From b58315ef220a130a44acbf528cd6885ddadad642 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Sat, 13 Oct 2007 15:22:37 +0000 Subject: [PATCH] - some new auxiliary lemmas - some improved proofs --- .../LAMBDA-TYPES/LambdaDelta-1/C/props.ma | 12 +- .../LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma | 19 +- .../LAMBDA-TYPES/LambdaDelta-1/clear/props.ma | 5 +- .../LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma | 6 +- .../LAMBDA-TYPES/LambdaDelta-1/definitions.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/drop/props.ma | 10 +- .../LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma | 3 +- .../LambdaDelta-1/{gz => ex0}/defs.ma | 2 +- .../LambdaDelta-1/{gz => ex0}/props.ma | 4 +- .../LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma | 48 +-- .../LAMBDA-TYPES/LambdaDelta-1/flt/props.ma | 34 +- .../LAMBDA-TYPES/LambdaDelta-1/leq/props.ma | 76 ++--- .../LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma | 100 +++--- .../LAMBDA-TYPES/LambdaDelta-1/lift/props.ma | 253 +++++++------- .../LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma | 134 ++++---- .../LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma | 10 +- .../LAMBDA-TYPES/LambdaDelta-1/llt/props.ma | 12 +- .../LambdaDelta-1/next_plus/props.ma | 13 +- .../LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma | 13 + .../LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma | 8 +- .../LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma | 37 +++ .../LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma | 59 ++-- .../LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma | 36 +- .../LAMBDA-TYPES/LambdaDelta-1/s/props.ma | 8 +- .../LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma | 212 ++++++------ .../LambdaDelta-1/subst0/props.ma | 6 +- .../LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma | 309 +++++++++--------- .../LAMBDA-TYPES/LambdaDelta-1/theory.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma | 150 ++++----- .../LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma | 24 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma | 307 +++++++++++++++++ .../LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma | 223 ++++++------- .../LambdaDelta-1/ty3/pr3_props.ma | 290 ++++++++-------- .../LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma | 16 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma | 161 +++++---- 35 files changed, 1426 insertions(+), 1178 deletions(-) rename helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/{gz => ex0}/defs.ma (95%) rename helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/{gz => ex0}/props.ma (99%) create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma index 979266883..30901423e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma @@ -25,18 +25,16 @@ theorem clt_cong: T).(clt (CHead c k t) (CHead d k t)))))) \def \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight -d))).(\lambda (_: K).(\lambda (t: T).(lt_le_S (plus (cweight c) (tweight t)) -(plus (cweight d) (tweight t)) (plus_lt_compat_r (cweight c) (cweight d) -(tweight t) H)))))). +d))).(\lambda (_: K).(\lambda (t: T).(plus_lt_compat_r (cweight c) (cweight +d) (tweight t) H))))). theorem clt_head: \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) \def \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight -c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_le_S (plus -(cweight c) O) (plus (cweight c) (tweight u)) (plus_le_lt_compat (cweight c) -(cweight c) O (tweight u) (le_n (cweight c)) (tweight_lt u))) (cweight c) -(plus_n_O (cweight c))))). +c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) +(plus_le_lt_compat (cweight c) (cweight c) O (tweight u) (le_n (cweight c)) +(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))). theorem clt_wf__q_ind: \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma index 4df511966..16adef5c3 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma @@ -29,12 +29,8 @@ theorem aplus_reg_r: (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(sym_eq A (asucc g (aplus g a2 (plus n h2))) -(asucc g (aplus g a1 (plus n h1))) (sym_eq A (asucc g (aplus g a1 (plus n -h1))) (asucc g (aplus g a2 (plus n h2))) (sym_eq A (asucc g (aplus g a2 (plus -n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g g (aplus g -a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq A (aplus g -a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))). +h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n +h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). theorem aplus_assoc: \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A @@ -51,14 +47,9 @@ g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda (n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) -(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_eq A (asucc g (aplus g -(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0)))) -(sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g -(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a -(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))) -H0))))) (plus n (S n0)) (plus_n_Sm n n0)))) h2)))) h1))). +(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) +n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) +(plus_n_Sm n n0)))) h2)))) h1))). theorem aplus_asucc: \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma index b01bf12e1..a1880711d 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma @@ -134,7 +134,6 @@ C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) (tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) -c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t)) (le_n_S (cweight c2) -(plus (cweight c) (tweight t)) (le_plus_trans (cweight c2) (cweight c) -(tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))) k H0))))))) c1). +c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 +(clear_gen_flat f c c2 t H1))))) k H0))))))) c1). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma index 86a83b5e7..d8a6ad246 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma @@ -49,9 +49,9 @@ T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 -c2)).(pc3_pr2_u c2 t3 t0 (csubt_pr2 g c1 t0 t3 H0 c2 H3) t4 (H2 c2 -H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 -t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall +c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 +(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 +t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). 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 809cb3235..620154782 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma @@ -42,7 +42,7 @@ include "llt/defs.ma". include "aprem/defs.ma". -include "gz/defs.ma". +include "ex0/defs.ma". include "wcpr0/defs.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma index a40e6a75b..029720727 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma @@ -200,12 +200,10 @@ k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k -x3))) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (sym_eq C (CHead x4 k x3) -(CHead x0 k x3) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C -CHead x0 x4 k k x3 x3 (H x0 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal -T x3))))) x5 (lift_inj x5 x3 h (r k n) H11))))) x1 H6)) x2 H3)))))) -(drop_gen_skip_l c0 x1 t h n k H1))))))) (drop_gen_skip_l c0 x2 t h n k -H2)))))))) d))))))) c). +x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h +H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n) +H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1))))))) +(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). theorem drop_conf_lt: \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma index 98f8ba300..f8ec287e3 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma @@ -162,8 +162,7 @@ H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 i))) (ptrans hds0 i)) v))))) (\lambda (x: C).(\lambda (H18: (drop1 (ptrans hds0 i) x e1)).(\lambda (H19: (getl (trans hds0 i) c3 (CHead x (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 -i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt -(S d) (S (trans hds0 i)) H16))) c2 c3 h H14 b x (lift1 (ptrans hds0 i) v) +i) d (blt_lt d (trans hds0 i) H16) c2 c3 h H14 b x (lift1 (ptrans hds0 i) v) H19) in (let H20 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S (trans hds0 i))) e2 x)) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma similarity index 95% rename from helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/defs.ma rename to helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma index b99cc1ee6..078545c74 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/defs.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/gz/defs". +set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex0/defs". include "A/defs.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma similarity index 99% rename from helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/props.ma rename to helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma index fe7112070..66350ed9c 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/gz/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/gz/props". +set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex0/props". -include "gz/defs.ma". +include "ex0/defs.ma". include "leq/defs.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma index 9872a1baf..218efc455 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma @@ -487,29 +487,31 @@ Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H43 \def (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S -O))) H41 (lift (S O) O (TLRef O)) (pc3_s (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x0 (lift (S O) -O (TLRef O)) H22)) (TLRef (plus O (S O))) (lift_lref_ge O (S O) O (le_n O))) -in (let H44 \def H43 in (ex2_ind T (\lambda (t: T).(pr3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef -(S O)) t)) P (\lambda (x14: T).(\lambda (H45: (pr3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (TLRef (S (S O))) x14)).(\lambda (H46: (pr3 (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(TLRef (S O)) x14)).(let H47 \def (eq_ind_r T x14 (\lambda (t: T).(eq T -(TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S -O))) x14 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort O) (TSort O) (S (S -O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) (clear_bind Abst -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef -O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) (getl_head (Bind Abst) O -(CHead (CSort O) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort -O)) (getl_refl Abst (CSort O) (TSort O)) (TSort O))))) (TLRef (S O)) +O))) H41 (lift (S O) O (TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift +(S O) O (TLRef O))) (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x0) H22)) (TLRef (plus O (S +O))) (lift_lref_ge O (S O) O (le_n O))) in (let H44 \def H43 in (ex2_ind T +(\lambda (t: T).(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda +(t: T).(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x14: +T).(\lambda (H45: (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) +x14)).(\lambda (H46: (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) x14)).(let +H47 \def (eq_ind_r T x14 (\lambda (t: T).(eq T (TLRef (S (S O))) t)) +(nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x14 H45 +(nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) +(getl_clear_bind Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) (clear_bind Abst (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) +(CHead (CSort O) (Bind Abst) (TSort O)) (S O) (getl_head (Bind Abst) O (CHead +(CSort O) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) +(getl_refl Abst (CSort O) (TSort O)) (TSort O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) x14 H46 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 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 a11df495d..a8d7daff3 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 @@ -24,21 +24,19 @@ theorem flt_thead_sx: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c (THead k u t))))) \def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_le_S -(plus (cweight c) (tweight u)) (plus (cweight c) (S (plus (tweight u) -(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S -(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight u) -(plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t)))))))). + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S (plus (tweight +u) (tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u) +(tweight t)) (le_plus_l (tweight u) (tweight t))))))). theorem flt_thead_dx: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c (THead k u t))))) \def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_le_S -(plus (cweight c) (tweight t)) (plus (cweight c) (S (plus (tweight u) -(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S -(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight t) -(plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t)))))))). + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S (plus (tweight +u) (tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u) +(tweight t)) (le_plus_r (tweight u) (tweight t))))))). theorem flt_shift: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c @@ -58,12 +56,8 @@ theorem flt_arith0: \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t (CHead c k t) (TLRef i))))) \def - \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: nat).(le_S_n (S -(plus (cweight c) (tweight t))) (plus (plus (cweight c) (tweight t)) (S O)) -(lt_le_S (S (plus (cweight c) (tweight t))) (S (plus (plus (cweight c) -(tweight t)) (S O))) (lt_n_S (plus (cweight c) (tweight t)) (plus (plus -(cweight c) (tweight t)) (S O)) (lt_x_plus_x_Sy (plus (cweight c) (tweight -t)) O))))))). + \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: +nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))). theorem flt_arith1: \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle @@ -89,12 +83,8 @@ c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) (H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda (_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight -t2)) (S O)) H (le_S_n (plus (cweight c2) (S O)) (plus (plus (cweight c2) -(tweight t2)) (S O)) (lt_le_S (plus (cweight c2) (S O)) (S (plus (plus -(cweight c2) (tweight t2)) (S O))) (le_lt_n_Sm (plus (cweight c2) (S O)) -(plus (plus (cweight c2) (tweight t2)) (S O)) (plus_le_compat (cweight c2) -(plus (cweight c2) (tweight t2)) (S O) (S O) (le_plus_l (cweight c2) (tweight -t2)) (le_n (S O)))))))))))))). +t2)) (S O)) H (plus_le_compat (cweight c2) (plus (cweight c2) (tweight t2)) +(S O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). theorem flt_wf__q_ind: \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma index ac9ef3ee8..0ad16f9e7 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma @@ -118,44 +118,44 @@ n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) (ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus (minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g -(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 -H10)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) -(aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 H9))))) (\lambda -(H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 -k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus (minus k k0) k0) -(\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n))) -H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A -(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) -H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 (sym_eq nat h0 h2 H6))) -H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A -(AHead a0 a5) (ASort h2 n2))).(\lambda (H5: (eq A (AHead a4 a6) a3)).((let H6 -\def (eq_ind A (AHead a0 a5) (\lambda (e: A).(match e in A return (\lambda -(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h2 n2) H4) in (False_ind ((eq A (AHead a4 a6) a3) \to ((leq -g a0 a4) \to ((leq g a5 a6) \to (leq g (ASort h1 n1) a3)))) H6)) H5 H2 -H3)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3))))))))))) -(\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: -((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: -A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: -A).((leq g a6 a7) \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g -(AHead a4 a6) a0)).(let H5 \def (match H4 in leq return (\lambda (a: -A).(\lambda (a7: A).(\lambda (_: (leq ? a a7)).((eq A a (AHead a4 a6)) \to -((eq A a7 a0) \to (leq g (AHead a3 a5) a0)))))) with [(leq_sort h1 h2 n1 n2 k -H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda -(H7: (eq A (ASort h2 n2) a0)).((let H8 \def (eq_ind A (ASort h1 n1) (\lambda -(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a4 a6) H6) in -(False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k) -(aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H8)) H7 H5))) | -(leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda (H7: (eq A (AHead a7 a9) -(AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) a0)).((let H9 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) (AHead a7 a9) -(AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) -\Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 (\lambda (a: -A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) \to ((leq g -a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9 +(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_trans k (S k) k0 (le_S +k k (le_n k)) H10))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g +(ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 +H9))))) (\lambda (H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) +(ASort h3 n3) k0 k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus +(minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g +(ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 +n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) +(aplus g (ASort h3 n3) k) H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 +(sym_eq nat h0 h2 H6))) H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) +\Rightarrow (\lambda (H4: (eq A (AHead a0 a5) (ASort h2 n2))).(\lambda (H5: +(eq A (AHead a4 a6) a3)).((let H6 \def (eq_ind A (AHead a0 a5) (\lambda (e: +A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow +False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind +((eq A (AHead a4 a6) a3) \to ((leq g a0 a4) \to ((leq g a5 a6) \to (leq g +(ASort h1 n1) a3)))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort h2 n2)) +(refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: +(leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3 +a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 +a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5 +a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def +(match H4 in leq return (\lambda (a: A).(\lambda (a7: A).(\lambda (_: (leq ? +a a7)).((eq A a (AHead a4 a6)) \to ((eq A a7 a0) \to (leq g (AHead a3 a5) +a0)))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A +(ASort h1 n1) (AHead a4 a6))).(\lambda (H7: (eq A (ASort h2 n2) a0)).((let H8 +\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda +(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (AHead a4 a6) H6) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq +A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 +a5) a0))) H8)) H7 H5))) | (leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda +(H7: (eq A (AHead a7 a9) (AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) +a0)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda +(_: A).A) with [(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) +(AHead a7 a9) (AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: +A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | +(AHead a _) \Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 +(\lambda (a: A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) +\to ((leq g a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9 a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4 a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A (AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma index e96bdc0ae..5a80f43e6 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma @@ -193,23 +193,23 @@ n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (n1: nat).(eq T (TLRef n0) (TLRef n1))) (let H5 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 d)) H (plus n0 h) H4) in (le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d -(le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H5)))) n (sym_eq nat n (plus -n0 h) H4))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T -t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d -t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d -(THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda -(t2: T).(eq T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: -T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) -t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow -(\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) -t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d -t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) -H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) -t1)))))))))))) t))))). +(lt_le_trans n0 (S (plus n0 h)) d (le_lt_n_Sm n0 (plus n0 h) (le_plus_l n0 +h)) (lt_le_S (plus n0 h) d H5))))) n (sym_eq nat n (plus n0 h) H4))))]) in +(H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: +T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef +n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq +T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 +t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq +T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k +t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: T).(\lambda +(_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to +(eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda +(H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H5 +\def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) +t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H5)))]) in (H4 +(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))). theorem lift_gen_lref_false: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n @@ -296,39 +296,39 @@ nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef n0) H3) in (eq_ind nat (plus n h) (\lambda (n1: nat).(eq T (TLRef n1) (TLRef n))) (let H5 \def (eq_ind_r nat n0 (\lambda (n1: nat).(lt n1 d)) H1 (plus n h) H4) in (le_false d n (eq T (TLRef -(plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l -n h) H5)))) n0 H4)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le -d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T -(TLRef (plus n h)) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in -(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? -t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with -[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef (plus -n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return -(\lambda (_: T).nat) with [(TSort _) \Rightarrow ((let rec plus (n1: nat) on -n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O \Rightarrow m | (S -p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | -(THead _ _ _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def -(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S -(plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H3) in -(eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) -(f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus -n0 h) H4))) (plus n0 h) H4)))]) in (H3 (refl_equal T (TLRef (plus n0 -h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef -(plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: -T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef -n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 -t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq -T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: -T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) -t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow -(\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) -t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d -t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) -H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) -t1)))))))))))) t))))). +(plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d (lt_le_trans +(plus n h) d (plus d h) H5 (le_plus_l d h)))))) n0 H4)))]) in (H3 (refl_equal +T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def (eq_ind T (lift h d +(TLRef n0)) (\lambda (t0: T).(eq T (TLRef (plus n h)) t0)) H0 (TLRef (plus n0 +h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 in eq return (\lambda +(t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T +(TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T +(TLRef (plus n h)) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda +(e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow +((let rec plus (n1: nat) on n1: (nat \to nat) \def (\lambda (m: nat).(match +n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h) +| (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow ((let rec plus (n1: +nat) on n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O +\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef +(plus n h)) (TLRef (plus n0 h)) H3) in (eq_ind nat (plus n h) (\lambda (_: +nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h +n0 n (sym_eq nat (plus n h) (plus n0 h) H4))) (plus n0 h) H4)))]) in (H3 +(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: +T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef +n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d +t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift +h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) +(\lambda (t2: T).(eq T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) +(lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq +return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h +d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with +[refl_equal \Rightarrow (\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift +h d t0) (lift h (s k d) t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H4) in (False_ind (eq +T (THead k t0 t1) (TLRef n)) H5)))]) in (H4 (refl_equal T (THead k (lift h d +t0) (lift h (s k d) t1)))))))))))) t))))). theorem lift_gen_head: \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 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 0051630c6..6a8cc0ac0 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 @@ -97,13 +97,9 @@ t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) -(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) -(lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift -O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1 -(lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) -(sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1)) -(lift_head k t0 t1 O d)))))))) t). +t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k +(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))) +(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). theorem lift_lref_gt: \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef @@ -171,30 +167,25 @@ z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind -b) t t0) (THead (Bind b) x0 x1) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind -b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K -(Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h (S d) -H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d -H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 -t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: -(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 -(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T -(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq -T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d -x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead -(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T -(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) -(THead (Flat f) x0 x1) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0) -(f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f)) -(sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) +(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b)) +(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t) +(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: +T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T +(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda +(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d +t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) +(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) +(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t) +(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: +T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda +(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq +T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda +(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) +(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T +(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 +x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). theorem lift_gen_lift: @@ -251,39 +242,37 @@ h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 -h1) (plus n h1) (le_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S -(plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat d2 n h1 h1 -H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt -(plus n h1) n0)) (lt_le_S (plus n h1) (plus (plus d2 h2) h1) -(plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2) -(plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 -d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: -(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n0: -nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) -h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1 -(le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) -(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda -(t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq -T (TLRef (minus (plus n h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T -(TLRef n) (lift h2 d2 t2))) (TLRef (minus n h2)) (eq_ind_r nat (plus (minus n -h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n -h2))))) (eq_ind_r T (TLRef (plus (minus n h2) h1)) (\lambda (t: T).(eq T -(TLRef (plus (minus n h2) h1)) t)) (refl_equal T (TLRef (plus (minus n h2) -h1))) (lift h1 d1 (TLRef (minus n h2))) (lift_lref_ge (minus n h2) h1 d1 -(le_trans d1 d2 (minus n h2) H (le_minus d2 n h2 H4)))) (minus (plus n h1) -h2) (le_minus_plus h2 n (le_trans_plus_r d2 h2 n H4) h1)) (eq_ind_r nat (plus -(minus n h2) h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef -(minus n0 h2))))) (eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) -h2)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T -TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) -(f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 -h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r -(minus n h2) h2)) (refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus -(minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 -(le_minus d2 (plus (minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2 -(le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n -(le_trans_plus_r d2 h2 n H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus -(plus n h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: +h1) (plus n h1) (plus_le_compat d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus +(plus d2 h2) h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (plus_lt_compat_r n +(plus d2 h2) h1 H4) (plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x +H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T +(TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 +\def (eq_ind nat (plus n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 +(plus d2 h1) x))) H2 (plus (minus (plus n h1) h2) h2) (le_plus_minus_sym h2 +(plus n h1) (le_plus_trans h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 +h2) H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 +T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) +(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n +h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) +(TLRef (minus n h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: +nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef +(plus (minus n h2) h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) +t)) (refl_equal T (TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n +h2))) (lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H +(le_minus d2 n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans +h2 (plus d2 h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) +h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) +(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: +T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus +n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat +plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus +(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2)) +(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2))) +(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus +(minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2 (le_minus d2 n h2 +H4) (le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n +(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n +h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift @@ -344,56 +333,55 @@ d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))) (lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1 -H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S -(S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm d1 d2 H1)))) H11)))) t H9) x0 H8)))) (H -x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S -d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift -h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind -T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus -d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t -t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead -(Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift -h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0) -(lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda -(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T -(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0 -x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 -d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) -(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T -t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 -x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 -d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0) -(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2))) -(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T -(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T -(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3: -T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2 -d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T -(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat -f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T -(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda -(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) -(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1 -d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 -x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift -h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1)) -(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2: -T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T -(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f) -x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 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). +H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) +(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 +(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T +(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def +(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift +h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) +(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: +T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T +(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: +T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: +T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) +(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x +(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 +h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) +x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda +(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t +t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1 +t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq +T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead +(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0 +(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift +h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2 +x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 +d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: +T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: +T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) +(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 +t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 +t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda +(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 +d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 +d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) +(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 +t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) +(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) +(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3) +(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2: +T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T +(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f) +x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2 +x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) +(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2 +d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 +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 lift_free: \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: @@ -490,26 +478,23 @@ nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) (plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T (TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) -(lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k) -(plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef -(plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef -n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus -(plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k)) -(\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef -(plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus -(plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h))) -(lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h)) -(le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n)) -(lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge -(plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S -(plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1 -(le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) -(lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: -nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift -h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall -(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h -(plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: +(lift_lref_lt (plus n k) h (plus d k) (plus_lt_compat_r n d k H1)))) (\lambda +(H1: (le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T +t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda +(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef +(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) +(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat +(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k +e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0))) +(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus +n k))) (lift_lref_ge (plus n k) h (plus d k) (plus_le_compat d n k k H1 (le_n +k)))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) (lift_lref_ge n k e +H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: +nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq +T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d +t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: +nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 +d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma index 19ce970db..0e041d02d 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma @@ -39,74 +39,74 @@ T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat (weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) (\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda -(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_S_n d (plus n h) -(le_n_S d (plus n h) (le_plus_trans d n h H0)))) (f n) (H n H0)) (lift h d -(TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to -nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat -(weight_map f (lift h d t0)) (weight_map f t0)))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to -nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat -(weight_map f (lift h d t1)) (weight_map f t1)))))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (f: ((nat \to nat))).(\lambda (H1: ((\forall -(m: nat).((le d m) \to (eq nat (f m) O))))).(K_ind (\lambda (k0: K).(eq nat -(weight_map f (lift h d (THead k0 t0 t1))) (weight_map f (THead k0 t0 t1)))) -(\lambda (b: B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) -d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f (THead (Bind -b) t0 t1)))) (B_ind (\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow -(S (plus (weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f -(lift h d t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map -f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void -\Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) -(lift h (S d) t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map -f t0) (weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S -(plus (weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S -(plus (weight_map f t0) (weight_map (wadd f O) t1)))]))) (eq_ind_r nat -(weight_map f t0) (\lambda (n: nat).(eq nat (S (plus n (weight_map (wadd f (S -n)) (lift h (S d) t1)))) (S (plus (weight_map f t0) (weight_map (wadd f (S -(weight_map f t0))) t1))))) (eq_ind_r nat (weight_map (wadd f (S (weight_map -f t0))) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) -(refl_equal nat (S (plus (weight_map f t0) (weight_map (wadd f (S (weight_map -f t0))) t1)))) (weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) -(H0 h (S d) (wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: -(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: -nat).(le d n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: -nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m -H3)))) (le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) -(eq_ind_r nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus -(weight_map f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd -f O) t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) -(weight_map (wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) -t1)) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f -t0) (weight_map (wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) -(refl_equal nat (weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h -(S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) +(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) +(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f +t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f +t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 +t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead +(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat +(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda +(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift +h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) +t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f +(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with +[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S +(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: +nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S +(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) +(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: +nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) +(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus +(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) +(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) +(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (wadd f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S -x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat -(wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat -(weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f -(lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) t1))))) -(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) -t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat -nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map (wadd f -O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat (weight_map -(wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd -f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: -nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) -(\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d -x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) -m H3)))) (le_gen_S d m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) -(lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat -f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat -(weight_map f t2) (weight_map f (THead (Flat f0) t0 t1)))) (f_equal nat nat S -(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus -(weight_map f t0) (weight_map f t1)) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map f t0) (weight_map f (lift h d t1)) (weight_map f -t1) (H h d f H1) (H0 h d f H1))) (lift h d (THead (Flat f0) t0 t1)) -(lift_head (Flat f0) t0 t1 h d))) k)))))))))) t). +n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda +(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda +(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) +(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r +nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map +f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) +t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 +nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map +(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat +(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h +(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat +(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd +f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le +d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x +H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) +(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus +(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f +t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd +f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) +(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: +nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S +n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: +nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d +m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h +d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s +(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f +(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) +(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) +(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k)))))))))) t). theorem lift_weight: \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma index 216c6d80b..db5d76536 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma @@ -31,13 +31,9 @@ PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 -(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(sym_eq T (lift n -n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 t))) (sym_eq T -(lift n n0 (lift1 p (lift1 is2 t))) (lift n n0 (lift1 (papp p is2) t)) -(sym_eq T (lift n n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 -t))) (f_equal3 nat nat T T lift n n n0 n0 (lift1 (papp p is2) t) (lift1 p -(lift1 is2 t)) (refl_equal nat n) (refl_equal nat n0) (sym_eq T (lift1 p -(lift1 is2 t)) (lift1 (papp p is2) t) (H is2 t)))))))))))) is1). +(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat +T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal +nat n) (refl_equal nat n0) (H is2 t)))))))) is1). theorem lift1_xhg: \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma index 96aab8ccf..d34488300 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma @@ -56,18 +56,14 @@ a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans theorem llt_head_sx: \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a1)) (S (plus (lweight -a1) (lweight a2))) (le_n_S (S (lweight a1)) (S (plus (lweight a1) (lweight -a2))) (le_n_S (lweight a1) (plus (lweight a1) (lweight a2)) (le_plus_l -(lweight a1) (lweight a2)))))). + \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) +(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). theorem llt_head_dx: \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a2)) (S (plus (lweight -a1) (lweight a2))) (le_n_S (S (lweight a2)) (S (plus (lweight a1) (lweight -a2))) (le_n_S (lweight a2) (plus (lweight a1) (lweight a2)) (le_plus_r -(lweight a1) (lweight a2)))))). + \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) +(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). theorem llt_wf__q_ind: \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma index 41139d520..e0f2654ac 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma @@ -52,11 +52,10 @@ theorem next_plus_lt: g n) h)))) \def \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(le_S_n (S n) -(next g n) (lt_le_S (S n) (S (next g n)) (lt_n_S n (next g n) (next_lt g -n))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g -(next g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next -g n0)) n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus -g (next g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus -g (next g n0) n)) (next_plus_next g (next g n0) n))))) h)). +nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) +(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next +g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) +n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next +g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next +g n0) n)) (next_plus_next g (next g n0) n))))) h)). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma index 701c00936..cd70ecf45 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma @@ -77,6 +77,19 @@ B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 (H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))). +theorem pc3_gen_abst_shift: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c +(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind +Abst) u) t1 t2))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def +(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (and_ind (pc3 c u u) +(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 +(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: +((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 +Abst u))) H0))))))). + theorem pc3_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall (d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma index c14f0f81a..3a6db7706 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma @@ -79,10 +79,10 @@ theorem pc3_ind_left__pc3_pc3_left: t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda (t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 -c t3 t4)).(pc3_pr2_u c t3 t0 H0 t4 H2))))))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 -t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 (pc3_pr2_x c t3 t0 H0) t4 -H2))))))) t1 t2 H)))). +c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda +(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 +(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))). theorem pc3_ind_left: \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 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 new file mode 100644 index 000000000..41136207b --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2". + +include "pc3/defs.ma". + +include "nf2/pr3.ma". + +theorem pc3_nf2: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c +t1) \to ((nf2 c t2) \to (eq T t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in +(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T +t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 +x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x +(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x +(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c +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))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma index 77f4c6d9e..5e8396c47 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma @@ -105,30 +105,29 @@ d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind Abbr) d) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S d) t0) (lift h (S d) t3) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in (eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) -(lift h d' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (lt_le_S O -(S d) (le_lt_n_Sm O d (le_O_n d))) h) d (eq_ind nat d (\lambda (n: nat).(eq -nat n d)) (refl_equal nat d) (minus d O) (minus_n_O d))))) (lift h d (THead -(Bind Abbr) u2 w)) (lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind -Abbr) u1 t0)) (lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (pr0 t0 t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h: -nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s -(Bind b) d) (lift (S O) O t0))) (\lambda (t: T).(pr0 t (lift h d t3))) -(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d -u) (lift h n (lift (S O) O t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O -(lift h d t0)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d -t3))) (pr0_zeta b H0 (lift h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift -h (plus (S O) d) (lift (S O) O t0)) (lift_d t0 h (S O) d O (le_O_n d))) (S d) -(refl_equal nat (S d))) (lift h d (THead (Bind b) u (lift (S O) O t0))) -(lift_head (Bind b) u (lift (S O) O t0) h d))))))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).(\lambda (H1: ((\forall (h: -nat).(\forall (d: nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u: -T).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Flat Cast) (lift h -d u) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(pr0 t (lift h d t3))) -(pr0_epsilon (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 h d) (lift h d -u)) (lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h -d))))))))) t1 t2 H))). +(lift h d' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (le_n_S O d +(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal +nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) +(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t0)) +(lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: +(not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 +t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) +(lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S +O) O t0))) (\lambda (t: T).(pr0 t (lift h d t3))) (eq_ind nat (plus (S O) d) +(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O +t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O (lift h d t0)) (\lambda (t: +T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t3))) (pr0_zeta b H0 (lift +h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S +O) O t0)) (lift_d t0 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) +(lift h d (THead (Bind b) u (lift (S O) O t0))) (lift_head (Bind b) u (lift +(S O) O t0) h d))))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (_: +(pr0 t0 t3)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) +t0)) (\lambda (t: T).(pr0 t (lift h d t3))) (pr0_epsilon (lift h (s (Flat +Cast) d) t0) (lift h d t3) (H1 h d) (lift h d u)) (lift h d (THead (Flat +Cast) u t0)) (lift_head (Flat Cast) u t0 h d))))))))) t1 t2 H))). theorem pr0_subst0_back: \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 @@ -1654,10 +1653,9 @@ x1)).(\lambda (H13: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H12 u) H13))))) H11)) (H2 v1 x0 i H10 v2 H4))) x H8) w1 H6)))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) -(S O) O H7 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) -(le_lt_n_Sm O i (le_O_n i)))))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +(S O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: @@ -1692,9 +1690,8 @@ x2)).(or_intror (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6)))) -(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_S_n (S O) (S i) -(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n -i)))))))))))) H5)) (subst0_gen_head (Bind b) v1 u (lift (S O) O t3) w1 i +(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_n_S O i (le_O_n +i))))))))) H5)) (subst0_gen_head (Bind b) v1 u (lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma index 6e91b63e9..848d216fa 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma @@ -2036,14 +2036,13 @@ T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))) (pr2_delta c d u i H8 u1 x3 H15 x8 H27) (pr2_free c x1 x4 H16) (pr2_free (CHead c (Bind x0) x4) x2 x5 H17))) x7 H25))))) -(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H24 (le_S_n (S O) (S i) -(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6 -H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat -Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s -(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat -Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s -(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H24 (le_n_S O i (le_O_n +i)))) x6 H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead +(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) +(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead +(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) +(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind @@ -2274,11 +2273,10 @@ T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H26))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O -H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm -O i (le_O_n i))))))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift -(S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda (H19: (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: +H24 (le_n_S O i (le_O_n i)))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl) +u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda +(H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 @@ -2495,11 +2493,10 @@ T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8 -H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S -O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n -i))))))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T x7 -(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 +H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i +(le_O_n i)))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T +x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s +(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda @@ -2733,8 +2730,7 @@ T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3 -x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) -(lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H24)))))) H23)) +x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i (le_O_n i)))) x7 H24)))))) H23)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22)) x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma index ceb02c249..7acdf612f 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma @@ -75,8 +75,7 @@ theorem s_le: \def \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_S_n (S i) (S j) (lt_le_S (S -i) (S (S j)) (lt_n_S i (S j) (le_lt_n_Sm i j H)))))))) (\lambda (_: +nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k). theorem s_lt: @@ -85,9 +84,8 @@ theorem s_lt: \def \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_S_n (S (S i)) (S j) (le_n_S -(S (S i)) (S j) (le_n_S (S i) j H))))))) (\lambda (_: F).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). +nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda +(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). theorem s_inj: \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma index 5a8baf190..e16f362b0 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma @@ -705,113 +705,113 @@ h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift (plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n -h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n) -(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (le_O_n d))) (subst0_lref -u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i H3))) (subst0_gen_lref -u x i (plus n h) H2)))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: +h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d +(plus O (S n)) (plus_le_compat O O d (S n) (le_n O) (le_S d n H1))) (le_O_n +d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i +H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) +i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: +T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: -nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t -t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i: -nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t0) x) \to -((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) -(\lambda (t2: T).(subst0 (minus i h) u t0 t2))))))))))).(\lambda (x: -T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: -(subst0 i u (lift h d (THead k t t0)) x)).(\lambda (H2: (le (plus d h) -i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) (\lambda (t2: T).(subst0 -i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) t0)) (lift_head k t t0 h -d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) -t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2))) (ex2 T (\lambda (t2: -T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u -(lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T -x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d -t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) -t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (H4: (ex2 T (\lambda -(u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i -u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2)) (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead -k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x (THead k x0 (lift h (s k -d) t0)))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(eq_ind_r T (THead k x0 -(lift h (s k d) t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift -h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) -(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 -(minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 (lift h (s k -d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) -t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h d x1))).(\lambda (H8: -(subst0 (minus i h) u t x1)).(eq_ind_r T (lift h d x1) (\lambda (t2: T).(ex2 -T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k d) t0)) (lift h d t3))) -(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift -h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift -h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) -(ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k x1 t0)) (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x1 -t0) (refl_equal T (lift h d (THead k x1 t0))) (subst0_fst u x1 t (minus i h) -H8 t0 k)) (THead k (lift h d x1) (lift h (s k d) t0)) (lift_head k x1 t0 h -d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) (\lambda (H4: (ex2 T (\lambda -(t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) -u (lift h (s k d) t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k -(lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) -t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 -(minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x -(THead k (lift h d t) x0))).(\lambda (H6: (subst0 (s k i) u (lift h (s k d) -t0) x0)).(eq_ind_r T (THead k (lift h d t) x0) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i -h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (s k -d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda -(t2: T).(eq T (THead k (lift h d t) x0) (lift h d t2))) (\lambda (t2: +nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda +(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 +t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda +(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) +(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) +t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x +(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) +u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda +(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s +k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda +(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) +(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: +T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u +(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda +(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u +(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 +(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 +(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda +(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: -(eq T x0 (lift h (s k d) x1))).(\lambda (H8: (subst0 (minus (s k i) h) u t0 -x1)).(eq_ind_r T (lift h (s k d) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead k (lift h d t) t2) (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d (THead k t x1)) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda -(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H9 \def (eq_ind_r -nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x1)) H8 (s k (minus i -h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: -T).(eq T (lift h d (THead k t x1)) (lift h d t2))) (\lambda (t2: T).(subst0 -(minus i h) u (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h d -(THead k t x1))) (subst0_snd k u x1 t0 (minus i h) H9 t))) (THead k (lift h d -t) (lift h (s k d) x1)) (lift_head k t x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s -k d) H6 (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le -k (plus d h) i H2) (plus (s k d) h) (s_plus k d h)))) x H5)))) H4)) (\lambda -(H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex2 T -(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) -u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T -x (THead k x0 x1))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(\lambda (H7: -(subst0 (s k i) u (lift h (s k d) t0) x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda -(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) -u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) -(\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x2: -T).(\lambda (H8: (eq T x1 (lift h (s k d) x2))).(\lambda (H9: (subst0 (minus -(s k i) h) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) -(\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead -k t t0) t2))) (\lambda (x3: T).(\lambda (H10: (eq T x0 (lift h d -x3))).(\lambda (H11: (subst0 (minus i h) u t x3)).(eq_ind_r T (lift h d x3) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 x1) (lift h d -t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind_r -T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k -(lift h d x3) t2) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u -(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x3 x2)) (\lambda (t2: +(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r +T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 +(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u +(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (let H12 \def (eq_ind_r nat (minus (s k -i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 (s k (minus i h)) (s_minus k i -h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h -d (THead k x3 x2)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u -(THead k t t0) t2)) (THead k x3 x2) (refl_equal T (lift h d (THead k x3 x2))) -(subst0_both u t x3 (minus i h) H11 k t0 x2 H12))) (THead k (lift h d x3) -(lift h (s k d) x2)) (lift_head k x3 x2 h d)) x1 H8) x0 H10)))) (H x0 i h d -H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind nat (s k (plus d h)) (\lambda -(n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) (s_plus k -d h)))) x H5)))))) H4)) (subst0_gen_head k u (lift h d t) (lift h (s k d) t0) -x i H3)))))))))))))) t1)). +(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift +h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u +(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) +(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k +d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) +(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) +(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T +(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 +(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda +(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: +(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) +x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) +(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T +(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 +(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d +t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) +t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda +(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) +(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) +(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) +(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t +t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: +nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r +d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) +(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) +(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 +(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t +x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) +(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) +(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s +k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift +h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u +(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) +x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq +T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) +t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda +(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T +(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead +k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) +x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda +(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t +t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda +(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t +x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T +(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead +k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T +(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda +(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d +(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d +t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 +\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 +(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T +(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal +T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 +H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) +x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind +nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i +H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u +(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma index 87dac1295..7b147c20e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma @@ -145,9 +145,9 @@ h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) (lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) -(lt_le_S (s k i0) (s k d) (s_lt k i0 d H4)) h) (minus d (S i0)) (minus_s_s k -d (S i0)))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d -(THead k u1 t0)) (lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). +(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d +(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) +(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). theorem subst0_lift_ge: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma index 37b74c00c..fd0ba2f0c 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma @@ -75,35 +75,33 @@ u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) f g H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t0)) (S (plus -(weight_map g u1) (weight_map g t0))) (le_lt_n_Sm (plus (weight_map f0 u2) -(weight_map f0 t0)) (plus (weight_map g u1) (weight_map g t0)) -(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t0) -(weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g H2))))))))) k))))))))) -(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: T).(\forall (t2: -T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 t2) \to -(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s -k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(le (weight_map f (THead k0 u0 t2)) (weight_map g (THead k0 u0 -t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s (Bind b0) -i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g +u1) (weight_map g t0)) (plus_le_compat (weight_map f0 u2) (weight_map g u1) +(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g +H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: +T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 +t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2) -(weight_map g t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead -(Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 t1))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to +(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g +t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map +f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) +(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda +(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: +nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) +\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: +T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to +(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 +t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda +(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le +(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: +((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: +nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f @@ -111,25 +109,24 @@ t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f (weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S (weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) (S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S -(S i)) O v)) (wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S -(weight_map f u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f -u0)) v (S i) f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda -(t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: -T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: -((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift -(S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) -t2)) (plus (weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_compat -(weight_map f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map -(wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: -nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O -v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) -O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda -(t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 +u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: +nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S +i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) +(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda +(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f +t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus +(weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_compat (weight_map f +u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) +(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f +g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda +(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) +(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: +T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g @@ -149,30 +146,29 @@ nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_le_S (plus (weight_map -f0 u0) (weight_map f0 t2)) (S (plus (weight_map g u0) (weight_map g t1))) -(le_lt_n_Sm (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g -u0) (weight_map g t1)) (plus_le_compat (weight_map f0 u0) (weight_map g u0) -(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 -H3)))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall -(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f -m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le -(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda -(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to -(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s -k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map -f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s -(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat +(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map +f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) +(plus_le_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) +(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) +(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda +(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall +(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt +(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map +g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: +T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le +(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead +k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind +(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v +t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2) +(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f -t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) +(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) (weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) @@ -186,25 +182,11 @@ t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) -(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (lt_le_S -(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S -(weight_map g u1)) (S i)) (eq_ind nat (weight_map f (lift (S i) O v)) -(\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f u2))) -(lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) -f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) -v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g -t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt -(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f -O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: +(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat +(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 +(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) +(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat @@ -216,18 +198,29 @@ u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall -(f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le -(f0 m) (g m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le -(weight_map f0 t2) (weight_map g t1)))))))).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t2)) (S (plus -(weight_map g u1) (weight_map g t1))) (le_lt_n_Sm (plus (weight_map f0 u2) +(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f +t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: +(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) +t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f +O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) +(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f +(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) +(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) +(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 +t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift +(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g +t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: +(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) -(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5))))))))))))) k)))))))) d u t +(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))). theorem subst0_weight_lt: @@ -263,9 +256,16 @@ m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S (weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S -(weight_map f u2) (weight_map g u1) (H1 f g H2 H3))) n))))))))) (\lambda (f: +g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S +(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 +H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt +(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) +t0)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd +f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) +(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd +f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus @@ -273,33 +273,26 @@ i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus f u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O -O (le_n O) n))))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd -f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) -(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd -f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) b)) (\lambda -(_: F).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda -(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0 -(lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 -t0)) (plus (weight_map g u1) (weight_map g t0)) (plus_lt_le_compat -(weight_map f0 u2) (weight_map g u1) (weight_map f0 t0) (weight_map g t0) (H1 -f0 g H2 H3) (weight_le t0 f0 g H2)))))))) k))))))))) (\lambda (k: K).(K_ind -(\lambda (k0: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall -(i: nat).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to +O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 +m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g +u1) (weight_map g t0)) (plus_lt_le_compat (weight_map f0 u2) (weight_map g +u1) (weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g +H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: +T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 +t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g +t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map +f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) +(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda +(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: +nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt -(weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: T).(\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map -f (THead k0 u0 t2)) (weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: -T).(\forall (i: nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind -b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: +\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) +\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 @@ -316,19 +309,17 @@ t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) (weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S (weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (lt_le_S (weight_map f u0) (S (weight_map g u0)) -(le_lt_n_Sm (weight_map f u0) (weight_map g u0) (weight_le u0 f g H2))) m)) -(lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S (S i)) O v)) -(wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f (lift (S i) O -v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f -u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) -f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le +u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: +nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S +i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) +(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda +(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f +t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) @@ -391,29 +382,26 @@ u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t1)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f -u2))) (wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_le f g H4 (S -(weight_map f u2)) (S (weight_map g u1)) (le_S (S (weight_map f u2)) -(weight_map g u1) (lt_le_S (weight_map f u2) (weight_map g u1) (H1 f g H4 -H5))) m)) (lt_le_S (weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) -O v)) (wadd g (S (weight_map g u1)) (S i)) (eq_ind nat (weight_map f (lift (S -i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S -(weight_map f u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f -u2)) v (S i) f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: -(subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt -(weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f t2) -(weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +u2))) (wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S +(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) +(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O +v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f +u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) +f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v +t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g +t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) t1)) (plus_lt_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) -(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (lt_le_S (weight_map (wadd f -O) (lift (S (S i)) O v)) (wadd g O (S i)) (eq_ind nat (weight_map f (lift (S -i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S -(S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))) (\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: +(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f +(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) +(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat @@ -424,10 +412,9 @@ i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n -O) m)))) (lt_le_S (weight_map (wadd f O) (lift (S (S i)) O v)) (wadd g O (S -i)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g -i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v -(S i) f)))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: +O) m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n +(g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O +v (S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) 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 c8f4922b1..daf2a250e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma @@ -20,7 +20,7 @@ include "subst0/tlt.ma". include "tau1/cnt.ma". -include "gz/props.ma". +include "ex0/props.ma". include "wcpr0/fwd.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma index c2dacafde..b75bc8053 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma @@ -101,39 +101,32 @@ t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le -(f n) (g n))))).(le_S_n (S (plus (weight_map f t0) (weight_map (wadd f O) -t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S -(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g -t0) (weight_map (wadd g O) t1))) (le_n_S (plus (weight_map f t0) (weight_map -(wadd f O) t1)) (plus (weight_map g t0) (weight_map (wadd g O) t1)) -(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f O) -t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda -(n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))))) (\lambda (t0: -T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) -(weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g -n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat +(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) +(plus (weight_map g t0) (weight_map (wadd g O) t1)) (plus_le_compat +(weight_map f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map +(wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: +nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: +((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: +nat).(le (f n) (g n)))) \to (le (weight_map f t0) (weight_map g +t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le (f +n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) +(plus (weight_map g t0) (weight_map (wadd g O) t1)) (plus_le_compat +(weight_map f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map +(wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: +nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) b)) (\lambda (_: F).(\lambda +(t0: T).(\lambda (H: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t0) +(weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f0: ((nat +\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g +n)))) \to (le (weight_map f0 t1) (weight_map g t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le -(f n) (g n))))).(le_S_n (S (plus (weight_map f t0) (weight_map (wadd f O) -t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S -(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g -t0) (weight_map (wadd g O) t1))) (le_n_S (plus (weight_map f t0) (weight_map -(wadd f O) t1)) (plus (weight_map g t0) (weight_map (wadd g O) t1)) -(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f O) -t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda -(n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))))) b)) (\lambda (_: -F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) \to (le (weight_map -f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f0: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) -(g n)))) \to (le (weight_map f0 t1) (weight_map g t1))))))).(\lambda (f0: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: -nat).(le (f0 n) (g n))))).(lt_le_S (plus (weight_map f0 t0) (weight_map f0 -t1)) (S (plus (weight_map g t0) (weight_map g t1))) (le_lt_n_Sm (plus -(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g -t1)) (plus_le_compat (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) -(weight_map g t1) (H f0 g H1) (H0 f0 g H1)))))))))))) k)) t). +(f0 n) (g n))))).(le_n_S (plus (weight_map f0 t0) (weight_map f0 t1)) (plus +(weight_map g t0) (weight_map g t1)) (plus_le_compat (weight_map f0 t0) +(weight_map g t0) (weight_map f0 t1) (weight_map g t1) (H f0 g H1) (H0 f0 g +H1))))))))))) k)) t). theorem weight_eq: \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to @@ -159,11 +152,9 @@ theorem weight_add_S: O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) \def \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) -(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(le_S_n (wadd (\lambda -(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (le_n_S (wadd (\lambda -(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (wadd_le (\lambda (_: -nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_O_n (S -m)) n)))))). +(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: +nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m +(le_O_n m)) n)))). theorem tlt_trans: \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to @@ -185,38 +176,21 @@ k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall \Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda (_: nat).O) u)) (S -(plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: -nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (le_n_S (S (weight_map -(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) -u))) t))) (le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map -(\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map -(\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) -u))) t))))))) (\lambda (u: T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda -(_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) O) t))) (le_n_S (S (weight_map (\lambda (_: -nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) (le_n_S (weight_map (\lambda (_: nat).O) u) -(plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: -nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) O) t))))))) (\lambda (u: T).(\lambda (t: -T).(le_S_n (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map -(\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))) -(le_n_S (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))) (le_n_S +(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S +(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: +nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: +nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map +(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map +(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) +u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda +(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda +(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) +t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))))))) b)) -(\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_S_n (S (weight_map -(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_: -nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda -(_: nat).O) t))) (le_n_S (weight_map (\lambda (_: nat).O) u) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) -(le_plus_l (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)))))))) k). +(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: +nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k). theorem tlt_head_dx: \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) @@ -247,31 +221,21 @@ u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus -(weight_map (\lambda (_: nat).O) u) n)))) (le_S_n (S (weight_map (\lambda (_: -nat).O) t)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda -(_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) -(le_n_S (weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map -(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))))) (weight_map -(wadd (\lambda (_: nat).O) O) t) (weight_add_O t)))) (\lambda (u: T).(\lambda -(t: T).(eq_ind_r nat (weight_map (\lambda (_: nat).O) t) (\lambda (n: -nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus (weight_map (\lambda -(_: nat).O) u) n)))) (le_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) -(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) (le_n_S (weight_map -(\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t))))) (weight_map (wadd (\lambda (_: -nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: -T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) -(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) (le_n_S (weight_map -(\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)))))))) k). +(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: +nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: +nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map +(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) +(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map +(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: +nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S +(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) +(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: +nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda +(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: +T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) +(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: +nat).O) t)))))) k). theorem tlt_wf__q_ind: \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma index ff4a0734e..4ad1efbd4 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma @@ -445,18 +445,18 @@ t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to T).(ty3 g c2 t t4)) P (\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda (H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def (ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def -(ty3_unique g c2 t0 t H15 x0 H9) in (H12 (pc3_s c2 x0 t H_y0) P))))))) -(ty3_gen_cast g c2 t0 t t3 H13))))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) -H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: -Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to -(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat -Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2 -(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: -T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat -Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 t)).(\lambda (_: (ty3 g c2 t -x0)).(H8 t H11 P))))) (ty3_gen_cast g c2 t0 t t3 H9))))))) H7)))) H5)) -(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: +(ty3_unique g c2 t0 t H15 x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) +H_y0) P))))))) (ty3_gen_cast g c2 t0 t t3 H13))))))) H11))))) (ty3_correct g +c2 t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to +(\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead +(Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) +t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 +(THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: +T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) +(\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2 +(THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 t)).(\lambda (_: +(ty3 g c2 t x0)).(H8 t H11 P))))) (ty3_gen_cast g c2 t0 t t3 H9))))))) H7)))) +H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma index aad32f384..358626faa 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma @@ -18,8 +18,28 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2". include "ty3/arity.ma". +include "pc3/nf2.ma". + include "nf2/arity.ma". +definition ty3_nf2_inv_abst_premise: + C \to (T \to (T \to Prop)) +\def + \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: +T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: +TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w +u)) \to False)))))))). + +theorem ty3_nf2_inv_abst_premise_csort: + \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise +(CSort m) w u))) +\def + \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda +(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) +wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs +(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d +(Bind Abst) wi) H False))))))))). + theorem ty3_nf2_inv_all: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T @@ -158,3 +178,290 @@ nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) H2)) H1)))))))). +theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux: + \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise +c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead +(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to +(ty3_nf2_inv_abst_premise c w2 u2)))))))) +\def + \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: +C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) +\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) +(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: +T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind +Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: +T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) +wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift +(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t +(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t +(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) +vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) +w1 u1) H0))))))))))))))). + +theorem ty3_nf2_inv_abst: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: +T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to +((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: +T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g +c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v +u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) +v)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: +T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c +t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w +u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in +(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq +T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c +w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) +(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: +T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 +c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) +u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind +Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: +T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t +(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 +(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 +g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r +T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) +w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda +(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex4_3_ind T T T +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w +t2) x)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c w t0)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) +w) u t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c +(Bind Abst) w) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T +(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda +(_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w +x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(\lambda (_: (ty3 g +(CHead c (Bind Abst) w) x2 x4)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda +(_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead (Bind Abst) w +u))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0 t0)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) +x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead +c (Bind Abst) x0) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T +(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) w) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda +(H14: (pc3 c (THead (Bind Abst) x0 x5) (THead (Bind Abst) w u))).(\lambda (_: +(ty3 g c x0 x6)).(\lambda (H16: (ty3 g (CHead c (Bind Abst) x0) x1 +x5)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x5 x7)).(and_ind (pc3 c x0 +w) (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x5 u))) (ex4_2 +T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) +(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda +(H18: (pc3 c x0 w)).(\lambda (H19: ((\forall (b: B).(\forall (u0: T).(pc3 +(CHead c (Bind b) u0) x5 u))))).(let H_y \def (pc3_nf2 c x0 w H18 H6 H1) in +(let H20 \def (eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) +x1 x5)) H16 w H_y) in (let H21 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead +c (Bind Abst) t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind +Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind +Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T +(THead (Bind Abst) w x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 +x5 H20 (H19 Abst w)) H21) x0 H_y)))))) (pc3_gen_abst c x0 w x5 u H14))))))))) +(ty3_gen_bind g Abst c x0 x1 (THead (Bind Abst) w u) H8))))))))) +(ty3_gen_bind g Abst c w u x H9)))) (ty3_correct g c (THead (Bind Abst) x0 +x1) (THead (Bind Abst) w u) H8)) t H5))))))) H4)) (\lambda (H4: (ex nat +(\lambda (n: nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t +(TSort n))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t (THead (Bind +Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x: +nat).(\lambda (H5: (eq T t (TSort x))).(let H6 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c t0 (THead (Bind Abst) w u))) H (TSort x) H5) in (eq_ind_r T +(TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T +t0 (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) +(pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c (THead (Bind Abst) w u) +x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TSort x) (THead (Bind +Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t H5)))) H4)) (\lambda +(H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads +(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c +ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (x0: TList).(\lambda (x1: nat).(\lambda (H5: +(eq T t (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (_: (nfs2 c +x0)).(\lambda (H7: (nf2 c (TLRef x1))).(let H8 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c t0 (THead (Bind Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) +H5) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 +T T (\lambda (v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 +in (unintro T u (\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) +(THead (Bind Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T +T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) +(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) +(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) +(unintro T w (\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) +x0 (TLRef x1)) (THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 +x) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) +x0 (TLRef x1)) (THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: +T).(ty3 g c t0 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind +Abst) t0) v x))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) +t0) v)))))))) (TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: +T).((ty3 g c (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to +((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: +T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda +(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind +Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda +(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 +T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x +v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C +T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O +t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind +Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef +x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O +x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind +Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 +(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind +Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda +(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c +(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) +(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda +(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: +(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let +H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind +(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind +Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) +(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: +T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: +T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to +((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: +T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: +(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda +(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind +Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) +t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: +T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead +(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: +T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) +(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 +(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let +H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let +H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T +(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) +(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat +Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads +(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c +x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: +(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T +(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat +Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) +(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil +(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 +g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v +x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) +H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads +(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3 +(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 +g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v +x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) +v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1)) +(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2 +(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons +t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: +T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind +Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) +v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat +Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t +H5))))))) H4)) H3))))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma index bc53829ae..d0f9999d4 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma @@ -294,20 +294,21 @@ T).(ty3 g c2 t5 (THead (Bind Void) u t8))) (let H32 \def (eq_ind_r T x0 (\lambda (t8: T).(ty3 g c2 t8 x1)) H30 t5 (lift_inj t5 x0 (S O) O H28)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Void (lift (S O) O x1) t4 -H31 x H25) t5 x1 H32 (pc3_pr2_x c2 x1 (THead (Bind Void) u (lift (S O) O x1)) -(pr2_free c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H23 x1 -x1 (pr0_refl x1) u))))) t3 H29))))))) H27)))))) b H18 (H5 (CHead c2 (Bind b) -u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H22 (H20 -(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S -O) O t5) (pr0_lift t6 t5 H19 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) -u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind -b)) t3 (pr0_refl t3)))))))) t7 (sym_eq T t7 t5 H17))) t2 H16)) u0 (sym_eq T -u0 u H15))) b0 (sym_eq B b0 b H14))) H13)) H12)) H11 H8 H9))) | (pr0_epsilon -t6 t7 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u0 t6) (THead -(Bind b) u t2))).(\lambda (H10: (eq T t7 t5)).((let H11 \def (eq_ind T (THead -(Flat Cast) u0 t6) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +H31 x H25) t5 x1 H32 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1)) +(pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead +(Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H23 x1 x1 (pr0_refl x1) +u)))))) t3 H29))))))) H27)))))) b H18 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c +c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H22 (H20 (CHead c2 (Bind +b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S O) O t5) +(pr0_lift t6 t5 H19 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 +(H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 +(pr0_refl t3)))))))) t7 (sym_eq T t7 t5 H17))) t2 H16)) u0 (sym_eq T u0 u +H15))) b0 (sym_eq B b0 b H14))) H13)) H12)) H11 H8 H9))) | (pr0_epsilon t6 t7 +H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u0 t6) (THead (Bind +b) u t2))).(\lambda (H10: (eq T t7 t5)).((let H11 \def (eq_ind T (THead (Flat +Cast) u0 t6) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H9) in (False_ind ((eq T t7 t5) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) H11)) H10 H8)))]) in (H8 (refl_equal T (THead (Bind b) u @@ -544,109 +545,109 @@ v2) (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O (THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead -(Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_s -(THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u -t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head w v2 -(pc1_pr0_r w v2 H17) (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) -O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead (Bind Abst) u t0) (THead (Bind -b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pr0_zeta b H16 (THead (Bind -Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) -(Flat Appl)))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) -(lift_bind Abst u t0 (S O) O)))))))))) (ty3_gen_bind g Abst (CHead c2 (Bind -b) u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H33))))))))))) -(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))))) (ty3_gen_bind g -Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind -Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 -(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) -H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) -(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind -T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return +(Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head +v2 w (pc1_pr0_x v2 w H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) +u t0))) (THead (Bind Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O +(THead (Bind Abst) u t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead +(Bind Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u +t0)) u2)) (Flat Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) +(S O) t0)) (lift_bind Abst u t0 (S O) O)))))))))) (ty3_gen_bind g Abst (CHead +c2 (Bind b) u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) +H33))))))))))) (ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 +(THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))))) +(ty3_gen_bind g Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 +t4) (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 +u2 H18 t3 t4 H19 (Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) +H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow +(\lambda (H9: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w +v))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H11 \def +(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to -((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_epsilon t3 t4 H6 u0) -\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) -w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) -u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead -(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in -(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) -(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 -t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 -t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let -H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda -(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow -(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 -t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat -Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g -c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl -t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat -Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda -(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T -(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat -Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to -((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 -t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat -Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead -(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T -(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda -(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 -t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T -(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv -g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat -Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 -t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 -H4 u2 H16)) (pc3_pr2_x c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) -(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast))))) (ty3_correct g c2 -t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 -(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda -(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return +False])])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (THead (Bind +Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w0) +\to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H11)) +H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 H7 u0) \Rightarrow (\lambda (H8: (eq T +(THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) w v))).(\lambda (H9: +(eq T t4 t2)).((let H10 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind +((eq T t4 t2) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 +(THead (Flat Appl) w (THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | +(pr0_epsilon t3 t4 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) +u0 t3) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def +(eq_ind T (THead (Flat Cast) u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T -(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 -t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 -H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast +\Rightarrow True])])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t4 +t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) +u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Appl) w v)) +(refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda +(t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 +c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda +(t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 +c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda +(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 +(THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda +(t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Flat +Cast) t3 t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 +t3))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat +Cast) t3 t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3 +t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 +t3)))) (\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead +(Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3))) +(ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl +t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp +u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead +(Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) +\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H11 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) +\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in (eq_ind K +(Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t5 t2) \to ((eq T +(THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 +(THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 t3)).(eq_ind T t3 +(\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat Cast) u2 t6) t4) \to +((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 +t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T +(THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 t6) \to (ty3 g c2 +t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T (THead (Flat Cast) u2 +t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda (t7: T).((pr0 t3 u2) +\to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 t3))))) (\lambda +(H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: +T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead (Flat Cast) +t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead +(Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3 +(pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat Cast) t0 u2) +(ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 t3 (H1 c2 H4 +t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 H4 u2 H16)) +(pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) (pc3_pr2_r c2 +(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) (pr2_thin_dx c2 t3 u2 +(pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3 +(pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 t3 +H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u +v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T +(THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return +(\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow +False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead +(Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 +(THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 +u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma index 3a17df985..4693788ce 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma @@ -167,154 +167,154 @@ T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O t) (eq_ind_r T t))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 (lift (S n) O t)) (plus h (minus n h)) -(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h))) -(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free -t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n -h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) -(ty3_abbr g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) -c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n -c0 (CHead d0 (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u -t)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 -d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: -T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda -(e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n -H4) in (let H6 \def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land -(le (plus x1 h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: +(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h +(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) +O t)) (lift_free t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus +O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) +(le_O_n x1))) (ty3_abbr g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 +(Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda +(H1: (getl n c0 (CHead d0 (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 +g d0 u t)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift +h x1 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: +T).(pc3 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 +t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef +n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x +\def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in (or_ind (land (lt n +x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n +h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) +(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1) (eq T x0 +(TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 -t2))) (\lambda (H7: (land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1) -(eq T x0 (TLRef n)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S -n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n -x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda -(t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: -nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 -H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus -x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind -Abst) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 -e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: -C).(\lambda (H11: (eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl -n e (CHead x3 (Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 -x3)).(let H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall -(x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) -\to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 -g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def -(eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) -H11) in (eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) -(refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda -(t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 -x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h -(minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda -(x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: -(ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: -nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h -(minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 -T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S -n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift -h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h -(minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda -(n0: nat).(pc3 c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O -(lift h (minus n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 -(S n)) x2))) (plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift -h (plus (S n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus -x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 -(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst -c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land -(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h) -n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le -(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T -(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T +t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T +(TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) +(lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind +nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n)))) +(lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl n e (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h +x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda +(x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h (minus x1 (S n)) +x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abst) x2))).(\lambda (H13: +(drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T u (\lambda (t0: +T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall +(e0: C).((drop h x5 d0 e0) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2) +t)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2) +H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift +h (minus x1 (S n)) x2) H11) in (eq_ind_r T (lift h (minus x1 (S n)) x2) +(\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O +t0))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14 x2 (minus +x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T +(\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: +T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S +n) O (lift h (minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) +t2))) (\lambda (x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S n)) x4) +t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S +n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift +(S n) O (lift h (minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) +t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S +n))) t2) (lift (S n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) +x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T +(lift (S n) O (lift h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift +(S n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind +nat x1 (\lambda (n0: nat).(pc3 c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) +(lift (S n) O (lift h (minus n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O +(lift h (minus x1 (S n)) x2))) (plus (S n) (minus x1 (S n))) (le_plus_minus +(S n) x1 H8)) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x2)) +(lift_d x2 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abst g +n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) +(getl_drop_conf_lt Abst c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) +H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n +h))))).(and_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: +T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 +(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T -(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O -u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 -O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) -(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h))) -(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free -u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n -h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) -(ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abst) u) -c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u t)).(\lambda -(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to -(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: B).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) u) t2 -t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T -(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: -T).(ty3 g e x0 t4)))))))))).(\lambda (t0: T).(\lambda (H5: (ty3 g (CHead c0 -(Bind b) u) t3 t0)).(\lambda (H6: ((\forall (x0: T).(\forall (x1: nat).((eq T -t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) -\to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t0)) -(\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H7: (eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: -C).(\lambda (H8: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda -(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq -T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) -z)))) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) -(\lambda (t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H9: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1 -x2))).(\lambda (H11: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind -b) x2 x3) (\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) -(THead (Bind b) u t3))) (\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def -(eq_ind T t2 (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 -(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) -\to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3)) -(\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let -H13 \def (eq_ind T t2 (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3)) -H3 (lift h (S x1) x3) H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3 -g (CHead c0 (Bind b) t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in -(let H15 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: -nat).((eq T (lift h (S x1) x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h -x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 -(Bind b) t4) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) -H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T u (\lambda (t4: -T).(\forall (x4: T).(\forall (x5: nat).((eq T t3 (lift h x5 x4)) \to (\forall -(e0: C).((drop h x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5: -T).(pc3 (CHead c0 (Bind b) t4) (lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0 -x4 t5))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u (\lambda -(t4: T).(ty3 g (CHead c0 (Bind b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let -H18 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t4 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x5 t5) t)) (\lambda (t5: T).(ty3 g e0 -x4 t5))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda -(t4: T).(ty3 g c0 t4 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2) -(\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind -b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20 -\def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) -(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H22: (ty3 g e x2 -x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e -(Bind b) x2) (drop_skip_bind h x1 c0 e H8 b x2)) in (ex2_ind T (\lambda (t4: -T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda -(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 +T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) +(lift (S n) O u))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift +(S (minus n h)) O u) (eq_ind_r T (lift (plus h (S (minus n h))) O u) (\lambda +(t0: T).(pc3 c0 t0 (lift (S n) O u))) (eq_ind nat (S (plus h (minus n h))) +(\lambda (n0: nat).(pc3 c0 (lift n0 O u) (lift (S n) O u))) (eq_ind nat n +(\lambda (n0: nat).(pc3 c0 (lift (S n0) O u) (lift (S n) O u))) (pc3_refl c0 +(lift (S n) O u)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus +x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n +h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free u (S (minus n h)) h O +x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n +H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abst g (minus n h) e +d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) +x0 H9))) H7)) H6)))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda +(t: T).(\lambda (H1: (ty3 g c0 u t)).(\lambda (H2: ((\forall (x0: T).(\forall +(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to +(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e +x0 t2)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda +(H3: (ty3 g (CHead c0 (Bind b) u) t2 t3)).(\lambda (H4: ((\forall (x0: +T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: C).((drop h +x1 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind +b) u) (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda +(t0: T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 t0)).(\lambda (H6: +((\forall (x0: T).(\forall (x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall +(e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t4: T).(pc3 +(CHead c0 (Bind b) u) (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x0 +t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H7: (eq T (THead +(Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H8: (drop h x1 c0 +e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) +y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda +(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: +T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: T).(ty3 g e +x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead +(Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1 x2))).(\lambda (H11: (eq +T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t4: +T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind b) u t3))) +(\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def (eq_ind T t2 (\lambda (t4: +T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 (lift h x5 x4)) \to (\forall +(e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda (t5: +T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0 +x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let H13 \def (eq_ind T t2 +(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3)) H3 (lift h (S x1) x3) +H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) +t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in (let H15 \def (eq_ind +T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S +x1) x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) +t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) t4) (lift h x5 +t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H12 (lift h x1 x2) H10) in +(let H16 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: +nat).((eq T t3 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 +(Bind b) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) t4) +(lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H6 (lift h x1 +x2) H10) in (let H17 \def (eq_ind T u (\lambda (t4: T).(ty3 g (CHead c0 (Bind +b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let H18 \def (eq_ind T u (\lambda +(t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 (lift h x5 x4)) \to +(\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t5: T).(pc3 c0 (lift +h x5 t5) t)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H2 (lift h x1 x2) H10) +in (let H19 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t)) H1 (lift h x1 +x2) H10) in (eq_ind_r T (lift h x1 x2) (\lambda (t4: T).(ex2 T (\lambda (t5: +T).(pc3 c0 (lift h x1 t5) (THead (Bind b) t4 t3))) (\lambda (t5: T).(ty3 g e +(THead (Bind b) x2 x3) t5)))) (let H20 \def (H18 x2 x1 (refl_equal T (lift h +x1 x2)) e H8) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t)) +(\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 +t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead +(Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda (_: (pc3 c0 (lift h x1 x4) +t)).(\lambda (H22: (ty3 g e x2 x4)).(let H23 \def (H15 x3 (S x1) (refl_equal +T (lift h (S x1) x3)) (CHead e (Bind b) x2) (drop_skip_bind h x1 c0 e H8 b +x2)) in (ex2_ind T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) +(lift h (S x1) t4) t3)) (\lambda (t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) +(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) +t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x5: +T).(\lambda (H24: (pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) x5) +t3)).(\lambda (H25: (ty3 g (CHead e (Bind b) x2) x3 x5)).(ex_ind T (\lambda +(t4: T).(ty3 g (CHead e (Bind b) x2) x5 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e -(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H24: (pc3 (CHead c0 -(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead -e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b) -x2) x5 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) -(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) -(\lambda (x6: T).(\lambda (H26: (ty3 g (CHead e (Bind b) x2) x5 -x6)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) -(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) -(THead (Bind b) x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S -x1) x5)) (\lambda (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3))) -(pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1 -(THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b -x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25))))) -H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1 -H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: +(THead (Bind b) x2 x3) t4))) (\lambda (x6: T).(\lambda (H26: (ty3 g (CHead e +(Bind b) x2) x5 x6)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) +(THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind +b) x2 x3) t4)) (THead (Bind b) x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 +x2) (lift h (S x1) x5)) (\lambda (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h +x1 x2) t3))) (pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) +H24) (lift h x1 (THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g +e x2 x4 H22 b x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 +H25))))) H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h +x1 H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma index d69a6dc4a..9809bc5ef 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma @@ -452,7 +452,7 @@ T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind -Abst) u x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c +Abst) u x0) x)).(\lambda (H2: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind Abst) u) t2 x0)).(\lambda (_: (ty3 g (CHead c (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) u t3) (THead (Bind Abst) u t2))))) (\lambda (_: @@ -461,16 +461,12 @@ T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (H6: +(H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (_: (ty3 g c u x4)).(\lambda (H7: (ty3 g (CHead c (Bind Abst) u) t1 x3)).(\lambda -(_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(and_ind (pc3 c u u) (\forall (b: -B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x3 t2))) (ex2 T (\lambda (w: -T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) -(\lambda (_: (pc3 c u u)).(\lambda (H10: ((\forall (b: B).(\forall (u0: -T).(pc3 (CHead c (Bind b) u0) x3 t2))))).(ex_intro2 T (\lambda (w: T).(ty3 g -c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x4 H6 -(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x3 H7 (H10 Abst u))))) -(pc3_gen_abst c u u x3 t2 H5))))))))) (ty3_gen_bind g Abst c u t1 (THead +(_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(let H_y \def (pc3_gen_abst_shift +c u x3 t2 H5) in (ex_intro2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: +T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x1 H2 (ty3_conv g (CHead c (Bind +Abst) u) t2 x0 H3 t1 x3 H7 H_y)))))))))) (ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))))) (ty3_gen_bind g Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2) H))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma index a9fb878ea..a470e0d0c 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma @@ -506,82 +506,82 @@ Appl) w t3) (THead (Flat Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w u0 H0 t3 x2 (ty3_sconv g c0 t3 x H16 (THead (Bind Abst) u0 t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 u0 x3 H20 Abst t x2 H21 x4 H22) H15)) (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g -c0 w u0 H0 v t H2) (pc3_s c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) -(THead (Flat Appl) w t3) (pc3_thin_dx c0 t3 (THead (Bind Abst) u0 t) H15 w -Appl)))))))))) (ty3_gen_bind g Abst c0 u0 t x1 H18)))) (ty3_correct g c0 v -(THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g -c0 v t3 H_y))))) t2 H13)) t0 (sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) -H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6) -\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (THead (Flat -Cast) v1 t0) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Flat Cast) -v2 t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) -(THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 -g c2 v1 v2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) -t2)))))) (\lambda (H10: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w -v))).(let H11 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (e: T).(match +c0 w u0 H0 v t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) t3 (ty3_unique g +c0 v (THead (Bind Abst) u0 t) H2 t3 H_y) w Appl))))))))) (ty3_gen_bind g Abst +c0 u0 t x1 H18)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) +(ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t3 H_y))))) t2 H13)) t0 +(sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) H10))) c1 (sym_eq C c1 c0 H6) +H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6) \Rightarrow (\lambda (H7: (eq +C c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w +v))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t3) t2)).(eq_ind C c0 (\lambda +(c2: C).((eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w v)) \to ((eq T +(THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2 t0 t3) +\to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H10: (eq T (THead +(Flat Cast) v1 t0) (THead (Flat Appl) w v))).(let H11 \def (eq_ind T (THead +(Flat Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: +F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead +(Flat Appl) w v) H10) in (False_ind ((eq T (THead (Flat Cast) v2 t3) t2) \to +((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w +v) t2)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) in (H5 (refl_equal +C c0) (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))) +(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 +t3)).(\lambda (H1: ((\forall (t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2 +t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: +((\forall (t4: T).((tau0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: +T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def +(match H4 in tau0 return (\lambda (c1: C).(\lambda (t: T).(\lambda (t5: +T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1 c0) \to ((eq T t (THead (Flat +Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) +t4)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H5: (eq C c1 +c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(\lambda (H7: +(eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) +(THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 +(THead (Flat Cast) t3 t2) t4)))) (\lambda (H8: (eq T (TSort n) (THead (Flat +Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) +t3 t2) H8) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead +(Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0 H5) H6 H7)))) | (tau0_abbr +c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T +(TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O w) +t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3 +t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2 (CHead d (Bind Abbr) v)) +\to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda +(H10: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T +(TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) +\Rightarrow False])) I (THead (Flat Cast) t3 t2) H10) in (False_ind ((eq T +(lift (S i) O w) t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d +v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) H11))) c1 (sym_eq C c1 c0 +H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 w H6) \Rightarrow (\lambda (H7: +(eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3 +t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c2: +C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O v) +t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g +c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead +(Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow -(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | -Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H10) in (False_ind -((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 -t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 c0 -H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat -Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: C).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda (H1: ((\forall -(t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2 t4))))).(\lambda (t0: T).(\lambda -(_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t4: T).((tau0 g c0 t3 t4) \to -(ty3 g c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat -Cast) t3 t2) t4)).(let H5 \def (match H4 in tau0 return (\lambda (c1: -C).(\lambda (t: T).(\lambda (t5: T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1 -c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0 -(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow -(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast) -t3 t2))).(\lambda (H7: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda -(_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g -n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H8: (eq T -(TSort n) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (TSort (next g -n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0 -H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C -c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda -(H9: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef -i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2 -(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) -t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Cast) t3 -t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 -t2) H10) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d -(Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) -t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 -w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) -(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind -C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T -(lift (S i) O v) t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d -v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T -(TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Cast) t3 t2) H10) in (False_ind ((eq T (lift (S i) O -v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 -g c0 (THead (Flat Cast) t3 t2) t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 -H6)))) | (tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 -c0)).(\lambda (H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 -t2))).(\lambda (H8: (eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda -(c2: C).((eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T -(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3 -g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v -t5) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v t5) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind -((eq T (THead (Bind b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6) -\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) -H7 H8 H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 +(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead +(Flat Cast) t3 t2) H10) in (False_ind ((eq T (lift (S i) O v) t4) \to ((getl +i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat +Cast) t3 t2) t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | +(tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda +(H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: +(eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead +(Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Bind b) v t6) +t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3 g c0 (THead (Flat +Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v t5) (THead (Flat +Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v t5) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) +\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T +(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6) \to (ty3 +g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 +H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t6) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T @@ -625,10 +625,9 @@ t3 t2) (THead (Flat Cast) v2 t6)) (\lambda (x0: T).(\lambda (H19: (ty3 g c0 t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) (THead (Flat Cast) x v2) (ty3_cast g c0 t6 v2 (ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t3) (ty3_cast g c0 t2 t3 H0 v2 H_y0) -(pc3_s c0 (THead (Flat Cast) v2 t3) (THead (Flat Cast) v2 t6) (pc3_thin_dx c0 -t6 t3 H17 v2 Cast))))) (ty3_correct g c0 t2 t6 H_y)))) (ty3_correct g c0 t3 -v2 H_y0))))))) t4 H14)) t5 (sym_eq T t5 t2 H13))) v1 (sym_eq T v1 t3 H12))) -H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) -(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 -H))))). +(pc3_thin_dx c0 t3 t6 (ty3_unique g c0 t2 t3 H0 t6 H_y) v2 Cast)))) +(ty3_correct g c0 t2 t6 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H14)) +t5 (sym_eq T t5 t2 H13))) v1 (sym_eq T v1 t3 H12))) H11))) c1 (sym_eq C c1 c0 +H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat +Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 H))))). -- 2.39.2