From f73bd1c1cdd504c2a991071505b2e4f541791a7f Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Mon, 26 May 2008 12:30:21 +0000 Subject: [PATCH] - some bugs fixed in the domain-based preorders on environments - some missing lemmas added --- .../LAMBDA-TYPES/Base-1/types/defs.ma | 8 + .../LAMBDA-TYPES/LambdaDelta-1/T/props.ma | 16 + .../LAMBDA-TYPES/LambdaDelta-1/aprem/fwd.ma | 23 +- .../LambdaDelta-1/arity/subst0.ma | 2 - .../LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma | 236 +- .../LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma | 28 +- .../LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma | 3 + .../LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma | 2327 ++++++++++++----- .../LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma | 793 ++++-- .../LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma | 846 +++--- .../LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma | 8 +- .../LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma | 189 +- .../LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma | 3 + .../LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma | 3 + .../LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma | 235 +- .../LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma | 737 ++++-- .../LambdaDelta-1/csubst0/clear.ma | 100 + .../LambdaDelta-1/csubst0/drop.ma | 244 ++ .../LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma | 190 ++ .../LambdaDelta-1/csubst0/getl.ma | 42 + .../LambdaDelta-1/csubst1/getl.ma | 2 - .../LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma | 15 +- .../LAMBDA-TYPES/LambdaDelta-1/csubt/csuba.ma | 39 + .../LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma | 4 +- .../LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma | 642 ++--- .../LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma | 252 +- .../LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma | 428 +-- .../LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma | 52 +- .../LAMBDA-TYPES/LambdaDelta-1/csubv/clear.ma | 188 ++ .../LAMBDA-TYPES/LambdaDelta-1/csubv/defs.ma | 30 + .../LAMBDA-TYPES/LambdaDelta-1/csubv/drop.ma | 112 + .../LAMBDA-TYPES/LambdaDelta-1/csubv/getl.ma | 84 + .../LAMBDA-TYPES/LambdaDelta-1/csubv/props.ma | 42 + .../LAMBDA-TYPES/LambdaDelta-1/definitions.ma | 2 + .../LAMBDA-TYPES/LambdaDelta-1/flt/props.ma | 10 + .../LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma | 22 + .../LAMBDA-TYPES/LambdaDelta-1/lift/props.ma | 49 + .../LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma | 3 +- .../LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma | 18 + .../LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma | 11 + .../LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma | 8 + .../LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma | 314 +-- .../LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma | 209 +- .../LAMBDA-TYPES/LambdaDelta-1/spare.ma | 406 --- .../LAMBDA-TYPES/LambdaDelta-1/theory.ma | 4 + .../LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma | 85 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma | 6 + .../LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma | 2 - .../LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma | 50 + .../LAMBDA-TYPES/LambdaDelta-1/ty3/fwd_nf2.ma | 286 ++ .../LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma | 90 +- 51 files changed, 6385 insertions(+), 3113 deletions(-) create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/csuba.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/clear.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/defs.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/drop.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/getl.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd_nf2.ma diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/types/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/types/defs.ma index cddd83fd9..6699002d5 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/types/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/types/defs.ma @@ -38,6 +38,14 @@ inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def | or4_intro2: P2 \to (or4 P0 P1 P2 P3) | or4_intro3: P3 \to (or4 P0 P1 P2 P3). +inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop +\def +| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4) +| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4) +| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4) +| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4) +| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4). + inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop): Prop \def | ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma index 575d918aa..9e1bb8cfd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma @@ -32,6 +32,22 @@ B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind False H0)). +theorem not_abbr_void: + not (eq B Abbr Void) +\def + \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | +Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind +False H0)). + +theorem not_abst_void: + not (eq B Abst Void) +\def + \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind +False H0)). + theorem thead_x_y_y: \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to (\forall (P: Prop).P)))) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/fwd.ma index 431b69d75..7a8e147ab 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/fwd.ma @@ -53,18 +53,17 @@ return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) -in (\lambda (H6: (eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(eq A a a1)) -(refl_equal A a1) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda -(i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i O) \to ((eq -A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda (H4: (eq -nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a4) -\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3 -a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A a4 -(AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 +in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: +A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i +O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda +(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let +H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) +with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 +a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e +in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ +a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A +a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A +a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma index 930e4c1ba..82894da35 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma @@ -20,8 +20,6 @@ include "LambdaDelta-1/fsubst0/fwd.ma". include "LambdaDelta-1/csubst0/getl.ma". -include "LambdaDelta-1/csubst0/props.ma". - include "LambdaDelta-1/subst0/dec.ma". include "LambdaDelta-1/subst0/fwd.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma index c898adc91..7b5623ac8 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma @@ -20,7 +20,7 @@ include "LambdaDelta-1/csuba/props.ma". include "LambdaDelta-1/arity/props.ma". -include "LambdaDelta-1/T/props.ma". +include "LambdaDelta-1/csubv/getl.ma". theorem csuba_arity: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 @@ -103,29 +103,59 @@ H2)))))))))) c1 t a H))))). theorem csuba_arity_rev: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a))))))) +t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 +t a)))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: (arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c2 -c)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: ((\forall -(c2: C).((csuba g c2 d) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda -(H3: (csuba g c2 c)).(let H4 \def (csuba_getl_abbr_rev g c d u i H0 c2 H3) in -(or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1))))) (arity g c2 -(TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 +A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 +a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: +(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda +(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u +a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to +(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 +c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i +H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 +(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d +u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity +g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) -(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: (getl i c2 (CHead x -(Bind Abbr) u))).(\lambda (H7: (csuba g x d)).(arity_abbr g c2 x u i H6 a0 -(H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x +(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf +c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 +(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let +H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 +(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 +(Bind x0) x2) H11)) in (let H13 \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 x1 (Bind x0) x2) (getl_mono +c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead +d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \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 x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 +(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda +(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c +(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda +(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def +(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def +(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16) +in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) +H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda @@ -135,49 +165,128 @@ A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: A).(\lambda (H6: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: -(csuba g x0 d)).(\lambda (H8: (arity g x0 x1 (asucc g x2))).(\lambda (H9: -(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H6 -x2 H8) a0 (arity_mono g d u x2 H9 a0 H1))))))))) H5)) H4)))))))))))) (\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u -(asucc g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to (arity g -c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(let H4 -\def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (ex2_ind C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) -(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x -(Bind Abst) u))).(\lambda (H6: (csuba g x d)).(arity_abst g c2 x u i H5 a0 -(H2 x H6))))) H4)))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity -g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c -(Bind b) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba -g c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) -u) (csuba_head g c2 c H5 (Bind b) u)))))))))))))))) (\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda -(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g -a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c -(Bind Abst) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: -(csuba g c2 c)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind -Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda -(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u a1))))).(\lambda +(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: +(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10: +(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7 +x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void) +x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c +H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2: +C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i +c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2: +C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c +(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u) +(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c +(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d +(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 +(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g +a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to +(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 +c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i +H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) +(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 +(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) +u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 +Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 +(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let +H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 +(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 +(Bind x0) x2) H11)) in (let H13 \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 Abst) u) (CHead x1 (Bind x0) x2) (getl_mono +c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead +d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \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 +Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 +(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda +(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c +(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda +(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def +(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def +(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) +in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) +H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl +i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let +H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in +(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: +C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef +i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda +(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d +(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) +(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in +(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in +C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d +(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 +(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall +(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda +(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 +a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to +((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: +C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b +H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c +H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g +a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to +(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda +(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: +C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst) +u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 +c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 +(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u) +(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall +(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda -(H3: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 (AHead a1 -a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(arity_appl g c2 u a1 -(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: -((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g -a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: -((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0))))).(\lambda (c2: -C).(\lambda (H4: (csuba g c2 c)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 -H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: -(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity -g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: -C).(\lambda (H3: (csuba g c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 -H2)))))))))) c1 t a H))))). +(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 +(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda +(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 +H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda +(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 +c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0: +T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g +c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda +(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2 +H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda +(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: +C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda +(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g +c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 +H2))))))))))) c1 t a H))))). theorem arity_appls_appl: \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c @@ -199,8 +308,9 @@ c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g (CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t -a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v -H))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: +a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) +(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst) +Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: ((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma index 8c424f5d8..83977c438 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma @@ -49,8 +49,18 @@ C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: -(arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u +(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: +(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba +g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) +u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) +(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: +((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) +(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda +(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) @@ -90,8 +100,18 @@ C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda -(e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: -(arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u +(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: +(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: +C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: +C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 +(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 +u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: +((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) +(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda +(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma index 6d09db0c9..7a23f50e7 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma @@ -20,6 +20,9 @@ inductive csuba (g: G): C \to (C \to Prop) \def | csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n)) | csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u)))))) +| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) | csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma index 8af712723..6b30e0e1b 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma @@ -147,41 +147,42 @@ Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g -c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead -d2 (Bind Void) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: +c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) -t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) +d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C +c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead +x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def +(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead +x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal +nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: +nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in +(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n +x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 +H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) +c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let +H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) +O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) +x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x -H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead -x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4)))) -(\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda -(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def -(csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda -(H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2 -C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 H7) in (ex2_ind C -(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) u))).(\lambda -(H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x -(drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) H8)) c2 -H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 +H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) +u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) +u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) +O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) +H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n H1)))))))))))) c1)))) i). theorem csuba_drop_abst: @@ -628,147 +629,149 @@ C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 -\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) -(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) -t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) +A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c +x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g +x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead -x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def -(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda -(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) -in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 +(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) -u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda +(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop +(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: +C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: +(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 +\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) +u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n +x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda +C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 -x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 -x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 -(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: +(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 +x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 +x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def +(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) +H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 +(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 +H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: +F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r +(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c +c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda +(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 +(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) +x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: +C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Void) -n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c -(Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) -u1))).(let H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g c d2))) (or (ex2 C (\lambda -(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c -x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H8 \def (H0 d1 u1 H4 g -x0 H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda +(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H9: (ex2 C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: +(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 +x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) -u1))).(\lambda (H11: (csuba g d1 x)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) +(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) +H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n -x0 (CHead x (Bind Abst) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda @@ -800,778 +803,1654 @@ Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 theorem csuba_drop_abst_rev: \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c2 c1) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))))))))) +c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))))))))) \def \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))) -(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 -(CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: -(csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 -(CHead d1 (Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in -(let H_x \def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in -(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H3: -(eq C c2 (CHead x (Bind Abst) u))).(\lambda (H4: (csuba g x d1)).(eq_ind_r C -(CHead x (Bind Abst) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop O O (CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abst) u)) H4) -c2 H3)))) H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: -C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) -\to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda -(d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n +O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: +C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind +Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 +c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 +(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x +\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or +(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C +(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) +u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O +(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x +(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind +Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x +(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) +x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O +(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind +Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: +((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 +(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or +(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S -n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))))) +(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop +(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind -Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: +Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: -(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq +nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) -u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: +\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) +H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) -O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba -g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u)) \to -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1)))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 (CHead -c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) -u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: +(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop +(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda +(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: +K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) +u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3: +(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c +(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c +(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) +t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) +u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in +(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda +(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def -(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (H8: (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba -g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: -C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x -c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) +(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) +t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) +(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x +H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind +Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H12: -(drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H13: (csuba g x0 d1)).(let -H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n -(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H12 (r (Bind Abst) -n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) +(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: +(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind +Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abst) u) H15 t) H13)))))) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop +(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) +H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc +g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) +(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t -x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C +(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: +(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind +Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) +(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda -(H14: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H15: (csuba g x -d1)).(let H16 \def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abst) u))) H14 -(r (Bind Abst) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H17 x1) H15)))))) -H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind -Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) +x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 +(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) +H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: +(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r +C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: +C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O +x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C +(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x +d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda +(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) +H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) +t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in -(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: -C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: -(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) -O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 -\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead -x0 (Bind Abst) u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal -nat (r (Bind Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) -(\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r -(Bind Void) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def -(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x -(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind -Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 \def (H c d1 u -H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal nat (r (Bind -Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in (ex_intro2 C +(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda +(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba +g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 +(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x +(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H +c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) +u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) +H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead +x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) +H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: +(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r +C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: +C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O +x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C +(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x +d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda +(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) +H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) +t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) +u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in +(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: +C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq +C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S +n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H +c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) +u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) +H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead +x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead -x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4)))) -(\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda -(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def -(csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 c))) (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) -x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) x1) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abst) -u))).(\lambda (H10: (csuba g x d1)).(ex_intro2 C (\lambda (d2: C).(drop (S n) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) +H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 +(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind +Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def +H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 +(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) +x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 +H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H9 x1) H10)))) +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat +f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) +u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 +x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 +(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n H1)))))))))))) c1)))) i). theorem csuba_drop_abbr_rev: \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))))))))) +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) \def \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n +G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: -T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: -G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 -(\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl -c1 (CHead d1 (Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 -u1 H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H3: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda +(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: +C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 +(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in +(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x -(Bind Abbr) u1))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind -Abbr) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) -u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind -Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abbr) u1)) H5)) c2 -H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H5: (csuba g x0 d1)).(\lambda (H6: (arity g x0 x1 -(asucc g x2))).(\lambda (H7: (arity g d1 u1 x2)).(eq_ind_r C (CHead x0 (Bind -Abst) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x +d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C +(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: +C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) +u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl +(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abst) x1)) H5 -H6 H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 -(Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to -(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g +x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 +u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C +(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: +C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 +(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda +(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: -C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind -Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort -n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort n0)) (eq nat (S n) O) -(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) (CSort n0))).(\lambda (H3: (eq -nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H5))))) +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C +c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C +(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop +O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) +(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind +Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 +x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) +(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall +(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall +(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall +(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: +nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) +(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: +(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort +n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O +c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) +(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let +H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) +in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda -(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abbr) -u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g c2 (CHead c k -t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) -O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop +(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda +(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: +K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) +u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda +(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop +(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: +B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead +d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: +(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba +g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 -(CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind -Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to -((drop (r (Bind b0) n) O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda -(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda +(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq +C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop +(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind +(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H5: (csuba g c2 -(CHead c (Bind Abbr) t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 -(Bind Abbr) u1))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 -\def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) -t))) (\lambda (d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H8: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: -C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) +t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x -(Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind -Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: +(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop +(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda +(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H12: -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) -u1))).(\lambda (H14: (csuba g x0 d1)).(let H15 \def (refl_equal nat (r (Bind -Abst) n)) in (let H16 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u1))) H13 (r (Bind Abst) n) H15) in (or_introl (ex2 C +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g -d2 d1)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H16 t) -H14))))))) H12)) (\lambda (H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 +(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 +x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) -x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 x1 (asucc g -x2))).(\lambda (H16: (arity g d1 u1 x2)).(let H17 \def (refl_equal nat (r -(Bind Abst) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O -x (CHead x0 (Bind Abst) x1))) H13 (r (Bind Abst) n) H17) in (or_intror (ex2 C +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n +x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) x1) H18 t) -H14 H15 H16))))))))))) H12)) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc -g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind +Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) +H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t a)))) (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind +Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc +g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) +(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g -x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t -x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C -(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H13 \def -(H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) +x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x +d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) +x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 +(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: +A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: +(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: +(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead +x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 +(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 +H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H14: (ex2 C (\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) +x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind +Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 +H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) +x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) +(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abbr) -u1))).(\lambda (H16: (csuba g x d1)).(let H17 \def (refl_equal nat (r (Bind -Abst) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 -(CHead x (Bind Abbr) u1))) H15 (r (Bind Abst) n) H17) in (or_introl (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abbr) u1) H18 x1) -H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x +d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 +(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: +(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: +(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 +H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (x4: -T).(\lambda (x5: A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) -x4))).(\lambda (H16: (csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g -x5))).(\lambda (H18: (arity g d1 u1 x5)).(let H19 \def (refl_equal nat (r -(Bind Abst) n)) in (let H20 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O -x0 (CHead x3 (Bind Abst) x4))) H15 (r (Bind Abst) n) H19) in (or_intror (ex2 -C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) x3 x4 x5 (drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H20 -x1) H16 H17 H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda -(H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda (H6: (drop (r (Bind Abst) -n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abst_rev g c c2 t -H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind +Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) +H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda +(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba +g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 +(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: +(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: +C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: -(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S -n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 -C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq +C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop +(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x -(CHead x0 (Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def -(refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n (\lambda -(n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) -in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) -u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x -(CHead x0 (Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: -(arity g x0 x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 -\def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def (eq_ind nat n -(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H12 (r (Bind -Abst) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: +(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop +(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda +(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 +(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 +x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n -x (CHead x0 (Bind Abst) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda -(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda -(H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or -(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda +x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind +Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) +H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 +(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda +T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H10 \def -(H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) +x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) +(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n -O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind -C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: -C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H13: -(csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) -u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x +d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 +(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: +(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: +(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) -n x (CHead x0 (Bind Abbr) u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 +H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind +Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) +H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda +(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: +C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x +c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x +H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 +(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) +(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) +(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x -(CHead x0 (Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: -(arity g x0 x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 -\def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def (eq_ind nat n -(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H12 (r (Bind -Abst) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 +(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 +x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n -x (CHead x0 (Bind Abst) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c -(Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) +x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind +Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) +H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat +f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C (\lambda +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 -c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda +c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 -H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 +H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S -n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) -u1))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Flat f) n -x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n) -O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda -(H12: (arity g x2 x3 (asucc g x4))).(\lambda (H13: (arity g d1 u1 -x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda +(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) +u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind -Abst) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 -(drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: +T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) +x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g +x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 +(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 +H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 +(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 +(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k +H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma index 9635a44a0..34d188209 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma @@ -59,71 +59,115 @@ u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow -True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 -(Bind Abbr) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 -(Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -H6)))))))))))) y c H0))) H))))). +d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1 +(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void +\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) +u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: +(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: +T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) +t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) +u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) +y c H0))) H))))). theorem csuba_gen_void: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g -(CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g +(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) \def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Void) u) c)).(insert_eq C (CHead d1 (Bind Void) u) -(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda -(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 -d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) -(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) +u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda +(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) +(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: +C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind +Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C +return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C +T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 +(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) +(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) 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 _ k0 _) \Rightarrow k0])) (CHead c1 k -u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \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 c1 k u0) (CHead d1 (Bind Void) u) H3) +u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \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 c1 k u) (CHead d1 (Bind Void) u1) H3) in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Void) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C -c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 -d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 -c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) -u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 -(refl_equal C (CHead c2 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C +T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda +(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind +Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda +(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind +C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g +c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) +Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6)))) +H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 +(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in +((let H6 \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 +c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1 +d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind +Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3: +T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C +c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda +(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead +d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba +g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) +H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc +g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 -(Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 -(Bind Abbr) u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))) +(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H6)))))))))))) y c H0))) H))))). theorem csuba_gen_abst: @@ -209,40 +253,62 @@ A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u -a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) -u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind -Abst) t) (CHead d1 (Bind Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 -\def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in -(let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 -d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 -(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) -H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 -d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) -u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) +(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b +Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind +Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind +Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | +Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind +Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind +b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 +C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind +b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 +(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc +g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t) +(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | +(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind +Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t +(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def +(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let +H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1)) +\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8) +in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in +(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity +g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: +A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 @@ -297,6 +363,18 @@ u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 +(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 +(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: +T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat +f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C @@ -361,89 +439,172 @@ H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: -T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ -_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead -c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst -b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: -T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let H12 \def (eq_ind C c1 -(\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) in (let H13 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C -T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c -c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 -(CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: +B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) +v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) +(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 +\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 c1 +(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void +b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: +C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 +H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9) +in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind +b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) +u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) +H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 +B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t +(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: +(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind +Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda +(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind +Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda +(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t +| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind +b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 +e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) +H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc +g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H13 Abst H9) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 -(Bind Abbr) u)) H14))))))))) H7)) H6)))))))))))) y c2 H0))) H)))))). +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let +H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 +\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to +(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) +H6)))))))))))) y c2 H0))) H)))))). theorem csuba_gen_abst_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))) +(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda -(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: +C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: +C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: +C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C +c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: +nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def +(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) -(CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) 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 _ k0 _) \Rightarrow k0])) (CHead c2 k -u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \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 c2 k u0) (CHead d1 (Bind Abst) u) H3) -in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Abst) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 -c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) -u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 -(refl_equal C (CHead c1 (Bind Abst) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda +(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C +(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k +u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | +(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) 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 _ k0 _) \Rightarrow k0])) +(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \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 c2 k u0) (CHead d1 +(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C +c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C +(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) +(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C +(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g +c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind +Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind +Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 +(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not +(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead +c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 +| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind +Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 +(Bind Abst) u) H4) in ((let H7 \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 c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in +(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def +(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11 +\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 +(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g +c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind +Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) +(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 +(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) +H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: +(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or +(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc +g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Abst) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -H6)))))))))))) c y H0))) H))))). +d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead +c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind +Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1))))) H6)))))))))))) c y H0))) H))))). theorem csuba_gen_void_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c @@ -485,7 +646,30 @@ d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 +(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in +((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in +((let H7 \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 +c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b +Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: +B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda +(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C +c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) +in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in +(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_: +False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13))))))) +H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g @@ -502,48 +686,56 @@ H6)))))))))))) c y H0))) H))))). theorem csuba_gen_abbr_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c -(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))))))) +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda (H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) -u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda +(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g c -y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind -Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or (ex2 C -(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda +(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C +(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind +Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C +return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2 +C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C -c2 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) +c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 @@ -553,75 +745,132 @@ u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \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 c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead +T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda (d2: +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H9 \def (eq_ind C c2 -(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C +c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C +(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 -(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda -(d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba +g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 +(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C +(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) +u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) +(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \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 c2 (Bind b) +u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda +(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 +Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) +u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in +(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in +(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) +(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: +A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq +C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: +C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) +u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C +(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) -(CHead d1 (Bind Abbr) u1) H5) in ((let H7 \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 c2 (Bind Abbr) u) (CHead d1 (Bind -Abbr) u1) H5) in (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u -(\lambda (t0: T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 -(\lambda (c0: C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 -(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c2 -(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_intror (ex2 C (\lambda -(d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: +T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) +u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | +(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind +Abbr) u1) H5) in ((let H7 \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 c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in +(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity +g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g +c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 -a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))) c1 t a -(refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) H6)))))))))))) c y -H0))) H))))). +a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8) +in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in +(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: +A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 +(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 +u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) +H6)))))))))))) c y H0))) H))))). theorem csuba_gen_flat_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall @@ -671,6 +920,18 @@ u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 +(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 +(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: +T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat +f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C @@ -735,32 +996,58 @@ H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: -T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) -u) (CHead e1 (Bind b1) v1))).(let H6 \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 (Bind Abbr) u) (CHead e1 (Bind b1) v1) 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 _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abbr])])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in -((let H8 \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 -c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abbr -b1)).(\lambda (H10: (eq C c3 e1)).(let H11 \def (eq_ind T u (\lambda (t0: -T).(arity g c3 t0 a)) H4 v1 H8) in (let H12 \def (eq_ind C c3 (\lambda (c: -C).(arity g c v1 a)) H11 e1 H10) in (let H13 \def (eq_ind C c3 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 -H10) in (let H14 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H10) -in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let +H5 \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 +(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B +(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 +(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \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 c3 (Bind b) u2) (CHead +e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3 +e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1 +H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C +c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda +(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) +H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 +B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t +(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: +(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \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 (Bind +Abbr) u) (CHead e1 (Bind b1) v1) 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 _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind +Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \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 c3 (Bind Abbr) u) (CHead e1 (Bind +b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 +e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) +in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) +in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csuba g e2 e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) +(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 +(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 +(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda +(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) +v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 +e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) +v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) +Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) H6)))))))))))) c2 y H0))) H)))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma index 6072645a0..8b7cab309 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma @@ -474,40 +474,51 @@ H2)))) H0))))))). theorem csuba_getl_abst_rev: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g -c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))))))))) +c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))))))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) -(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (x: +(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda -(H4: (clear (CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 -(Bind Abst) u) n H4 (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda +(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: -(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 -(Bind Abst) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to -((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u)) \to (\forall (c2: -C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda -(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 -(Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) +(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear +(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) +n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) +\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i +c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: +(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: +K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind +Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: +B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear +(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) +(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in +((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 +in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with @@ -519,177 +530,297 @@ t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u -H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda -(x1: C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H18: -(csuba g x1 d1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 -(Bind Abst) u) (CHead x1 (Bind Abst) u) H17 (clear_bind Abst x1 u)) H18)))) -H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead -x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind -Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c -(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n -O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (ex2 C -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead -x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 -\def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) -(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 -(CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) -f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 -(CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 -(CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda -(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst) -u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2 -u H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda -(H16: (csuba g x3 d1)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 -c)) H13 (CHead x3 (Bind Abst) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl -O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x3 -(getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H17) H16))))) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) -\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O -x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 -x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B -C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind -b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead -x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x2: B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) -x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def +H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C +(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: +C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: +(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i +c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 +(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18 +(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: +C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void) +x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 +(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1 +x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: +(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) +t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: +C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) +\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop +n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or +(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) +t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C +x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 +(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind +Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11 +\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) +u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u))) +(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead +d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12: +(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x +\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2 +C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead +d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) +u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) +u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: +C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: +C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void) +x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 +x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18) +H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: +((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: +C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: +(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: +(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in +(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 +(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: +B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind +x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x5: C).(\lambda (H15: -(csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x -\def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in -(ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 x3)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x6: B).(\lambda (x7: -C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) -x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: -C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 -x7 H19) in (ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x9: -C).(\lambda (H22: (getl n x7 (CHead x9 (Bind Abst) u))).(\lambda (H23: (csuba -g x9 d1)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 -(CHead x9 (Bind Abst) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) -H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). +C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba +g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def +(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C +T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind +b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 +x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: +T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g +x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead +x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl +n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl +(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl +(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 +H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl +(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: +C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void) +x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 +(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) +H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) +x H1 H2)))) H0))))))). theorem csuba_getl_abbr_rev: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba -g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) -(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 +(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: -(clear x (CHead d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) -\to ((clear c (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 -c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda -(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) -(CHead d1 (Bind Abbr) u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 -(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear -x0 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) +(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead +d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c +(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear -(CHead x0 k t) (CHead d1 (Bind Abbr) u1))).(K_ind (\lambda (k0: K).((drop i O -c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: +(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) +u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) +\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) -t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) -u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead -d1 (Bind Abbr) u1) t H6)) in ((let H8 \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 d1 (Bind -Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) -u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B -Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba -g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 -(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: -B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def -(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1 -H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in -(or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 +(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) +u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear +(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 +c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H17: (ex2 C (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: +B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear +(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1) +(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) +in ((let H8 \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 d1 (Bind Abbr) u1) (CHead x0 (Bind b) +t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind +Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) +u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 +x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r +T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let +H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) +u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O +c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def +(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) -u1))).(\lambda (H19: (csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C +(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 +(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i +c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda @@ -700,90 +831,131 @@ A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) +u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x1: +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 -x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or_intror (ex2 C +x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 -(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) -H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) +(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind +Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead +x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) +H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda +(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda -(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u1) -(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u1) t H6) f t) in (let H11 \def -(csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u1) -H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) -u1))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 -u1 H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop +n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 +(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: +(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g +c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead +x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def +(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind +Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) +t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: +C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) +(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H15: -(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: +C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: +(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 +\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead -x3 (Bind Abbr) u1))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C -x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u1) H16) in -(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: +C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba +g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead +x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda @@ -793,130 +965,202 @@ a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(getl -O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: +A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba +g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity +g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 +(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O +c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H16: -(eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba g x3 d1)).(\lambda -(H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity g d1 u1 x5)).(let -H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) -x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) -x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14)))))) H11)))))))) +x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind +Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 +(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in +(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 +(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda +(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S -n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 -x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B -C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind -b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead -x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 +(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let +H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T +(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) +v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 +(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 -(CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) -t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) -H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) -x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 -x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda -(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g -d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl -(S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 +(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 +\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind +C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: +C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: +(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let +H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: +T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g +x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead +x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x6: B).(\lambda (x7: -C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) -x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: -C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 -x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: +(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) +u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H22: -(ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 -(CHead x9 (Bind Abbr) u1))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u1) n H23) -H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: +A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: +(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda +(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) -(\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: (getl n -x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: (csuba g x9 d1)).(\lambda -(H25: (arity g x9 x10 (asucc g x11))).(\lambda (H26: (arity g d1 u1 -x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 -(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) H21)))))))) -H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) -H0))))))). +(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T +(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S +n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: +(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 +d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro +C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 +(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) +H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) +x H1 H2)))) H0))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma index 7f0630a13..ce99fdddd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma @@ -26,9 +26,11 @@ a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))). theorem csubc_arity_trans: \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -(\forall (t: T).(\forall (a: A).((arity g c2 t a) \to (arity g c1 t a))))))) +((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to +(arity g c1 t a)))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c2 t -a)).(csuba_arity_rev g c2 t a H0 c1 (csubc_csuba g c1 c2 H)))))))). +c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda +(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 +H) H0)))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma index f266b8bc8..69f700b8b 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma @@ -26,93 +26,142 @@ e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def -(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or_ind (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K +(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 +C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g +e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g -(CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C (\lambda (c3: C).(eq C c2 -(CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e c3)))).(ex2_ind C +a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e -c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead -e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind b) -u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x (Bind b) u) (\lambda -(c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead -e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x (Bind b) -u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)) (CHead x (Bind b) -u) (clear_bind b x u) (csubc_head g e x H4 (Bind b) u)) c2 H3)))) H2)) -(\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: +c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda +(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: +C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 +(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x +(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda +(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: +C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind +b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind +b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 -e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind b) (Bind -Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H5: -(csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: (sc3 g -x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 C -(\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) -e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return +(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda +(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind +b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda +(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: +(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 +C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) +u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) (Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 -H6 x1 H7)) b H8)) c2 H4))))))))) H2)) H1)))))))) (\lambda (e: C).(\lambda (c: -C).(\lambda (_: (clear e c)).(\lambda (H1: ((\forall (c2: C).((csubc g e c2) -\to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c -e2))))))).(\lambda (f: F).(\lambda (u: T).(\lambda (c2: C).(\lambda (H2: -(csubc g (CHead e (Flat f) u) c2)).(let H_x \def (csubc_gen_head_l g e c2 u -(Flat f) H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C c2 -(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind -Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 -(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e -u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) -(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) -(\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) -(\lambda (c3: C).(csubc g e c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 -(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x: C).(\lambda -(H5: (eq C c2 (CHead x (Flat f) u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C -(CHead x (Flat f) u) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) -(\lambda (e2: C).(csubc g c e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c -e2)) (ex2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: -C).(csubc g c e2))) (\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda -(H9: (csubc g c x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) -u) e2)) (\lambda (e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) -H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: +H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda +(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e +c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind +x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B +x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc +g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0: +K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | +(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void +(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda +(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead +e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) +(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) +(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: +((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: +T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x +\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind +(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: +C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: +A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: +C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C +(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e +c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda +(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: +C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) +u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda +(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c +e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda +(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: +C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) +(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c +x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda +(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) +H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: +A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda -(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind -Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: -(csubc g e x0)).(\lambda (_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 -x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 -\def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: -K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I -(Bind Abst) H5) in (False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind -Abbr) x1) e2)) (\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) -H3))))))))))) c1 e1 H)))). +(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 +e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: +(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda +(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C +(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 +e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) +(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in +(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) +(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4: +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: +C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda +(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) +x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0 +Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) +(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: +C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match +ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat +_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: +C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) +c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma index df5b989c7..73f8956e4 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma @@ -27,6 +27,9 @@ c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda (n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda (v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: +B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma index 9f18fca8d..9d22e520e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma @@ -20,6 +20,9 @@ inductive csubc (g: G): C \to (C \to Prop) \def | csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n)) | csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall (k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v)))))) +| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) | csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall (v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma index 3a1a9b10c..42983d318 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma @@ -52,56 +52,86 @@ c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l -g c c2 t k H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C +g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex2 C (\lambda -(e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda -(H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) (\lambda (c3: -C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) -(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: C).(drop (S n) O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H5: (eq C -c2 (CHead x k t))).(\lambda (H6: (csubc g c x)).(eq_ind_r C (CHead x k t) -(\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: -C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k n) (drop_gen_drop k c e1 t n -H1) x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r k n) O -x e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) -O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: -C).(\lambda (H8: (drop (r k n) O x x0)).(\lambda (H9: (csubc g e1 -x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda -(e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 H8 t) H9)))) H7))) c2 H5)))) -H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(drop (S n) -O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: A).(\lambda (H5: (eq K k (Bind Abst))).(\lambda (H6: (eq C -c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (csubc g c x0)).(\lambda (_: -(sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C (CHead -x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 -e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H10 \def (eq_ind K k (\lambda -(k0: K).(drop (r k0 n) O c e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H5) -in (let H11 \def (eq_ind K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1) -\to (\forall (c3: C).((csubc g (CHead c k0 t) c3) \to (ex2 C (\lambda (e2: -C).(drop n O c3 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) -H5) in (let H_x0 \def (H e1 (r (Bind Abst) n) H10 x0 H7) in (let H12 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(drop n O x0 e2)) (\lambda (e2: C).(csubc -g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H13: (drop -n O x0 x)).(\lambda (H14: (csubc g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop -(S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x -(drop_drop (Bind Abbr) n x0 x H13 x1) H14)))) H12))))) c2 H6))))))))) H4)) -H3)))))))) h))))))) c1)). +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) +(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) +(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 +(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: +C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: +C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c +x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop +(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k +n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C +(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) +(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: +C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x +x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) +O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 +H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind +Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c +t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 +C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k +(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda +(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 +g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C +(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) +(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) +(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k +(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g +(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda +(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r +(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: +C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: +C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g +e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) +e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 +x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c +c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 +e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda +(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: +(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C +(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) +(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) +(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k +(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g +(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda +(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r +(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: +C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: +C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g +e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) +e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) +H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)). theorem drop_csubc_trans: \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall @@ -169,26 +199,30 @@ x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def -(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C +(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (H10: (ex2 C -(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 -c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: -C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x: -C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x0 -x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop -h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) -c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in -(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g -c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: +a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) +(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda +(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k +x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) +(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: +(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda +(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r +k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def +H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: +C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) +(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r @@ -229,8 +263,41 @@ x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g (asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g -x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) H9))) t -H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). +x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda +(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: +C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) +x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: +(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind +Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 +x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda +(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r +k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: +nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to +(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: +C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) +x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda +(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind +Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 +(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) +c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def +H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc +g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) +(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) +c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g +c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) +x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) +n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 +x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n +x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) +(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). theorem csubc_drop_conf_rev: \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall @@ -298,15 +365,19 @@ k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def -(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C +(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -x0 x1))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C +x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq +C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: +T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda +(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda @@ -358,6 +429,38 @@ C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) (lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) -H17)))))) H18))) k H11))) e1 H12))))))))) H10)) H9))) t H4))))))))) -(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). +H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T +(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: +T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: +C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) +(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 +(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: +(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 +(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 +c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let +H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c +k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 +(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda +(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) +H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 +(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: +C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 +(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h +H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 +x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind +x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n +x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S +n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind +x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) +(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n +x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) +H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma index 18435fe3e..2ff7d012e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma @@ -36,161 +36,307 @@ in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) (CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 -v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead -c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in -(False_ind (eq C (CHead c2 (Bind Abbr) w) (CHead c1 (Bind Abst) v)) -H6)))))))))))) y x H0))) H)))). +c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort +n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) 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 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 +(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: +(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def +(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) +w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))). theorem csubc_gen_head_l: \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: -K).((csubc g (CHead c1 k v) x) \to (or (ex2 C (\lambda (c2: C).(eq C x (CHead -c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: +K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x +(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))))))) +(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T +(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 +c2))))))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) -(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or (ex2 C (\lambda (c2: +(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))) -(\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda (c: -C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c2: -C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead +c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k +(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 +c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda +(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c1 k v))).(let H2 \def -(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead c1 k v) H1) in (False_ind (or (ex2 C (\lambda (c2: C).(eq C -(CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2 -(Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) -c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))) -H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 -w))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 -v0) (CHead c1 k v))).(let H4 \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 v0) (CHead c1 k v) H3) in ((let H5 \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 v0) -(CHead c1 k v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K -k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or (ex2 C -(\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: +v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 +(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: +T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) +(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ +_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C +(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g +c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort +n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g +a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: +T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: +C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: +C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: +K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k +v))).(let H4 \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 v0) (CHead c1 k v) H3) in ((let H5 \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 v0) (CHead c1 k v) H3) in +((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead +c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq +C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C +(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead +c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc +g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g +a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 +w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 +(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) (eq_ind_r K k -(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 -k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) -(let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or -(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g -c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 +(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k +(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let +H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in +(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda +(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind +b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) +(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 +H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda +(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 +C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 +c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc -g c c2)) H1 c1 H8) in (or_introl (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) -(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: +a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not +(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead +c0 (Bind Void) u1) (CHead c1 k v))).(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 (Bind Void) u1) (CHead c1 k v) H4) +in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) +\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7 +\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 c0 +(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) +k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: +C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead +c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) -w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) -(ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) (\lambda -(c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 H7) v0 -H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: -(csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C +C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind +b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c +c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 +(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 +(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) +u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 +(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) +(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind +Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C +(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda +(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) +(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B +b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda +(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C +(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) +H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 +c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind +Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: +(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 +\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 +(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind +Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 +k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K +(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 +(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C +c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def +(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 -(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) -c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 -w))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) -c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 w)).(\lambda (H5: (eq C -(CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 \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 (Bind Abst) v0) (CHead c1 k v) H5) -in ((let H7 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) -\Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in ((let H8 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind Abst) -k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda (t: -T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 (\lambda -(c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind C c0 -(\lambda (c: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: C).(eq -C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead +c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g +(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 +g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind +C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K +k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda (c: C).(csubc g -c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 -(CHead c1 k0 v)) \to (or (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: -T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda -(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))))) H13 (Bind -Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (c3: -C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) (\lambda (c3: C).(csubc g -c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C -(CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda -(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))))) (or_intror -(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) -v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda -(_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: +w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) +(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) +(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda +(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: +w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: +C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))) c2 w a -(refl_equal K (Bind Abst)) (refl_equal C (CHead c2 (Bind Abbr) w)) H14 H12 -H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) H)))))). +(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead +c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda +(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) +(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g +(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 +g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 +(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) +H)))))). theorem csubc_gen_sort_r: \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to @@ -212,109 +358,236 @@ in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) (CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 -c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 -v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead -c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in -(False_ind (eq C (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) w)) -H6)))))))))))) x y H0))) H)))). +c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort +n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\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 c1 +(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 +(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: +(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def +(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) +v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))). theorem csubc_gen_head_r: \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: -K).((csubc g x (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: C).(eq C x (CHead -c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: +K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x +(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))))) +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T +(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2))))))))))) \def \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) -(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or (ex2 C (\lambda (c1: +(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) -(\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda (c: -C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: -C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead +c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K +k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda +(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k w))).(let H2 \def -(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead c2 k w) H1) in (False_ind (or (ex2 C (\lambda (c1: C).(eq C -(CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) (CHead c1 -(Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) -c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))) -H2)))) (\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 -c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: -C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind -Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 -w))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 -v) (CHead c2 k w))).(let H4 \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 v) (CHead c2 k w) H3) in ((let H5 \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 v) (CHead c2 k -w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow -t])) (CHead c0 k0 v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0 -k)).(\lambda (H8: (eq C c0 c2)).(eq_ind_r T w (\lambda (t: T).(or (ex2 C -(\lambda (c3: C).(eq C (CHead c1 k0 t) (CHead c3 k w))) (\lambda (c3: +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead +c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K +k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k +w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda +(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) +(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind +Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) +(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a +c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq +C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: +C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: +C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: -A).(eq C (CHead c1 k0 t) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) (eq_ind_r K k -(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: +K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let +H4 \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 +v) (CHead c2 k w) H3) in ((let H5 \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 v) (CHead c2 k w) H3) in ((let H6 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) +(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 +c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead +c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) +(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 +g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 +g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k +(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 -w))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) -\to (or (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w)))))))) H2 c2 H8) in (let H10 \def (eq_ind C -c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) in (or_introl (ex2 C (\lambda -(c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 -c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C -(CHead c1 k w) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex_intro2 C (\lambda (c3: C).(eq C -(CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2)) c1 -(refl_equal C (CHead c1 k w)) H10)))) k0 H7) v H6)))) H5)) H4))))))))) -(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda -(H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 +c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda +(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) +C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))).(\lambda (v: +(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 +H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) +in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) +(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: +T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda +(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) +(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k +w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) +H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: +C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(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 (Bind b) +u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e +in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead +_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let +H7 \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 c0 +(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda +(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead +c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda +(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: +C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda +(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) +v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 +c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g +c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 +(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) +(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: +C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda +(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) +v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind +b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 +c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 +C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda +(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) +v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind +Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: +C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) +v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind +Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: +B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead +c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K +(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K +(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda +(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k +w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C @@ -328,43 +601,61 @@ with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in -(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or +(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w)))))))) H2 c2 H10) in (let H14 \def (eq_ind C c0 (\lambda -(c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K k (\lambda -(k0: K).((eq C c2 (CHead c2 k0 w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 -(CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))))) H13 -(Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda -(c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 w))) (\lambda (c3: +A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda +(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind +C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K +k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind +Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 +c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) +c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 +w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C +c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: +K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 +w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda +(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda +(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) +v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 +v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: +C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda -(_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda -(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))))) (or_intror (ex2 -C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 -(Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g -a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 -w))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: -A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))) c1 v a (refl_equal -K (Bind Abbr)) (refl_equal C (CHead c1 (Bind Abst) v)) H14 H3 H12)) k -H9))))))))) H7)) H6)))))))))))) x y H0))) H)))))). +(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) +v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 +v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda +(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) +(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 +g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: +A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead +c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) +H)))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma index 7a1d5e009..cf1e0e68e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma @@ -14,6 +14,8 @@ (* This file was automatically generated: do not edit *********************) +include "LambdaDelta-1/csubst0/props.ma". + include "LambdaDelta-1/csubst0/fwd.ma". include "LambdaDelta-1/clear/fwd.ma". @@ -1025,3 +1027,101 @@ T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 (CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1). +theorem csubst0_clear_trans: + \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 +i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1)))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or +(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1: +C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1 +u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2) +e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear +(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda +(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2) +(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1: +C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind +b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2)) +(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) +(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda +(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear +(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1 +u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda +(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear +(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2 +(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: +T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2: +C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 +v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda +(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0: +K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C +(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead +c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u) +e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3 +(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c)) +(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear +(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0 +(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind +b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 +(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3 +(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2 +(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead +c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in +(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 +v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u) +e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear +(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear +(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f +u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda +(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x +e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead +c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f +u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 +i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0 +c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 +e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind +(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1) +e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: +C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear +(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c: +C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s +(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1))))) +(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C +(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: +C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 +(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind +b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4 +H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f: +F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2 +(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 +e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 +i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda +(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda +(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or +(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 +e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x: +C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror +(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 +e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k +H3))))))))))))) i v c1 c2 H))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma index dab261bc6..b18b7c5e6 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma @@ -6030,3 +6030,247 @@ x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n). +theorem csubst0_drop_lt_back: + \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O +c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) +v e1 e2)) (\lambda (e1: C).(drop n O c1 e1)))))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) +\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) +\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda +(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 +c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: +nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) +(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C +(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) +(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O +c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl +c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt +n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 +c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda +(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v +c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c +e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda +(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda +(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O +(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to +(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: +C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda +(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda +(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) +O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3: +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda +(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: +T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead +c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s +k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t +x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 +(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall +(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S +n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 +(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) +H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) +n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S +n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v +e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda +(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to +(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C +(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 +n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) +O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda +(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind +b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 +H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c +e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop +(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) +(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12 +t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda +(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) +(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead +c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s +k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c +x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 +(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall +(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S +n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 +(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) +H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) +n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S +n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v +e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda +(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to +(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C +(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 +n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) +O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda +(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 +H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2 +C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 +O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O +(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop +(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 +(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x +e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind +b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14)) +H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c +e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop +(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def +H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus +x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0) +O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) +(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat +f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat +f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 +(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop +(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda +(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 +(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0) +O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) +(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0 +c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i +H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0) +O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: +T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k +x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t +x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda +(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind +nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c +c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) +(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat +i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2) +(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall +(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to +((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O +(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda +(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) +x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) +O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 +e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S +n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x +\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in +(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 +e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) +t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c +e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C +(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) +(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t) +e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16: +(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror +(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 +(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) +e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 +c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda +(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in +(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c +e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl +(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 +(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) +t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda +(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) +t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda +(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c +x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 +(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x +H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 +(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c +c2 t v i H1))))))))))) c1)))))) n). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma index 585127337..3052659c0 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma @@ -261,3 +261,193 @@ u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))). +theorem csubst0_gen_S_bind_2: + \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall +(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to +(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x +(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) +(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: +T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 +(Bind b) v1)))))))))))) +\def + \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) +v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x +c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda +(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i +v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: +C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: +(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x +y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T +(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind +b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C +x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 +c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) +(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda +(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) +\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: +T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) +(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead +c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 +v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: +C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat +(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) +v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) +(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K +(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) +\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 +(Bind b) v2) H4) in ((let H7 \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 c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: +(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: +C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i +v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T +u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K +k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K +(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) +(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 +u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 +c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 +(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e +in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) +\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda +(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: +T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead +c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda +(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: +C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T +(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind +b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) +u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: +nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: +(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead +c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) +(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq +nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) +v2))).(let H6 \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 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K +(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) +\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 +(Bind b) v2) H5) in ((let H8 \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 c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: +(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: +T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i +v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C +c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) +\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead +c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: +C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda +(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) +(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda +(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 +(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 +v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: +C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let +H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda +(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) +H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to +((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i +v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 +(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 +v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: +C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in +(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i +H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda +(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind +b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 +c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead +c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 +(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) +H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: +C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq +nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda +(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) +v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C +c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 +c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) +v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C +(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \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 k u2) (CHead c2 (Bind b) v2) H6) +in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) +(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \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 c0 k u2) (CHead c2 +(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 +c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C +c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 +v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def +(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 +\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 +\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) +in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 +i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) +v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C +(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 +k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda +(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 +| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 +(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to +(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 +(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead +c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda +(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 +(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T +(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind +b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 +c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: +C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 +(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) +H7)))))))))))))) y0 v x y H1))) H0))) H))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma index 218664c0f..32accdc4e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma @@ -1101,3 +1101,45 @@ H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))). +theorem csubst0_getl_lt_back: + \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 +e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)))))))))))) +\def + \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def +(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) +(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x +\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in +(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) +(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) +(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 +(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda +(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or +(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) +(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 +(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def +(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in +(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) +(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) +(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind +C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 +e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: +(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl +n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: +C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 +H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma index e04a34a34..499d2e13e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma @@ -18,8 +18,6 @@ include "LambdaDelta-1/csubst1/props.ma". include "LambdaDelta-1/csubst0/getl.ma". -include "LambdaDelta-1/csubst0/props.ma". - include "LambdaDelta-1/subst1/props.ma". include "LambdaDelta-1/drop/props.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma index 2541790a9..f0cd01b2e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma @@ -60,11 +60,12 @@ u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind -Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C -(\lambda (e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind -Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind Abst) -t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind -Abbr) u) (csubt_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 -(clear_gen_bind Abst c3 e1 t H3))))))))))) c1 c2 H)))). +(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda +(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) +t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: +C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 +(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 +H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 +c2 H)))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/csuba.ma new file mode 100644 index 000000000..50131b20a --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/csuba.ma @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/ty3/arity.ma". + +theorem csubt_csuba: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba +g c1 c2)))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 +c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda +(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda +(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda +(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: +B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: +T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u +t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A +(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g +a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda +(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g +x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) +H4))))))))))) c1 c2 H)))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma index 520e3bd70..8b92e30f2 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma @@ -24,6 +24,6 @@ inductive csubt (g: G): C \to (C \to Prop) \def (b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) | csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csubt g (CHead c1 (Bind Abst) t) -(CHead c2 (Bind Abbr) u))))))). +(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g +(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma index 23e56e0f3..d34cd09a1 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma @@ -103,17 +103,18 @@ d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u -t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 -(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csubt -g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 -(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) -c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))). +u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u +t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda +(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) +u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 +O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) +u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O +c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) +u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 +c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 +H5)))))))))))))) c1 c2 H0)))))) n))). theorem csubt_drop_abbr: \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g @@ -203,339 +204,378 @@ x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g -c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O -(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) -u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) -O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: -C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind -Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 -(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 -d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 -H4))))))))))))) c1 c2 H0)))))) n)). +c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind +Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 +(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda +(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind +Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 +(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). theorem csubt_drop_abst: \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t)))))))))))) +O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) \def \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) -t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (H: (csubt g c1 c2)).(\lambda (d1: C).(\lambda (t: -T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind -C c1 (\lambda (c: C).(csubt g c c2)) H (CHead d1 (Bind Abst) t) -(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def -(csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: C).(eq C c2 -(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) (ex3_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C (\lambda (e2: C).(eq C c2 (CHead -e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda -(e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) -(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 -(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g +c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 +(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H +(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in +(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: +C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) +(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O +O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C +(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt +g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) +(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead -d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) -u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop O O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) t))) x H5 -(drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) (\lambda (H3: (ex3_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (v2: +u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead +d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) +(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead -d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g x0 x1 t)).(eq_ind_r -C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) -(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt -g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) -x1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) x1)) H6)) c2 H4)))))) H3)) -H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 +T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead +x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1 +x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O +O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind +Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 +(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind +Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda 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C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 +(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 +u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 -(Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda +n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 +u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) -u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead -c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt -g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) -u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2) -H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind -Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda -(H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop -(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) -(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda -(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) -t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead -d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) -(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O -(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: -T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) -(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: -T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csubt g d1 -x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C +c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda +(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 +t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) +(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda +(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 +u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead +d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: +T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: +(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: +T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind +Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g +d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: +(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: -C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 -(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) -H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead -d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) -(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: -T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: -(csubt g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) -x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: +(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 +g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 +(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: +(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) +t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: -T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: -T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind -Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) -c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)). +(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind +Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 +(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 +t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda +(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 +t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S +n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) +(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda +(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: +(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) +x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 +t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 +g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) +H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 +(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma index 78ab95a2b..06c651505 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma @@ -72,131 +72,143 @@ v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (H4: (eq C (CHead c1 -(Bind Abst) t) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 -(Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind -Abbr) v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind -Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) -H5)))))))))) y c2 H0))) H))))). +T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u +t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) +v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C +(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) +(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))). theorem csubt_gen_abst: \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead -e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda +e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1))))))))) +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) \def \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 -e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: -(csubt g y c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead -e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C c0 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 -(Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ -_ _) \Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or -(ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda -(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C -(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) +e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind +Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g +(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or +(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: +C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: +(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) +v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 +(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: +T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt -g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 +g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: -K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) -v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 k u) (CHead e1 (Bind Abst) v1) 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 _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 -(Bind Abst) v1) H3) in ((let H6 \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 c1 k u) (CHead e1 (Bind Abst) v1) H3) in (\lambda -(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 -(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or -(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) -(\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: -T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 -(Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: -C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C -(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt -g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind -Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: +(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 +(Bind Abst) v1) 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 _ k0 _) +\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 +\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 c1 k u) +(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda +(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: +C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g +e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda +(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind +Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) +(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda +(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind +Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl +(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: +T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: +(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead -e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 +(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T +u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda -(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C -c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead -e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda -(e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u -t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind +Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: +T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u +t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) +v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H4) in ((let H6 \def +(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind Abst) v1) H4) in (\lambda (H7: (eq C c1 e1)).(let H8 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H3 v1 H6) in (let H9 \def +Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 +\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def +(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def +(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 +C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H7) in -(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H7) in -(or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind +C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda +(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda +(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) +(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind +Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt +g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind -Abbr) u)) H10 H8))))))) H5)))))))))) y c2 H0))) H))))). +Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))). theorem csubt_gen_flat: \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall @@ -251,15 +263,15 @@ C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u -t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v))).(let -H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) -H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) -(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))) y c2 -H0))) H)))))). +e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u +t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) +(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: +C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: +C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))). theorem csubt_gen_bind: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall @@ -343,30 +355,32 @@ H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u -t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 -(Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Abst -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c3 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c: +e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u +t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) +t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ +_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) +\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead +c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst +b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: +C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H9) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) -in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) +H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) +in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csubt g e1 e2))))))) H11 Abst H8) in (ex2_3_intro B C T (\lambda (b2: +(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H12)))))))) H6)) -H5)))))))))) y c2 H0))) H)))))). +e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) +H7)) H6))))))))))) y c2 H0))) H)))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma index 280fdcaea..fc4f15e7e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma @@ -140,47 +140,50 @@ theorem csubt_getl_abst: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall (n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t)))))))))))) +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) \def \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 -x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).(C_ind (\lambda (c: -C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: -C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort -n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) -t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) +(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead +d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 (CHead x0 k t0))).(\lambda -(H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) t))).(K_ind (\lambda (k0: -K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear (CHead x0 k0 t0) (CHead d1 -(Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 +(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) +t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt +g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) +\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 +(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) +t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear +(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 +c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n +c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | @@ -201,194 +204,217 @@ H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) -(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 -(Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))) (\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop -n O c2 (CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 +(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead -x1 (Bind Abst) t) (CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) -H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2 +(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 +(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) +(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda +(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 -x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O +c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: (ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 H17 (getl_intro n c2 -(CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 (clear_bind Abbr x1 -x2)) H19))))))) H16)) (csubt_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) -H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) -t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead d1 (Bind Abst) -t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 -(Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: -nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall -(c2: C).((csubt g x1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead -x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csubt g x1 c2)).(let H10 -\def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) H9 (CHead x0 (Flat f) t0) -(drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in (let H_y \def (clear_flat x0 -(CHead d1 (Bind Abst) t) (clear_gen_flat f x0 (CHead d1 (Bind Abst) t) t0 H6) -f t0) in (let H11 \def (csubt_clear_conf g (CHead x0 (Flat f) t0) c2 H10 -(CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead -d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) -t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H12: (csubt -g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: (clear c2 x2)).(let H14 \def -(csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x2 -(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) (ex3_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead -e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda -(e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) -(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) t))).(\lambda -(H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 -c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) -(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 -(getl_intro O c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) -(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead -e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda -(e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g -e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g x3 x4 -t)).(let H19 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 -(Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt -g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro -O c2 (CHead x3 (Bind Abbr) x4) c2 (drop_refl c2) H19) H18)))))))) H15)) -H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 +H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 +(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 +d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 +(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead +d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n +O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl -n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat -f) t0))).(\lambda (c2: C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def -(drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat -f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda +(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: +(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) +H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in +(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 +(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g +(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C +(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: +C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: +C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: +(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C +(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt +g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda +(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) +(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 +(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind +Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 +(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 +(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O +c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: +(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind +Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 +t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind +Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 +(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) +H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 +(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: +(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: +(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) +in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 +(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 +C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 (Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 -(Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def -(csubt_gen_bind g x2 x3 x5 x4 H15) in (ex2_3_ind B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead -d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x6: -B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind -x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda -(c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 -H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 -u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl -(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 -(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n0 x7 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t))) -(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead -d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 x7 -(CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) +x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 +x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) -(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 +(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def +(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) +in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 +(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead +d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 +(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 +C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: +(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 +x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C +T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: -T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 -x9)).(\lambda (H24: (getl n0 x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: -(ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g +d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: +C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 +x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda +(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead -d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x9 x10 -H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n0 H24) -H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 -H4))))))) x H1 H2)))) H0))))))). +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) +x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) +H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma index 6334713f6..11be53e89 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma @@ -46,33 +46,35 @@ Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 -u0 u)))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda +c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 +u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) +(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) +(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -u))))).(ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abst) u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda -(x: C).(\lambda (H6: (csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind -Abst) u))).(ty3_abst g n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex3_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: -C).(\lambda (u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: -C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: +u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: +(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g +n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 -u0 u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u -H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csubt g c -c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall -(c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda -(c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 c2 H4) b t0 t3 -(H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) u))))))))))))))) -(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w -u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 w -u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind +c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 +u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda +(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: +T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 +g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) +x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g +n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: +C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: +T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda +(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 +t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 +c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) +u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: +(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 +w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: @@ -90,5 +92,5 @@ t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: (ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead -c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H))))))))). +c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/clear.ma new file mode 100644 index 000000000..57e380c5b --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/clear.ma @@ -0,0 +1,188 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/csubv/defs.ma". + +include "LambdaDelta-1/clear/fwd.ma". + +theorem csubv_clear_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 +(Bind b2) v2)))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall +(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) +v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda +(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) +v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) +v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear +c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: +T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) +v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) +(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 +(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) +(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 +H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void +c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda +(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) +v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear +(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0 +(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3: +C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 +(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1 +Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0: +B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1) +v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | +(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) +v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 +(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | +(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) +v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq +B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B +C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda +(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) +(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda +(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) +v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4))))))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: +((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear +c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2: +F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: +C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 +(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 +(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3))))) +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 +(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4 +(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) +v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2))))))) +H3))))))))))))))) c1 c2 H))). + +theorem csubv_clear_conf_void: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: +C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v2: T).(clear c2 (CHead d2 (Bind Void) v2)))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c +(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 +(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: +T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort +(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead +d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: +(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 +(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 +(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: +C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 +(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) +\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1) +(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind +Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind +Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: +C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) +v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) +(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 +(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: +((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to +(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: +C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda +(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear +(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0) +(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 +H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0) +(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 +H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 +(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda +(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear +(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r +B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match +(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda +(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with +[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: +T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear +c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2: +F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: +T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void) +v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0) +v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 +(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead +d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv +d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 +(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 +c2 H))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/defs.ma new file mode 100644 index 000000000..69cc18f45 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/C/defs.ma". + +inductive csubv: C \to (C \to Prop) \def +| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n)) +| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind +Void) v2)))))) +| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall +(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2))))))))) +| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 +(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/drop.ma new file mode 100644 index 000000000..88a2fcdb4 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/drop.ma @@ -0,0 +1,112 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/csubv/props.ma". + +include "LambdaDelta-1/drop/fwd.ma". + +theorem csubv_drop_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: +C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 +e2)) (\lambda (e2: C).(drop h O c2 e2)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h +O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O +c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda +(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq +nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O +(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h +O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) +(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) +(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: +C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) +(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort +n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to +(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 +e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: +nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda +(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) +e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C +(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c +e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C +(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop +O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same +c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl +(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 +O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: +(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind +Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in +(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 +e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O +(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 +x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 +e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 +(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall +(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: +B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h +O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead +c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: +C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead +c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 +C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind +b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) +e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind +b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) +v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: +nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind +b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) +e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 +h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: +C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 +(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h +H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to +(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 +e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat +f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) +e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O +(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) +v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) +e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) +(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) +v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 +(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda +(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) +e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x +\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in +(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: +C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda +(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda +(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda +(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) +v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h +H2)))))))))))) c1 c2 H))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/getl.ma new file mode 100644 index 000000000..f78859458 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/getl.ma @@ -0,0 +1,84 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/csubv/clear.ma". + +include "LambdaDelta-1/csubv/drop.ma". + +include "LambdaDelta-1/getl/fwd.ma". + +theorem csubv_getl_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl +i c2 (CHead d2 (Bind b2) v2))))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: +B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i +c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind +b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: +C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda +(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: +C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) +v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in +(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead +d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda +(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) +in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda +(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) +v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: +(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro +B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) +(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind +b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 +H9))))))) H7)))))) H4)))))) H1))))))))). + +theorem csubv_getl_conf_void: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: +C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) +\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: +C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: +C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 +(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i +H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e +(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 +(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda +(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 +c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) +(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 +(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda +(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 +H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 +(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) +v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 +x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 +(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/props.ma new file mode 100644 index 000000000..857a4dca3 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubv/props.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/csubv/defs.ma". + +include "LambdaDelta-1/T/props.ma". + +theorem csubv_bind_same: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall +(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) +v2))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: +B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 +(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: +T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void +H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst +(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1: +T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))). + +theorem csubv_refl: + \forall (c: C).(csubv c c) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: +nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda +(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead +c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) +(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma index 97bb1ee9f..bcb0c72b0 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma @@ -28,6 +28,8 @@ include "LambdaDelta-1/cnt/defs.ma". include "LambdaDelta-1/cimp/defs.ma". +include "LambdaDelta-1/csubv/defs.ma". + include "LambdaDelta-1/subst/defs.ma". include "LambdaDelta-1/subst1/defs.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma index 20356f946..e6e148647 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma @@ -84,6 +84,16 @@ c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). +theorem flt_trans: + \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 +t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt +c1 t1 c3 t3)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: +T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 +t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))). + theorem flt_wf__q_ind: \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C \to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma index af9a9fb97..11e130cc2 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma @@ -64,6 +64,28 @@ k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). +theorem getl_gen_2: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 +B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind +b) v))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 +c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: +B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) +(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x +c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B +C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind +b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: +(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: +C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind +x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda +(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: +B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c +(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) +H3))))) H0))))). + theorem getl_gen_flat: \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma index 20649f9bf..748f9a280 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma @@ -379,6 +379,55 @@ H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2))))))))))))) t1). +theorem lifts_inj: + \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: +nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) +\def + \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h +d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts +h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: +nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda +(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: +nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) +(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match +ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | +(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in +(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: +T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: +nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq +TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t +t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d +t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) +(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with +[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in +(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda +(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList +(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t +t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList +(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d +t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList +return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow +(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda +(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d +t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def +(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda +(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) +(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | +(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in +lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h +d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift +h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) +(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H +t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). + theorem lift_free: \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma index 6549ddb52..e212a16db 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma @@ -191,5 +191,6 @@ theorem nf2_gen_void: T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t -(pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))). +(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) +P))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma index 17e87d94d..a801ea1bb 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma @@ -84,6 +84,24 @@ b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 H3)))))) H2)))))))). +theorem nfs2_tapp: + \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) +\to (land (nfs2 c ts) (nf2 c t))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: +TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: +(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True +(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I +H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c +(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c +t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c +(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: +(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let +H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c +t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj +(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) +H6))) H4))))) H1)))))) ts))). + theorem nf2_appls_lref: \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma index 22b552be5..de03e8cb1 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma @@ -33,3 +33,14 @@ t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 H_y0))))))))) H2))))))). +theorem pc3_nf2_unfold: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c +t2) \to (pr3 c t1 t2))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: +T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: +T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def +(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: +T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma index 867542f5a..159e6d5e7 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma @@ -162,6 +162,14 @@ theorem pc3_pr2_u2: t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x c t1 t0 H) t2 H0)))))). +theorem pc3_pr3_conf: + \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall +(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t +t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c +t2 t H0) t1 H)))))). + theorem pc3_head_12: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma index 4a240f898..35f388d27 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma @@ -262,36 +262,36 @@ O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y -(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x) -t))) t0 H3))) H2))) H1))) b)) (\lambda (f: F).(F_ind (\lambda (f0: F).(or -(\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to (eq T (THead (Flat f0) -t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat f0) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2))))) -(let H_x \def (binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T +(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void +not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: +F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) +t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def +H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: +T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq +T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat +Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda +(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead -(Bind b) w u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 -(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda -(H2: (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind b) w u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: -T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))) (or (\forall (t2: -T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) -(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H3: (eq T t0 -(THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 (\lambda (t2: T).(or -(\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda (t3: T).((eq -T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 t3))))) H0 -(THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda -(t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t t2) t3) \to (eq T -(THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Flat -Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead -(Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall (t2: T).((pr0 -(THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or +u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T +(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat +Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 +(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T +(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: +T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind +x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t +t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq +T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: +T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall +(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: @@ -357,95 +357,35 @@ Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S -O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void not_void_abst t t -(pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) x0 H4) t0 H3)))))) -H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in -(or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: -T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or -(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat -Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) -t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let -H5 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T -(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to -(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead -(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) -\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t +O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst +Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl +x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq +T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 -(THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 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 t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T -t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) -(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 -t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t -x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def -(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 -t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead -(Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: -T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead -(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) -(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 +t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 +t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 +(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda +(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall +(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) +(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda +(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead -(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 -x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead -(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall -(t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in -(let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: -Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind -Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl -(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 -x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: -(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 t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T -t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: +(_: T).(\lambda (t3: T).(pr0 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 t0 (THead (Bind @@ -456,51 +396,111 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: -B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T -t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda -(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let -H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: -Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) -x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind -x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t -(THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O -x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) -(\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall -(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) -(\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: -Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 -(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat -Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t -x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat Appl) t t0) -(THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (H12 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) -(\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) -(\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: +(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda +(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t +u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) +t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead +(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 +x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def +(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def +(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 +H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) +in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) +t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: +T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t +t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq +T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 +x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t +x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda +(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 +(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead +(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: +T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) +w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in +(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat +Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind +Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t +(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 +H10))))))))) H8)) (\lambda (H8: (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 t0 (THead (Bind b) +y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat +Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))))).(ex6_6_ind 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 t0 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift +(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda +(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: +T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) +(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not +(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda +(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) +x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 +x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) +x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def +(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 +t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda +(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead +(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) +in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat +Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) +(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 +x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) +H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 +t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t +t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T +t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror +(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat +Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) +t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) +(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead +(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat +Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x +(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x +(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in +(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat +Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda +(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) +(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead +(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) +t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma index 549441f26..b84dc67bd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma @@ -54,17 +54,21 @@ C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind -Abbr) H10) in (let H11 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0 +Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1 -is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3 -(Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 +C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq +C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc +g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind @@ -99,87 +103,128 @@ in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) -H11)))))) H8)))))) H5)))))))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g -a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 d) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is -u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1 -c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let H5 \def H0 in (let -H_x \def (drop1_getl_trans is c d1 H3 Abst d u i H5) in (let H6 \def H_x in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: -C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 (ptrans is i) u)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (H7: (drop1 -(ptrans is i) x d)).(\lambda (H8: (getl (trans is i) d1 (CHead x (Bind Abst) -(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x -(Bind Abst) (lift1 (ptrans is i) u)) (trans is i) H8 c2 H4) in (let H9 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: -C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 -(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H10: (getl (trans is i) c2 -x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) (lift1 (ptrans is i) u)) -x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind -Abst) H11) in (let H12 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0 +(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: +B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq +K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: +(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is +i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind +Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1 +is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5)))))))))))))))) +(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: +(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: +PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g +(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: +PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g +d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d +u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is +i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 +(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: +C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i) +d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def +(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is +i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans +is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans +is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda +(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) +(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 +(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C +(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) +(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans +is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 +w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 +C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) +(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1 -is (TLRef i))) (\lambda (H13: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3 -(Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 -(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abst) -(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H16 \def (eq_ind -C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind Abst) -(lift1 (ptrans is i) u)) H14) in (let H_y \def (sc3_abst g a0 TNil) in -(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2 -(trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T -(lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c -is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i)) -(lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 (ptrans is i) u) (trans is i) -H16) I) (lift1 is (TLRef i)) (lift1_lref is i))))))) H13)) (\lambda (H13: -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind -Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C -x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g -(asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a1: A).(sc3 g a1 c3 w)))))).(ex5_3_ind C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) +c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C +x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x +x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) +H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def +(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: +T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef +(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 +t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 +H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 +(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is +i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: A).(\lambda (_: (eq K (Bind Abst) (Bind Abst))).(\lambda -(H15: (eq C x0 (CHead x1 (Bind Abbr) x2))).(\lambda (_: (csubc g x -x1)).(\lambda (H17: (sc3 g (asucc g x3) x (lift1 (ptrans is i) u))).(\lambda -(H18: (sc3 g x3 x1 x2)).(let H19 \def (eq_ind C x0 (\lambda (c0: C).(getl -(trans is i) c2 c0)) H10 (CHead x1 (Bind Abbr) x2) H15) in (let H_y \def -(sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: -T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 (let H_y0 \def (arity_lift1 g -(asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def (sc3_arity_gen g x -(lift1 (ptrans is i) u) (asucc g x3) H17) in (sc3_repl g x3 c2 (lift (S -(trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S (trans is i)) O (getl_drop -Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g x3 a0 (arity_mono g x (lift1 -(ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) H_y0))))) H19) (lift1 is -(TLRef i)) (lift1_lref is i)))))))))))) H13)) H12)))))) H9)))))) -H6))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity -g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 -c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 -c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g -d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead -(Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) -(H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) -(Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) -(csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is -(THead (Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: +w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq +K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is +(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: +(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) +x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x +(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def +(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind +Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef +(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 +(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let +H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in +(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S +(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g +x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) +H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) +(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: +B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq +K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: +(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is +i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind +Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1 +is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6))))))))))))))))) +(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda +(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: +((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: +C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 +a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 +(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 +(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5: +(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y +\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u) +(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) +(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is) +(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head +g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead +(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma index e77507354..25afeec1d 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma @@ -16,92 +16,6 @@ include "LambdaDelta-1/theory.ma". -theorem lifts_inj: - \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: -nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) -\def - \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h -d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts -h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: -nat).(\lambda (H: (eq TList TNil TNil)).H))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList TNil -(lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) (lifts h d t0)))).(let -H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList return -(\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) -\Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in (False_ind -(eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: nat).(\forall (d: -nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq TList t0 -ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: TList).(\forall (h: -nat).(\forall (d: nat).((eq TList (lifts h d (TCons t t0)) (lifts h d t1)) -\to (eq TList (TCons t t0) t1))))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let -H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) (\lambda (ee: -TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil -\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in (False_ind -(eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda (t2: -TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList (TCons -(lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t t0) -t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons -(lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)))).(let H2 -\def (f_equal TList T (\lambda (e: TList).(match e in TList return (\lambda -(_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: ((nat \to nat))) -(d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i -| false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow (THead k (lref_map -f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda (x: nat).(plus x -h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) -(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList -TList (\lambda (e: TList).(match e in TList return (\lambda (_: TList).TList) -with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) (ts0: TList) on -ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | (TCons t3 ts1) -\Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in lifts) h d t0) | -(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons -(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h -d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2))) -(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1 -(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). - -theorem nfs2_tapp: - \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) -\to (land (nfs2 c ts) (nf2 c t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: -TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: -(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True -(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I -H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c -(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c -t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c -(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: -(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let -H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c -t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj -(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) -H6))) H4))))) H1)))))) ts))). - -theorem pc3_nf2_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t2) \to (pr3 c t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: -T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def -(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: -T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). - -theorem pc3_pr3_conf: - \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall -(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t -t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c -t2 t H0) t1 H)))))). - axiom pc3_gen_appls_sort_abst: \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) @@ -122,323 +36,3 @@ TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads (Flat Appl) ws (TSort n))) \to False)))))))) . -inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def -| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c -TNil u))) -| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: -TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). - -theorem tys3_gen_nil: - \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T -(\lambda (u0: T).(ty3 g c u u0)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil -u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: -TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda -(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq -TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: -T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList -TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda -(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: -TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to -(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t -ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: -TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil -\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind -(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). - -theorem tys3_gen_cons: - \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall -(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts -u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda -(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) -(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) -(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind -g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to -(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: -T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t -ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList -return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) -\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) -(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: -(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 -u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) -(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t -ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) -\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal -TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: -TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) -(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def -(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land -(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList -ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind -T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 -g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). - -theorem ty3_gen_appl_nf2: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g -c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in -(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 -x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) -w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) -(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda -(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def -(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: -T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead -(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 -g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) -(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c -(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind -Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c -(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c -(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def -(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c -x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 -(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 -(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda -(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: -T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c -(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead -(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) -(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w -Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead -(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind -Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 -x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 -(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) -(ty3_gen_appl g c w v x H))))))). - -theorem ty3_inv_lref_nf2_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c -(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to -((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t -u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c -u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda -(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: -T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift -(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 -c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to -((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 -t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda -(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 -\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 -\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to -(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: -T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y -\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 -H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq -T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: -T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) -u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in -(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 -(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y -d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 -(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq -T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) -O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: -T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda -(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S -i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) -t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 -(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda -(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 -u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) -\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to -(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq -T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) -w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead -(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: -T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T -t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 -t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda -(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to -((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) -t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 -(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T -u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). - -theorem ty3_inv_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c -(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: -T).(eq T u (lift (S i) O u0)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: -(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). - -theorem ty3_inv_appls_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: -T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to -((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S -i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) -u1)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t -(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t -(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: -(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c -u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in -(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) -(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def -(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r -T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) -O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda -(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) -(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall -(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef -i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) -u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c -(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t -(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T -T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind -Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) -u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: -T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) -u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat -Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t -x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def -(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) -x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 -c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) -(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def -(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c -(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) -O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O -u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O -x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead -(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S -i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c -(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t -Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). - -theorem ty3_inv_lref_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c -(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i -j))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda -(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda -(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 -H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift -(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S -i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 -in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x -(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x -(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt -j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda -(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S -i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) -j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) -H2))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma index e2c568a37..ba6c06704 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma @@ -32,6 +32,10 @@ include "LambdaDelta-1/ex1/props.ma". include "LambdaDelta-1/ty3/sty0.ma". +include "LambdaDelta-1/csubt/csuba.ma". + +include "LambdaDelta-1/ty3/fwd_nf2.ma". + include "LambdaDelta-1/ty3/nf2.ma". include "LambdaDelta-1/wf3/props.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma index 4f6b4cf13..3be6c0d3b 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma @@ -119,49 +119,50 @@ x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 -(arity_bind g Void not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void -not_void_abst c0 u x H5 t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) -H4)))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: -(ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) -(\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind -Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: -A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A -(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: -A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) -(\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u -(asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v -a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 -A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: -A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) -(\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 -(THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t -(asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A -(asucc g x0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u -(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind -Abst) u) t a2))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) +(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0 +H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4 +(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda +(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda +(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 +u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v +(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v +a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g +a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) +(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity +g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat +Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda +(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def +H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity +g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g +c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat +Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda +(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) +(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in +(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 +a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) +(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) +(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda +(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g +a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) +(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: +(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) +(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in +(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A +x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A -(asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g -x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def -(sym_eq A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g -x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) -(\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 -(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) -w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: -(eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def -(eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 -(asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v -a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead -(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w -(THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl -g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g -x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind -Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) -H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A -(\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g +t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 +x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 +(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) +in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 +x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) +a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u +t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 +(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc +g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc +g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) +H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: +A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma index d30e8bdd2..f72531c50 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma @@ -41,3 +41,9 @@ t))))))))) \to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) (THead (Flat Cast) t0 t2))))))). +inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def +| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c +TNil u))) +| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: +TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma index 7a71c7d35..c85e929eb 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma @@ -18,8 +18,6 @@ include "LambdaDelta-1/ty3/props.ma". include "LambdaDelta-1/pc3/fsubst0.ma". -include "LambdaDelta-1/csubst0/props.ma". - include "LambdaDelta-1/getl/getl.ma". theorem ty3_fsubst0: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma index 5a88b676b..69f001666 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma @@ -849,3 +849,53 @@ Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). +theorem tys3_gen_nil: + \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T +(\lambda (u0: T).(ty3 g c u u0)))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil +u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: +TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda +(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq +TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: +T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList +TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda +(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: +TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to +(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t +ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: +TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil +\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind +(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). + +theorem tys3_gen_cons: + \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall +(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts +u))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda +(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) +(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) +(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind +g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to +(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: +T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t +ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList +return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) +\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) +(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: +(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 +u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) +(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t +ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList +return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) +\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal +TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: +TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) +(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def +(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land +(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList +ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind +T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 +g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd_nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd_nf2.ma new file mode 100644 index 000000000..52e6c661f --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd_nf2.ma @@ -0,0 +1,286 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "LambdaDelta-1/ty3/arity_props.ma". + +include "LambdaDelta-1/pc3/nf2.ma". + +include "LambdaDelta-1/nf2/fwd.ma". + +theorem ty3_gen_appl_nf2: + \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: +T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: +T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda +(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) +x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) +x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g +c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in +(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 +x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) +w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v +(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) +(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda +(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def +(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: +T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead +(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: +T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 +g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) +(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c +(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind +Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c +(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda +(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) +x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c +(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def +(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: +T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c +x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind +b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 +(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 +(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda +(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: +T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: +T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c +(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead +(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) +(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w +Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead +(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind +Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 +x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 +(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) +(ty3_gen_appl g c w v x H))))))). + +theorem ty3_inv_lref_nf2_pc3: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c +(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to +((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda +(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t +u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c +u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda +(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: +T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift +(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 +c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T +(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda +(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to +((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T +(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 +t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda +(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 +\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 +\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to +(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: +T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def +(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y +\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 +H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq +T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: +T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) +u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in +(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g +d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: +T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S +i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: +(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: +(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef +i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) +O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 +(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl +n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g +d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: +T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S +i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: +(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: +(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef +i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) +O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 +(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl +n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 +(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y +d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 +(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq +T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) +O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: +T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda +(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S +i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef +i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: +B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) +u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) +t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 +(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O +u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda +(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 +u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T +(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T +(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda +(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda +(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) +\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O +u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead +(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to +(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq +T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) +w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead +(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead +(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: +T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T +t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 +t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda +(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to +((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T +(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T +(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) +t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 +(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 +t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T +u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). + +theorem ty3_inv_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c +(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: +T).(eq T u (lift (S i) O u0)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: +(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). + +theorem ty3_inv_appls_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: +T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to +((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S +i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) +u1)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: +TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t +(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: +T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t +(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: +(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c +u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in +(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: +T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) +(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def +(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r +T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) +O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda +(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) +(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) +(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall +(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef +i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) +(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) +u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead +(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c +(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t +(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T +T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind +Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: +T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) +u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: +T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) +u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat +Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t +x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def +(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) +x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 +c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) +(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) +x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def +(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c +(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) +O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O +u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift +(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O +x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead +(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) +(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S +i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c +(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c +(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t +Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). + +theorem ty3_inv_lref_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c +(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i +j))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda +(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda +(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 +H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift +(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S +i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 +in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x +(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x +(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt +j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda +(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S +i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) +j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) +H2))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma index dc7c1df85..9c1ecb711 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma @@ -459,63 +459,63 @@ T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w (lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d -x3))).(\lambda (H14: (ty3 g a x2 x3)).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s (Bind Abst) d) u0 t t3))) (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w -(THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H15: (eq T -(lift (S O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H16: (subst1 d u0 u -x4)).(\lambda (H17: (subst1 (s (Bind Abst) d) u0 t x5)).(let H18 \def (sym_eq -T (lift (S O) d x1) (THead (Bind Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda -(y: T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: +x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind +Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: +T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead +(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S +O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u +x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S +O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: +T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) -x6 x7))).(\lambda (H20: (eq T x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 -(lift (S O) (S d) x7))).(let H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 -(s (Bind Abst) d) u0 t t0)) H17 (lift (S O) (S d) x7) H21) in (let H23 \def -(eq_ind T x4 (\lambda (t0: T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20) -in (let H24 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead -(Bind Abst) x6 x7) H19) in (let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g -a x0 (THead (Bind Abst) t0 x7))) H24 x3 (subst1_confluence_lift u x6 u0 d H23 -x3 H13)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 -(THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) -x2 x0) (THead (Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead -(Flat Appl) (lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d -u0 (THead (Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 -(Flat Appl) v (lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) -(lift_flat Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d -x2) (lift (S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S -O) d x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind +(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) +x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 +(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 +(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 +(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 +\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 +x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead +(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in +(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat +Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 +(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead +(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) +(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead +(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v +(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat +Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift +(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead +(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d +x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t -(lift (S O) (S d) x7) H22) (lift (S O) d (THead (Bind Abst) x3 x7)) +(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) (lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead (Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) -(ty3_appl g a x2 x3 H14 x0 x7 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S -O) d H18)))))))) (subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d -H9))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda -(H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e +(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S +O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda +(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 +t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl +d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to +(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -- 2.39.2