP_152 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) in (let TMP_153 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u0 u3))) in (let TMP_158 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_154 +\def (S n) in (let TMP_155 \def (Flat f) in (let TMP_156 \def (CHead c4 +TMP_155 u2) in (let TMP_157 \def (CHead e2 k0 u3) in (getl TMP_154 TMP_156 +TMP_157))))))) in (let TMP_159 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e1 e2))) in (let TMP_160 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) +in (let TMP_161 \def (ex3_2 C T TMP_158 TMP_159 TMP_160) in (let TMP_172 \def +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 +x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_166 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_162 \def (S n) in (let +TMP_163 \def (Flat f) in (let TMP_164 \def (CHead c4 TMP_163 u2) in (let +TMP_165 \def (CHead e2 k0 u3) in (getl TMP_162 TMP_164 TMP_165))))))) in (let +TMP_167 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_168 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_169 \def +(Flat f) in (let TMP_170 \def (CHead x0 k0 x1) in (let TMP_171 \def +(getl_head TMP_169 n c4 TMP_170 H6 u2) in (ex3_2_intro C T TMP_166 TMP_167 +TMP_168 x0 x1 TMP_171 H7 H8)))))))))))) in (ex3_2_ind C T TMP_151 TMP_152 +TMP_153 TMP_161 TMP_172 H5))))))))))))))))))))) in (let TMP_174 \def (K_ind +TMP_117 TMP_144 TMP_173 k) in (nat_ind TMP_18 TMP_110 TMP_174 +h)))))))))))))))) in (wcpr0_ind TMP_5 TMP_12 TMP_175 c1 c2 H)))))). theorem wcpr0_getl_back: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: @@ -349,116 +503,178 @@ k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) \def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k -u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 -(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r -T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) -(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 -(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 -(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) -(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 -e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) +(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (let TMP_2 \def +(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (getl h +c TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in +(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c: +C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: +K).(\lambda (H0: (getl h c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2: +C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (getl h c TMP_6)))) in +(let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let +TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in (let TMP_10 \def +(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7 +TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_175 \def (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall +(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c4 (CHead +e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c3 (CHead +e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(let TMP_18 +\def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: +K).((getl n (CHead c4 k u2) (CHead e1 k0 u3)) \to (let TMP_15 \def (\lambda +(e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c3 k u1) in (let TMP_14 \def +(CHead e2 k0 u4) in (getl n TMP_13 TMP_14))))) in (let TMP_16 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_17 \def (\lambda (_: +C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T TMP_15 TMP_16 TMP_17))))))))) +in (let TMP_110 \def (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: +K).(\lambda (H3: (getl O (CHead c4 k u2) (CHead e1 k0 u0))).(let TMP_24 \def +(\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 k0 u0)) \to (let TMP_21 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_19 \def (CHead c3 k1 u1) in +(let TMP_20 \def (CHead e2 k0 u3) in (getl O TMP_19 TMP_20))))) in (let +TMP_22 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_23 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in (ex3_2 C T TMP_21 +TMP_22 TMP_23)))))) in (let TMP_80 \def (\lambda (b: B).(\lambda (H4: (clear +(CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let TMP_25 \def (\lambda (e: +C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) +in (let TMP_26 \def (CHead e1 k0 u0) in (let TMP_27 \def (Bind b) in (let +TMP_28 \def (CHead c4 TMP_27 u2) in (let TMP_29 \def (CHead e1 k0 u0) in (let +TMP_30 \def (clear_gen_bind b c4 TMP_29 u2 H4) in (let H5 \def (f_equal C C +TMP_25 TMP_26 TMP_28 TMP_30) in (let TMP_31 \def (\lambda (e: C).(match e +with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) in (let +TMP_32 \def (CHead e1 k0 u0) in (let TMP_33 \def (Bind b) in (let TMP_34 \def +(CHead c4 TMP_33 u2) in (let TMP_35 \def (CHead e1 k0 u0) in (let TMP_36 \def +(clear_gen_bind b c4 TMP_35 u2 H4) in (let H6 \def (f_equal C K TMP_31 TMP_32 +TMP_34 TMP_36) in (let TMP_37 \def (\lambda (e: C).(match e with [(CSort _) +\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) in (let TMP_38 \def (CHead e1 +k0 u0) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead c4 TMP_39 u2) +in (let TMP_41 \def (CHead e1 k0 u0) in (let TMP_42 \def (clear_gen_bind b c4 +TMP_41 u2 H4) in (let H7 \def (f_equal C T TMP_37 TMP_38 TMP_40 TMP_42) in +(let TMP_78 \def (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 +c4)).(let TMP_43 \def (Bind b) in (let TMP_50 \def (\lambda (k1: K).(let +TMP_47 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_44 \def (Bind b) in +(let TMP_45 \def (CHead c3 TMP_44 u1) in (let TMP_46 \def (CHead e2 k1 u3) in +(getl O TMP_45 TMP_46)))))) in (let TMP_48 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) in (let TMP_49 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u0))) in (ex3_2 C T TMP_47 TMP_48 TMP_49))))) in (let TMP_58 \def (\lambda +(t: T).(let TMP_55 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_51 \def +(Bind b) in (let TMP_52 \def (CHead c3 TMP_51 u1) in (let TMP_53 \def (Bind +b) in (let TMP_54 \def (CHead e2 TMP_53 u3) in (getl O TMP_52 TMP_54))))))) +in (let TMP_56 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let +TMP_57 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))) in (ex3_2 C T +TMP_55 TMP_56 TMP_57))))) in (let TMP_66 \def (\lambda (c: C).(let TMP_63 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_59 \def (Bind b) in (let +TMP_60 \def (CHead c3 TMP_59 u1) in (let TMP_61 \def (Bind b) in (let TMP_62 +\def (CHead e2 TMP_61 u3) in (getl O TMP_60 TMP_62))))))) in (let TMP_64 \def +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) in (let TMP_65 \def (\lambda +(_: C).(\lambda (u3: T).(pr0 u3 u2))) in (ex3_2 C T TMP_63 TMP_64 TMP_65))))) +in (let TMP_71 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_67 \def (Bind +b) in (let TMP_68 \def (CHead c3 TMP_67 u1) in (let TMP_69 \def (Bind b) in +(let TMP_70 \def (CHead e2 TMP_69 u3) in (getl O TMP_68 TMP_70))))))) in (let +TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) in (let TMP_73 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) in (let TMP_74 \def +(getl_refl b c3 u1) in (let TMP_75 \def (ex3_2_intro C T TMP_71 TMP_72 TMP_73 +c3 u1 TMP_74 H0 H2) in (let TMP_76 \def (eq_ind_r C c4 TMP_66 TMP_75 e1 H9) +in (let TMP_77 \def (eq_ind_r T u2 TMP_58 TMP_76 u0 H7) in (eq_ind_r K TMP_43 +TMP_50 TMP_77 k0 H8)))))))))))))) in (let TMP_79 \def (TMP_78 H6) in (TMP_79 +H5)))))))))))))))))))))))))) in (let TMP_106 \def (\lambda (f: F).(\lambda +(H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_81 \def (CHead +e1 k0 u0) in (let TMP_82 \def (drop_refl c4) in (let TMP_83 \def (CHead e1 k0 +u0) in (let TMP_84 \def (clear_gen_flat f c4 TMP_83 u2 H4) in (let TMP_85 +\def (getl_intro O c4 TMP_81 c4 TMP_82 TMP_84) in (let H5 \def (H1 O e1 u0 k0 +TMP_85) in (let TMP_87 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_86 +\def (CHead e2 k0 u3) in (getl O c3 TMP_86)))) in (let TMP_88 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_89 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_93 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_90 \def (Flat f) in (let TMP_91 \def (CHead c3 +TMP_90 u1) in (let TMP_92 \def (CHead e2 k0 u3) in (getl O TMP_91 +TMP_92)))))) in (let TMP_94 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in +(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_105 \def +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 +x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_100 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_97 \def (Flat f) in (let +TMP_98 \def (CHead c3 TMP_97 u1) in (let TMP_99 \def (CHead e2 k0 u3) in +(getl O TMP_98 TMP_99)))))) in (let TMP_101 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 e1))) in (let TMP_102 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u3 u0))) in (let TMP_103 \def (CHead x0 k0 x1) in (let TMP_104 \def +(getl_flat c3 TMP_103 O H6 f u1) in (ex3_2_intro C T TMP_100 TMP_101 TMP_102 +x0 x1 TMP_104 H7 H8))))))))))) in (ex3_2_ind C T TMP_87 TMP_88 TMP_89 TMP_96 +TMP_105 H5))))))))))))))))) in (let TMP_107 \def (CHead c4 k u2) in (let +TMP_108 \def (CHead e1 k0 u0) in (let TMP_109 \def (getl_gen_O TMP_107 +TMP_108 H3) in (K_ind TMP_24 TMP_80 TMP_106 k TMP_109))))))))))) in (let +TMP_117 \def (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall +(u3: T).(\forall (k1: K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to +(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead +e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: +T).(\forall (k1: K).((getl (S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (let +TMP_114 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_111 \def (S n) in +(let TMP_112 \def (CHead c3 k0 u1) in (let TMP_113 \def (CHead e2 k1 u4) in +(getl TMP_111 TMP_112 TMP_113)))))) in (let TMP_115 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_116 \def (\lambda (_: +C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T TMP_114 TMP_115 +TMP_116))))))))))) in (let TMP_144 \def (\lambda (b: B).(\lambda (n: +nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl +n (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda +(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: +K).(\lambda (H4: (getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let +TMP_118 \def (Bind b) in (let TMP_119 \def (CHead e1 k0 u0) in (let TMP_120 +\def (getl_gen_S TMP_118 c4 TMP_119 u2 n H4) in (let H5 \def (H1 n e1 u0 k0 +TMP_120) in (let TMP_122 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_121 +\def (CHead e2 k0 u3) in (getl n c3 TMP_121)))) in (let TMP_123 \def (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_124 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_129 \def (\lambda (e2: +C).(\lambda (u3: T).(let TMP_125 \def (S n) in (let TMP_126 \def (Bind b) in +(let TMP_127 \def (CHead c3 TMP_126 u1) in (let TMP_128 \def (CHead e2 k0 u3) +in (getl TMP_125 TMP_127 TMP_128))))))) in (let TMP_130 \def (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_131 \def (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_132 \def (ex3_2 C T TMP_129 +TMP_130 TMP_131) in (let TMP_143 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 +e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_137 \def (\lambda (e2: C).(\lambda +(u3: T).(let TMP_133 \def (S n) in (let TMP_134 \def (Bind b) in (let TMP_135 +\def (CHead c3 TMP_134 u1) in (let TMP_136 \def (CHead e2 k0 u3) in (getl +TMP_133 TMP_135 TMP_136))))))) in (let TMP_138 \def (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 e1))) in (let TMP_139 \def (\lambda (_: C).(\lambda (u3: +T).(pr0 u3 u0))) in (let TMP_140 \def (Bind b) in (let TMP_141 \def (CHead x0 +k0 x1) in (let TMP_142 \def (getl_head TMP_140 n c3 TMP_141 H6 u1) in +(ex3_2_intro C T TMP_137 TMP_138 TMP_139 x0 x1 TMP_142 H7 H8)))))))))))) in +(ex3_2_ind C T TMP_122 TMP_123 TMP_124 TMP_132 TMP_143 H5)))))))))))))))))))) +in (let TMP_173 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +(CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat -f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 2103 -END *) +(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_145 \def (S n) +in (let TMP_146 \def (Flat f) in (let TMP_147 \def (CHead e1 k0 u0) in (let +TMP_148 \def (getl_gen_S TMP_146 c4 TMP_147 u2 n H4) in (let H5 \def (H1 +TMP_145 e1 u0 k0 TMP_148) in (let TMP_151 \def (\lambda (e2: C).(\lambda (u3: +T).(let TMP_149 \def (S n) in (let TMP_150 \def (CHead e2 k0 u3) in (getl +TMP_149 c3 TMP_150))))) in (let TMP_152 \def (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) in (let TMP_153 \def (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u0))) in (let TMP_158 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_154 +\def (S n) in (let TMP_155 \def (Flat f) in (let TMP_156 \def (CHead c3 +TMP_155 u1) in (let TMP_157 \def (CHead e2 k0 u3) in (getl TMP_154 TMP_156 +TMP_157))))))) in (let TMP_159 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e2 e1))) in (let TMP_160 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) +in (let TMP_161 \def (ex3_2 C T TMP_158 TMP_159 TMP_160) in (let TMP_172 \def +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 +x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_166 +\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_162 \def (S n) in (let +TMP_163 \def (Flat f) in (let TMP_164 \def (CHead c3 TMP_163 u1) in (let +TMP_165 \def (CHead e2 k0 u3) in (getl TMP_162 TMP_164 TMP_165))))))) in (let +TMP_167 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_168 +\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_169 \def +(Flat f) in (let TMP_170 \def (CHead x0 k0 x1) in (let TMP_171 \def +(getl_head TMP_169 n c3 TMP_170 H6 u1) in (ex3_2_intro C T TMP_166 TMP_167 +TMP_168 x0 x1 TMP_171 H7 H8)))))))))))) in (ex3_2_ind C T TMP_151 TMP_152 +TMP_153 TMP_161 TMP_172 H5))))))))))))))))))))) in (let TMP_174 \def (K_ind +TMP_117 TMP_144 TMP_173 k) in (nat_ind TMP_18 TMP_110 TMP_174 +h)))))))))))))))) in (wcpr0_ind TMP_5 TMP_12 TMP_175 c2 c1 H)))))).