matita/matita/contribs/lambdadelta/.depend
matita/matita/contribs/lambdadelta/nodes
matita/matita/contribs/lambdadelta/token
-matita/matita/contribs/lambdadelta/2A1
+matita/matita/contribs/lambdadelta/2A
matita/matita/contribs/lambdadelta/*/*_probe.txt
matita/matita/contribs/lambdadelta/*/web/*_sum.tbl
] -H /2 width=1 by conj/
qed-.
-(* Basic_2A1: removed theorems 6:
- snv_fwd_da snv_fwd_lstas snv_cast_scpes
+(* Basic_2A1: removed theorems 3:
shnv_cast shnv_inv_cast snv_shnv_cast
*)
(* *)
(**************************************************************************)
-include "basic_2/dynamic/cnv_cpm_tdeq_conf.ma".
-include "basic_2/dynamic/cnv_cpms_tdeq.ma".
-(*
-include "basic_2/dynamic/cnv_cpm_trans.ma".
include "basic_2/dynamic/cnv_cpm_conf.ma".
-*)
+include "basic_2/dynamic/cnv_cpms_tdeq_conf.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Sub confluence propery with t-bound rt-computation for terms *************)
-fact cnv_cpms_tdeq_strip_lpr_aux (a) (h) (o) (G0) (L0) (T0):
+fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (a) (h) (o) (G0) (L0) (T0):
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
- ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
-#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
-@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
-[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
- <minus_O_n <minus_n_O
- elim (cnv_cpm_tdeq_conf_lpr … H0T1 0 T1 … H1T12 H2T12 … HL01 … HL02) // -L0 -H2T12
- <minus_O_n <minus_n_O #T #H1T1 #H2T1 #H1T2 #H2T2
- /3 width=5 by cpm_cpms, ex4_intro/
-| #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH
- #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
- elim (cnv_cpm_tdeq_conf_lpr … H0T0 … H1T03 H2T03 … H1T02 H2T02 … L0 … HL02) -T0 //
- #T0 #H1T30 #H2T30 #H1T20 #H2T20
- elim (IH … H1T30 H2T30 … HL01 … HL02) -L0 -T3
- #T3 #H1T13 #H2T13 #H1T03 #H2T03
- <minus_plus >arith_l3
- /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
-]
-qed-.
-
-fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
+ ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
-#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
-generalize in match IH1; generalize in match IH2;
-@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
-[ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
- <minus_O_n <minus_n_O
- elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T12 H0T1 H2T12 0 T1 … HL02 … HL01) // -L0 -H2T12
- <minus_O_n <minus_n_O #T #H1T2 #H2T2 #H1T1 #H2T1
- /3 width=5 by cpm_cpms, ex4_intro/
-| #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH #IH2 #IH1
- #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
- elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T02 H0T0 H2T02 … H1T03 H2T03 … HL02 L0) -H0T0 -H2T03 //
- #T4 #H1T24 #H2T24 #H1T34 #H2T34
- elim (IH … H1T34 H2T34 … HL01 … HL02) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -L0 -T0 -T3 (**)
- #T3 #H1T13 #H2T13 #H1T43 #H2T43
- <minus_plus >arith_l3
- /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
-]
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+#a #h #o #G #L0 #T0 #IH2 #IH1 #HT0
+#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
+#L1 #HL01 #L2 #HL02
+elim (cnv_cpms_tdeq_conf_lpr_aux … IH2 IH1 … H1T01 … H1T02 … HL01 … HL02) -IH2 -IH1 -H1T01 -H1T02 -HL01 -HL02
+/2 width=3 by ex2_intro/
qed-.
-(*
-fact cnv_cpms_conf_lpr_refl_refl_aux (h) (G0) (L1) (L2) (T0:term):
- ∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[h] T & ⦃G0,L2⦄ ⊢ T0 ➡*[h] T.
-/2 width=3 by ex2_intro/ qed-.
-
-fact cnv_cpms_conf_lpr_refl_step_aux (a) (h) (o) (G0) (L0) (T0) (m21) (m22):
+fact cnv_cpms_conf_lpr_refl_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m21) (m22):
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
⦃G0,L0⦄ ⊢ T0 ![a,h] →
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_step_step_aux (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+fact cnv_cpms_conf_lpr_step_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+ ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → T0 ≛[h,o] X1 → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 → X1 ≛[h,o] T1 →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ((∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ∀m21,m22.
+ ∀X2. ⦃G0,L0⦄ ⊢ X1 ➡[m21,h] X2 → (X1 ≛[h,o] X2 → ⊥) →
+ ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-m12,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m12-(m21+m22),h]T
+ ) →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
+#a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
+#X1 #H1X01 #H2X01 #T1 #H1XT1 #H2XT1 #X2 #H1X02 #H2X02 #T2 #HXT2
+#L1 #HL01 #L2 #HL02 #IH
+lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1
+lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X02 … L0 ?) // #HX2
+elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … H1X01 … H1X02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
+cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
+lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
+lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ20 … L0 ?) // #HZ0
+elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
+elim (tdeq_dec h o X1 Z0) #H2XZ
+[ -IH
+ elim (cnv_cpms_conf_lpr_tdeq_tdeq_aux … HX1 … H1XT1 H2XT1 … HXZ10 H2XZ … L1 … L0) [2,3: // |4,5: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
+| -H1XT1 -H2XT1
+ elim (cpms_tdneq_fwd_step_sn_aux … HXZ10 HX1 H2XZ) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
+ -HXZ10 -H2XZ #n1 #n2 #X0 #H1X10 #H2X10 #HXZ0 #Hn
+ elim (IH … H1X10 H2X10 … HXZ0 … L1 … L0) [2,3: // |4,5: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
+ >Hn -n1 -n2 -X0 -IH
+]
+#Z1 #HTZ1 #HZ01
+elim (IH1 … HZ01 … HZ02 L1 … L2) // -L0 -T0 -X1 -X2 -Z0 #Z #HZ01 #HZ02
+lapply (cpms_trans … HTZ1 … HZ01) -Z1 <arith_l4 #HT1Z
+lapply (cpms_trans … HTZ2 … HZ02) -Z2 <arith_l4 #HT2Z
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (n1) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+ ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-(m21+m22),h] T.
+#a #h #o #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
+#T1 #H1T01 #H2T01
+generalize in match m22; generalize in match m21; -m21 -m22
+generalize in match IH1; generalize in match IH2;
+@(cpms_tdeq_ind_sn … H1T01 HT0 H2T01 IH1 IH2) -n1 -T0
+[ #HT1 #IH2 #IH1 #m21 #m22
+ #X2 #HX02 #HnX02 #T2 #HXT2 #L1 #HL01 #L2 #HL02
+ <minus_O_n <minus_n_O
+ @(cnv_cpms_conf_lpr_refl_tdneq_sub … IH2 IH1) -IH2 -IH1 /2 width=4 by/
+| #m11 #m12 #T0 #X1 #H1X01 #HT0 #H2X01 #H1XT1 #_ #H2XT1 #IH #IH2 #IH1 #m21 #m22
+ #X2 #HX02 #HnX02 #T2 #HXT2 #L1 #HL01 #L2 #HL02
+ @(cnv_cpms_conf_lpr_step_tdneq_sub … IH2 IH1 … IH) -IH2 -IH1 -IH /2 width=4 by/
+]
+qed-.
+
+fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
(∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
⦃G0,L0⦄ ⊢ T0 ![a,h] →
∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛[h,o] X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
#a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
#X1 #HX01 #HnX01 #T1 #HXT1 #X2 #HX02 #HnX02 #T2 #HXT2
#L1 #HL01 #L2 #HL02
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_aux (a) (h) (o):
- ∀G0,L0,T0. 𝐏[h,o]⦃T0⦄ →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpms_conf_lpr a h G1 L1 T1.
-#a #h #o #G #L #T #H0 #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
+fact cnv_cpms_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr a h G L T.
+#a #h #o #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
#HT0 #n1 #T1 #HT01 #n2 #T2 #HT02 #L1 #HL01 #L2 #HL02 destruct
-elim (cpms_fwd_tdpos_sn … HT0 H0 … HT01) *
-elim (cpms_fwd_tdpos_sn … HT0 H0 … HT02) *
--H0 -HT01 -HT02
-[ #H21 #H22 #H11 #H12 destruct -a -o -L0
- <minus_n_O
- /2 width=1 by cnv_cpms_conf_lpr_refl_refl_aux/
-| #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #H11 #H12 destruct
- <minus_n_O <minus_O_n
- @(cnv_cpms_conf_lpr_refl_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
-| #H21 #H22 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
- <minus_n_O <minus_O_n
- @ex2_commute @(cnv_cpms_conf_lpr_refl_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
-| #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
- @(cnv_cpms_conf_lpr_step_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
+elim (tdeq_dec h o T0 T1) #H2T01
+elim (tdeq_dec h o T0 T2) #H2T02
+[ @(cnv_cpms_conf_lpr_tdeq_tdeq_aux … IH2 IH1) -IH2 -IH1 /2 width=1 by/
+| elim (cpms_tdneq_fwd_step_sn_aux … HT02 HT0 H2T02 IH1 IH2) -HT02 -H2T02
+ #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 destruct
+ @(cnv_cpms_conf_lpr_tdeq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
+| elim (cpms_tdneq_fwd_step_sn_aux … HT01 HT0 H2T01 IH1 IH2) -HT01 -H2T01
+ #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
+ @ex2_commute @(cnv_cpms_conf_lpr_tdeq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
+| elim (cpms_tdneq_fwd_step_sn_aux … HT01 HT0 H2T01 IH1 IH2) -HT01 -H2T01
+ elim (cpms_tdneq_fwd_step_sn_aux … HT02 HT0 H2T02 IH1 IH2) -HT02 -H2T02
+ #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
+ @(cnv_cpms_conf_lpr_tdneq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
]
qed-.
-*)
(* Properties with restricted rt-computation for terms **********************)
fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (o) (G) (L) (T1):
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛[h,o] T2 → ⊥) →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛[h,o] T2 → ⊥) →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
∃∃n1,n2,T0. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛[h,o] T0 → ⊥ & ⦃G, L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/cnv_cpm_tdeq_conf.ma".
+include "basic_2/dynamic/cnv_cpms_tdeq.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Sub confluence propery with restricted rt-transition for terms ***********)
+
+fact cnv_cpms_tdeq_strip_lpr_aux (a) (h) (o) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
+#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
+[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
+ <minus_O_n <minus_n_O
+ elim (cnv_cpm_tdeq_conf_lpr … H0T1 0 T1 … H1T12 H2T12 … HL01 … HL02) // -L0 -H2T12
+ <minus_O_n <minus_n_O #T #H1T1 #H2T1 #H1T2 #H2T2
+ /3 width=5 by cpm_cpms, ex4_intro/
+| #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH
+ #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
+ elim (cnv_cpm_tdeq_conf_lpr … H0T0 … H1T03 H2T03 … H1T02 H2T02 … L0 … HL02) -T0 //
+ #T0 #H1T30 #H2T30 #H1T20 #H2T20
+ elim (IH … H1T30 H2T30 … HL01 … HL02) -L0 -T3
+ #T3 #H1T13 #H2T13 #H1T03 #H2T03
+ <minus_plus >arith_l3
+ /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
+]
+qed-.
+
+fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
+#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+generalize in match IH1; generalize in match IH2;
+@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
+[ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
+ <minus_O_n <minus_n_O
+ elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T12 H0T1 H2T12 0 T1 … HL02 … HL01) // -L0 -H2T12
+ <minus_O_n <minus_n_O #T #H1T2 #H2T2 #H1T1 #H2T1
+ /3 width=5 by cpm_cpms, ex4_intro/
+| #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH #IH2 #IH1
+ #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
+ elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T02 H0T0 H2T02 … H1T03 H2T03 … HL02 L0) -H0T0 -H2T03 //
+ #T4 #H1T24 #H2T24 #H1T34 #H2T34
+ elim (IH … H1T34 H2T34 … HL01 … HL02) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -L0 -T0 -T3 (**)
+ #T3 #H1T13 #H2T13 #H1T43 #H2T43
+ <minus_plus >arith_l3
+ /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
+]
+qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Main preservation properties *********************************************)
-(*
+
(* Basic_2A1: uses: snv_preserve *)
lemma cnv_preserve (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] →
∧∧ IH_cnv_cpms_conf_lpr a h G L T
& IH_cnv_cpm_trans_lpr a h G L T.
#a #h #G #L #T #HT
-elim (tdpos_total h … T) #o
+letin o ≝ (sd_O h)
lapply (cnv_fwd_fsb … o … HT) -HT #H
-@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH #Ho
+@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH
@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ]
@(aux … o … G L T) // #G0 #L0 #T0 #H
elim (IH … H) -IH -H //
-
-
-lemma snv_preserve: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
- ∧∧ IH_da_cpr_lpr h o G L T
- & IH_snv_cpr_lpr h o G L T
- & IH_snv_lstas h o G L T
- & IH_lstas_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_fwd_aaa … HT) -HT
-#A #HT @(aaa_ind_fpbg h o … HT) -G -L -T -A
-#G #L #T #A #_ #IH -A @and4_intro
-[ letin aux ≝ da_cpr_lpr_aux | letin aux ≝ snv_cpr_lpr_aux
-| letin aux ≝ snv_lstas_aux | letin aux ≝ lstas_cpr_lpr_aux
-]
-@(aux … G L T) // #G0 #L0 #T0 #H elim (IH … H) -IH -H //
-qed-.
-
-theorem da_cpr_lpr: ∀h,o,G,L,T. IH_da_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
-qed-.
-
-theorem snv_cpr_lpr: ∀h,o,G,L,T. IH_snv_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
qed-.
-theorem snv_lstas: ∀h,o,G,L,T. IH_snv_lstas h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/
+theorem cnv_cpms_conf_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_conf_lpr a h G L T.
+#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
qed-.
-theorem lstas_cpr_lpr: ∀h,o,G,L,T. IH_lstas_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/
+(* Basic_2A1: uses: snv_cpr_lpr *)
+theorem cnv_cpm_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpm_trans_lpr a h G L T.
+#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
qed-.
(* Advanced preservation properties *****************************************)
-lemma snv_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G #L1 #T1 #HT1 #T2 #H
-@(cprs_ind … H) -T2 /3 width=5 by snv_cpr_lpr/
-qed-.
-
-lemma da_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
-#h #o #G #L1 #T1 #HT1 #d #Hd #T2 #H
-@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/
-qed-.
-
-lemma lstas_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #o #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ]
-#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
-elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (lstas_cpr_lpr … o … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2
-[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1
-/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
-qed-.
-
-lemma lstas_cpcs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, o] →
- ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, o] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 →
- ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #o #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1
-elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2
-lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/
+(* Basic_2A1: uses: snv_cprs_lpr *)
+lemma cnv_cpms_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_trans_lpr a h G L T.
+#a #h #G #L1 #T1 #HT1 #n #T2 #H
+@(cpms_ind_dx … H) -n -T2 /3 width=6 by cnv_cpm_trans_lpr/
qed-.
-*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/da_aaa.ma".
+include "basic_2/computation/scpds_aaa.ma".
+include "basic_2/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma snv_fwd_da: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d.
+#h #o #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_da/
+qed-.
+
+lemma snv_fwd_lstas: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
+ ∀d. ∃U. ⦃G, L⦄ ⊢ T •*[h, d] U.
+#h #o #G #L #T #H #d elim (snv_fwd_aaa … H) -H
+#A #HT elim (aaa_lstas h … HT d) -HT /2 width=2 by ex_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lsubd_da.ma".
+include "basic_2/dynamic/snv_aaa.ma".
+include "basic_2/dynamic/snv_scpes.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+fact da_cpr_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h o G1 L1 T1.
+#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #s #_ #_ #_ #_ #d #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
+ lapply (da_inv_sort … H2) -H2
+ lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
+| #i #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
+ elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #d1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
+ lapply (drop_mono … H … HLK1) -H #H destruct
+ elim (cpr_inv_lref1 … H3) -H3
+ [1,3: #H destruct
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
+ |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
+ lapply (drop_mono … H … HLK1) -H #H destruct
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
+ lapply (drop_fwd_drop2 … HLK2) -V2
+ /4 width=8 by da_lift, fqup_fpbg/
+ ]
+| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+ elim (snv_inv_bind … H1) -H1 #_ #HT1
+ lapply (da_inv_bind … H2) -H2
+ elim (cpr_inv_bind1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
+ | #T2 #HT12 #HT2 #H1 #H2 destruct
+ /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, drop_drop/
+ ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #b1 #W1 #U1 #d1 #HV1 #HT1 #HVW1 #HTU1
+ lapply (da_inv_flat … H2) -H2 #Hd
+ elim (cpr_inv_appl1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fqup_fpbg/
+ | #b #V2 #W #W2 #U #U2 #HV12 #HW2 #HU2 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HU
+ lapply (da_inv_bind … Hd) -Hd #Hd
+ elim (scpds_inv_abst1 … HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -d1
+ elim (snv_fwd_da … HV1) #d1 #Hd1
+ elim (snv_fwd_da … HW) #d0 #Hd0
+ lapply (cprs_scpds_div … HW3 … Hd0 … 1 HVW1) -W3 #H
+ elim (da_scpes_aux … IH3 IH2 IH1 … Hd0 … Hd1 … H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
+ <minus_n_O #H destruct >(plus_minus_k_k d1 1) in Hd1; // -H1 #Hd1
+ lapply (IH1 … HV1 … Hd1 … HV12 … HL12) -HV1 -Hd1 -HV12 [ /2 by fqup_fpbg/ ]
+ lapply (IH1 … Hd0 … HW2 … HL12) -Hd0 /2 width=1 by fqup_fpbg/ -HW
+ lapply (IH1 … HU … Hd … HU2 (L2.ⓛW2) ?) -IH1 -HU -Hd -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
+ /4 width=6 by da_bind, lsubd_da_trans, lsubd_beta/
+ | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -d1 -HV1
+ elim (snv_inv_bind … HT1) -HT1 #_
+ lapply (da_inv_bind … Hd) -Hd
+ /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
+ ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
+ lapply (da_inv_flat … H2) -H2 #Hd
+ elim (cpr_inv_cast1 … H3) -H3
+ [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
+ | /3 width=7 by fqup_fpbg/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fsb_aaa.ma".
+include "basic_2/dynamic/snv_da_lpr.ma".
+include "basic_2/dynamic/snv_lstas.ma".
+include "basic_2/dynamic/snv_lstas_lpr.ma".
+include "basic_2/dynamic/snv_lpr.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Main preservation properties *********************************************)
+
+theorem da_cpr_lpr: ∀h,o,G,L,T. IH_da_cpr_lpr h o G L T.
+#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
+qed-.
+
+theorem snv_lstas: ∀h,o,G,L,T. IH_snv_lstas h o G L T.
+#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/
+qed-.
+
+theorem lstas_cpr_lpr: ∀h,o,G,L,T. IH_lstas_cpr_lpr h o G L T.
+#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/
+qed-.
+
+(* Advanced preservation properties *****************************************)
+
+lemma da_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
+#h #o #G #L1 #T1 #HT1 #d #Hd #T2 #H
+@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/
+qed-.
+
+lemma lstas_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #o #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (lstas_cpr_lpr … o … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2
+[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
+qed-.
+
+lemma lstas_cpcs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, o] →
+ ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, o] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #o #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1
+elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2
+lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fpbg_fpbs.ma".
+include "basic_2/equivalence/scpes_cpcs.ma".
+include "basic_2/equivalence/scpes_scpes.ma".
+include "basic_2/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Inductive premises for the preservation results **************************)
+
+definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
+
+definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
+
+definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+
+definition IH_snv_lstas: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
+ ∀d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
+ ∀U. ⦃G, L⦄ ⊢ T •*[h, d2] U → ⦃G, L⦄ ⊢ U ¡[h, o].
+
+(* Properties for the preservation results **********************************)
+
+fact snv_cprs_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
+#h #o #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
+@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
+qed-.
+
+fact da_cprs_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
+#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
+qed-.
+
+fact da_scpds_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d1. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
+ ∀T2,d2. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d2] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ d2 ≤ d1 ∧ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d1-d2.
+#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12
+lapply (da_mono … H … Hd1) -H #H destruct
+lapply (lstas_da_conf … HT1 … Hd1) #Hd12
+lapply (da_cprs_lpr_aux … IH2 IH1 … Hd12 … HT2 … HL12) -IH2 -IH1 -HT2 -HL12
+/3 width=8 by fpbg_fpbs_trans, lstas_fpbs, conj/
+qed-.
+
+fact da_scpes_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] →
+ ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] →
+ ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 →
+ ∀d21,d22. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 →
+ ∧∧ d21 ≤ d11 & d22 ≤ d12 & d11 - d21 = d12 - d22.
+#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2
+elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd11 … HT1 … L) -Hd11 -HT1 //
+elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd12 … HT2 … L) -Hd12 -HT2 //
+/3 width=7 by da_mono, and3_intro/
+qed-.
+
+fact lstas_cprs_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (IH1 … Hd21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
+[2: /3 width=12 by da_cprs_lpr_aux/
+|3: /3 width=10 by snv_cprs_lpr_aux/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
+] -G0 -L0 -T0 -T1 -T -d1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
+qed-.
+
+fact scpds_cpr_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀U1,d. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, o, d] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
+lapply (IH2 … H01 … HTd1 … HT12 … HL12) -L0 -T0 // -T1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/
+qed-.
+
+fact scpes_cpr_lpr_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
+ ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, o] →
+ ∀d1,d2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ U1 •*⬌*[h, o, d1, d2] U2.
+#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
+elim (scpds_cpr_lpr_aux … IH2 IH1 … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
+elim (scpds_cpr_lpr_aux … IH2 IH1 … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
+elim (cprs_conf … H1 … H2) -T0
+/3 width=5 by scpds_div, scpds_cprs_trans/
+qed-.
+
+fact lstas_scpds_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
+ ∀G,L,T. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, o] →
+ ∀d,d1. d1 ≤ d → ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀T1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T •*➡*[h, o, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d2-d1, d1-d2] T2.
+#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2
+lapply (da_mono … H … HTd) -H #H destruct
+lapply (lstas_da_conf … HT1 … HTd) #HTd1
+lapply (lstas_da_conf … HTX … HTd) #HXd
+lapply (da_cprs_lpr_aux … IH3 IH2 … HXd … HXT2 L ?)
+[1,2,3: /3 width=8 by fpbg_fpbs_trans, lstas_fpbs/ ] #HTd2
+elim (le_or_ge d1 d2) #Hd12 >(eq_minus_O … Hd12)
+[ elim (da_lstas … HTd2 0) #X2 #HTX2 #_ -IH4 -IH3 -IH2 -IH1 -H0 -HT -HTd -HXd
+ /5 width=6 by lstas_scpds, scpds_div, cprs_strap1, lstas_cpr, lstas_conf_le, monotonic_le_minus_l, ex4_2_intro/
+| elim (da_lstas … HTd1 0) #X1 #HTX1 #_
+ lapply (lstas_conf_le … HTX … HT1) // #HXT1 -HT1
+ elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … HXd … HXT1 … HXT2 L) -IH3 -IH2 -IH1 -HXd -HXT1 -HXT2
+ /4 width=8 by cpcs_scpes, cpcs_cpr_conf, fpbg_fpbs_trans, lstas_fpbs, lstas_cpr, monotonic_le_minus_l/
+]
+qed-.
+
+fact scpes_le_aux: ∀h,o,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] →
+ ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] →
+ ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 →
+ ∀d21,d22,d. d21 + d ≤ d11 → d22 + d ≤ d12 →
+ ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21+d, d22+d] T2.
+#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20
+elim (da_lstas … Hd11 (d21+d)) #X1 #HTX1 #_
+elim (da_lstas … Hd12 (d22+d)) #X2 #HTX2 #_
+lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd11 … HTX1 … HT10) -HT10
+[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_k_k_commutative #HX1 ]
+lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd12 … HTX2 … HT20) -HT20
+[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_k_k_commutative #HX2 ]
+lapply (lstas_scpes_trans … Hd11 … HTX1 … HX1) [ // ] -Hd11 -HTX1 -HX1 -H1 #H1
+lapply (lstas_scpes_trans … Hd12 … HTX2 … HX2) [ // ] -Hd12 -HTX2 -HX2 -H2 #H2
+/2 width=4 by scpes_canc_dx/ (**) (* full auto fails *)
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma snv_cast_scpes: ∀h,o,G,L,U,T. ⦃G, L⦄ ⊢ U ¡[h, o] → ⦃G, L⦄ ⊢ T ¡[h, o] →
+ ⦃G, L⦄ ⊢ U •*⬌*[h, o, 0, 1] T → ⦃G, L⦄ ⊢ ⓝU.T ¡[h, o].
+#h #o #G #L #U #T #HU #HT * /2 width=3 by snv_cast/
+qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/da_aaa.ma".
-include "basic_2/computation/scpds_aaa.ma".
-include "basic_2/dynamic/snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Forward lemmas on atomic arity assignment for terms **********************)
-
-lemma snv_fwd_aaa: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
-#h #o #G #L #T #H elim H -G -L -T
-[ /2 width=2 by aaa_sort, ex_intro/
-| #I #G #L #K #V #i #HLK #_ * /3 width=6 by aaa_lref, ex_intro/
-| #a * #G #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2 by aaa_abbr, aaa_abst, ex_intro/
-| #a #G #L #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 * #B #HV * #X #HT
- lapply (scpds_aaa_conf … HV … HVW0) -HVW0 #HW0
- lapply (scpds_aaa_conf … HT … HTU0) -HTU0 #H
- elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct
- lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4 by aaa_appl, ex_intro/
-| #G #L #U #T #U0 #_ #_ #HU0 #HTU0 * #B #HU * #A #HT
- lapply (scpds_aaa_conf … HU … HU0) -HU0 #HU0
- lapply (scpds_aaa_conf … HT … HTU0) -HTU0 #H
- lapply (aaa_mono … H … HU0) -U0 #H destruct /3 width=3 by aaa_cast, ex_intro/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma snv_fwd_da: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d.
-#h #o #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_da/
-qed-.
-
-lemma snv_fwd_lstas: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
- ∀d. ∃U. ⦃G, L⦄ ⊢ T •*[h, d] U.
-#h #o #G #L #T #H #d elim (snv_fwd_aaa … H) -H
-#A #HT elim (aaa_lstas h … HT d) -HT /2 width=2 by ex_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lsubd_da.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_scpes.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on degree assignment for terms ********************************)
-
-fact da_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h o G1 L1 T1.
-#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #s #_ #_ #_ #_ #d #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
- lapply (da_inv_sort … H2) -H2
- lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
-| #i #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
- elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #d1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
- lapply (drop_mono … H … HLK1) -H #H destruct
- elim (cpr_inv_lref1 … H3) -H3
- [1,3: #H destruct
- lapply (fqup_lref … G1 … HLK1)
- elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
- |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
- lapply (drop_mono … H … HLK1) -H #H destruct
- lapply (fqup_lref … G1 … HLK1)
- elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
- lapply (drop_fwd_drop2 … HLK2) -V2
- /4 width=8 by da_lift, fqup_fpbg/
- ]
-| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH2
- elim (snv_inv_bind … H1) -H1 #_ #HT1
- lapply (da_inv_bind … H2) -H2
- elim (cpr_inv_bind1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
- | #T2 #HT12 #HT2 #H1 #H2 destruct
- /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, drop_drop/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #b1 #W1 #U1 #d1 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat … H2) -H2 #Hd
- elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fqup_fpbg/
- | #b #V2 #W #W2 #U #U2 #HV12 #HW2 #HU2 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HU
- lapply (da_inv_bind … Hd) -Hd #Hd
- elim (scpds_inv_abst1 … HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -d1
- elim (snv_fwd_da … HV1) #d1 #Hd1
- elim (snv_fwd_da … HW) #d0 #Hd0
- lapply (cprs_scpds_div … HW3 … Hd0 … 1 HVW1) -W3 #H
- elim (da_scpes_aux … IH3 IH2 IH1 … Hd0 … Hd1 … H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
- <minus_n_O #H destruct >(plus_minus_k_k d1 1) in Hd1; // -H1 #Hd1
- lapply (IH1 … HV1 … Hd1 … HV12 … HL12) -HV1 -Hd1 -HV12 [ /2 by fqup_fpbg/ ]
- lapply (IH1 … Hd0 … HW2 … HL12) -Hd0 /2 width=1 by fqup_fpbg/ -HW
- lapply (IH1 … HU … Hd … HU2 (L2.ⓛW2) ?) -IH1 -HU -Hd -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
- /4 width=6 by da_bind, lsubd_da_trans, lsubd_beta/
- | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -d1 -HV1
- elim (snv_inv_bind … HT1) -HT1 #_
- lapply (da_inv_bind … Hd) -Hd
- /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
- lapply (da_inv_flat … H2) -H2 #Hd
- elim (cpr_inv_cast1 … H3) -H3
- [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
- | /3 width=7 by fqup_fpbg/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fsb_aaa.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* forward lemmas on "qrst" strongly normalizing closures *********************)
-
-lemma snv_fwd_fsb: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → ⦥[h, o] ⦃G, L, T⦄.
-#h #o #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/multiple/fqus_alt.ma".
-include "basic_2/computation/scpds_lift.ma".
-include "basic_2/dynamic/snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Relocation properties ****************************************************)
-
-lemma snv_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, o] → ∀L,b,l,k. ⬇[b, l, k] L ≘ K →
- ∀U. ⬆[l, k] T ≘ U → ⦃G, L⦄ ⊢ U ¡[h, o].
-#h #o #G #K #T #H elim H -G -K -T
-[ #G #K #s #L #b #l #k #_ #X #H
- >(lift_inv_sort1 … H) -X -K -l -k //
-| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #b #l #k #HLK #X #H
- elim (lift_inv_lref1 … H) * #Hil #H destruct
- [ elim (drop_trans_le … HLK … HK0) -K /2 width=2 by ylt_fwd_le/ #X #HL0 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #L0 #W #HLK0 #HVW #H destruct
- /3 width=9 by snv_lref/
- | lapply (drop_trans_ge … HLK … HK0 ?) -K
- /3 width=9 by snv_lref, drop_inv_gen/
- ]
-| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #b #l #k #HLK #X #H
- elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
- /4 width=5 by snv_bind, drop_skip/
-| #a #G #K #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L #b #l #k #HLK #X #H
- elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
- elim (lift_total W0 l k)
- elim (lift_total U0 (l+1) k)
- /4 width=17 by snv_appl, scpds_lift, lift_bind/
-| #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #b #l #k #HLK #X #H
- elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
- elim (lift_total U0 l k)
- /3 width=12 by snv_cast, scpds_lift/
-]
-qed.
-
-lemma snv_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, o] → ∀K,b,l,k. ⬇[b, l, k] L ≘ K →
- ∀T. ⬆[l, k] T ≘ U → ⦃G, K⦄ ⊢ T ¡[h, o].
-#h #o #G #L #U #H elim H -G -L -U
-[ #G #L #s #K #b #l #k #_ #X #H
- >(lift_inv_sort2 … H) -X -L -l -k //
-| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #b #l #k #HLK #X #H
- elim (lift_inv_lref2 … H) * #Hil #H destruct
- [ elim (drop_conf_le … HLK … HL0) -L /2 width=2 by ylt_fwd_le/ #X #HK0 #H
- elim (drop_inv_skip1 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K0 #V #HLK0 #HVW #H destruct
- /3 width=12 by snv_lref/
- | lapply (drop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
- ]
-| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #b #l #k #HLK #X #H
- elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
- /4 width=5 by snv_bind, drop_skip/
-| #a #G #L #W #W1 #U #U1 #d #_ #_ #HW1 #HU1 #IHW #IHU #K #b #l #k #HLK #X #H
- elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
- elim (scpds_inv_lift1 … HW1 … HLK … HVW) -HW1 #W0 #HW01 #HVW0
- elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU0
- elim (lift_inv_bind2 … H) -H #Y #U0 #HY #HU01 #H destruct
- lapply (lift_inj … HY … HW01) -HY #H destruct
- /3 width=6 by snv_appl/
-| #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #b #l #k #HLK #X #H
- elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
- elim (scpds_inv_lift1 … HWU1 … HLK … HVW) -HWU1 #U0 #HU01 #HVU0
- elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #HX #HTU0
- lapply (lift_inj … HX … HU01) -HX #H destruct
- /3 width=5 by snv_cast/
-]
-qed-.
-
-(* Properties on subclosure *************************************************)
-
-lemma snv_fqu_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I1 #G1 #L1 #V1 #H
- elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
- lapply (drop_inv_O2 … H) -H #H destruct //
-|2: *
-|5,6: /3 width=8 by snv_inv_lift/
-]
-[1,3: #a #I #G1 #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
-|2,4: * #G1 #L1 #V1 #T1 #H
- [1,3: elim (snv_inv_appl … H) -H //
- |2,4: elim (snv_inv_cast … H) -H //
- ]
-]
-qed-.
-
-lemma snv_fquq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_inv_gen … H) -H [|*]
-/2 width=5 by snv_fqu_conf/
-qed-.
-
-lemma snv_fqup_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by fqup_strap1, snv_fqu_conf/
-qed-.
-
-lemma snv_fqus_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_gen … H) -H [|*]
-/2 width=5 by snv_fqup_conf/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_lift.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_scpes.ma".
-include "basic_2/dynamic/lsubsv_snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact snv_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h o G1 L1 T1.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #s #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
- >(cpr_inv_sort1 … H2) -X //
-| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (fqup_lref … G1 … HLK1) #HKL
- elim (cpr_inv_lref1 … H2) -H2
- [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
- | * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
- lapply (drop_mono … H … HLK1) -HLK1 -H #H destruct
- lapply (drop_fwd_drop2 … HLK2) -HLK2 /4 width=8 by fqup_fpbg, snv_lift/
- ]
-| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- elim (cpr_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpr_pair/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
- /4 width=10 by fqup_fpbg, snv_inv_lift, lpr_pair, drop_drop/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
- elim (cpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 -HV12 #XV #HVW2 #HXV
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HTU1 -HT12 #X #HTU2 #H
- elim (cprs_inv_abst1 … H) -H #XW #U2 #HXW #_ #H destruct -IH1 -IH3 -IH2 -L1
- elim (cprs_conf … HXV … HXW) -W1 #W2 #HXV #HXW
- lapply (scpds_cprs_trans … HVW2 … HXV) -XV
- lapply (scpds_cprs_trans … (ⓛ{a}W2.U2) … HTU2 ?)
- /2 width=7 by snv_appl, cprs_bind/
- | #b #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW10 #HT10
- elim (scpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30 -d0
- elim (snv_fwd_da … HV1) #d #HV1d
- elim (snv_fwd_da … HW10) #d0 #HW10d
- lapply (cprs_scpds_div … HW130 … HW10d … 1 HVW1) -W30 #HVW10
- elim (da_scpes_aux … IH4 IH1 IH2 … HW10d … HV1d … HVW10) /2 width=1 by fqup_fpbg/
- #_ #Hd <minus_n_O #H destruct >(plus_minus_k_k d 1) in HV1d; // -Hd #HV1d
- lapply (scpes_cpr_lpr_aux … IH2 IH3 … HVW10 … HW120 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW10 #HVW20
- lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1d #HV2d
- lapply (IH2 … HW10d … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10d #HW20d
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH1 … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10 #HW20
- lapply (IH1 … HT10 … HT120 … (L2.ⓛW20) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT10 #HT20
- @snv_bind /2 width=1 by snv_cast_scpes/
- @(lsubsv_snv_trans … HT20) -HT20
- @(lsubsv_beta … (d-1)) //
- @shnv_cast [1,2: // ] #d0 #Hd0
- lapply (scpes_le_aux … IH4 IH1 IH2 IH3 … HW20d … HV2d … d0 … HVW20) -IH4 -IH3 -IH2 -IH1 -HW20d -HV2d -HVW20
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
- elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
- elim (scpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
- elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 #XV #HXV0 #HXVW1
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HXT2 #H
- elim (cprs_inv_abst1 … H) -H #W #U2 #HW3 #_ #H destruct -U3
- lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ #HW2
- lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ #HV0
- lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
- lapply (snv_lift … HV0 (L2.ⓓW2) (Ⓕ) … HV02) /2 width=1 by drop_drop/ -HV0 #HV2
- elim (lift_total XV 0 1) #XW #HXVW
- lapply (scpds_lift … HXV0 (L2.ⓓW2) (Ⓕ) … HV02 … HXVW) /2 width=1 by drop_drop/ -V0 #HXWV2
- lapply (cprs_lift … HXVW1 (L2.ⓓW2) (Ⓕ) … HW13 … HXVW) /2 width=1 by drop_drop/ -W1 -XV #HXW3
- elim (cprs_conf … HXW3 … HW3) -W3 #W3 #HXW3 #HW3
- lapply (scpds_cprs_trans … HXWV2 … HXW3) -XW
- lapply (scpds_cprs_trans … (ⓛ{a}W3.U2) … HXT2 ?) /2 width=1 by cprs_bind/ -W
- /3 width=6 by snv_appl, snv_bind/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
- elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
- elim (cpr_inv_cast1 … H2) -H2
- [ * #W2 #T2 #HW12 #HT12 #H destruct
- elim (snv_fwd_da … HW1) #d #HW1d
- lapply (scpds_div … HWU1 … HTU1) -U1 -d #HWT1
- lapply (scpes_cpr_lpr_aux … IH2 IH3 … HWT1 … HW12 … HT12 … HL12) /2 width=1 by fqup_fpbg/
- lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -L1
- /2 width=1 by snv_cast_scpes/
- | #H -IH3 -IH2 -HW1 -U1
- lapply (IH1 … H … HL12) /2 width=1 by fqup_fpbg/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_lift.ma".
-include "basic_2/dynamic/snv_scpes.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on nat-iterated stratified static type assignment for terms ***)
-
-fact snv_lstas_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_lstas h o G1 L1 T1.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #s #HG0 #HL0 #HT0 #_ #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
- >(lstas_inv_sort1 … H2) -X //
-| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #T #H2 destruct -IH4 -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HLK0 #HX0
- elim (da_inv_lref … Hd1) -Hd1 * #K1 [ #V1 | #W1 #d0 ] #HLK1 [ #Hd1 | #Hd0 #H ]
- lapply (drop_mono … HLK0 … HLK1) -HLK0 #H0 destruct
- elim (lstas_inv_lref1 … H2) -H2 * #K #Y #X [3,6: #d ] #HLK #HYX [1,2: #HXT #H0 |3,5: #HXT |4,6: #H1 #H2 ]
- lapply (drop_mono … HLK … HLK1) -HLK #H destruct
- [ lapply (le_plus_to_le_c … Hd21) -Hd21 #Hd21 |3: -Hd21 ]
- lapply (fqup_lref … G1 … HLK1) #H
- lapply (drop_fwd_drop2 … HLK1) /4 width=8 by snv_lift, snv_lref, fqup_fpbg/
-| #p #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- lapply (da_inv_bind … Hd1) -Hd1 #Hd1
- elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fqup_fpbg, snv_bind/
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat … Hd1) -Hd1 #Hd1
- elim (lstas_inv_appl1 … H2) -H2 #T0 #HT10 #H destruct
- lapply (IH1 … HT1 … Hd1 … HT10) /2 width=1 by fqup_fpbg/ #HT0
- lapply (lstas_scpds_aux … IH1 IH4 IH3 IH2 … Hd1 … HT10 … HTU1) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fqup_fpbg/ -T1 -d1 #H
- elim (scpes_inv_abst2 … H) -H /3 width=6 by snv_appl, scpds_cprs_trans/
-| #U1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
- elim (snv_inv_cast … H1) -H1
- lapply (da_inv_flat … Hd1) -Hd1
- lapply (lstas_inv_cast1 … H2) -H2 /3 width=8 by fqup_fpbg/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_scpes.ma".
-include "basic_2/dynamic/lsubsv_lstas.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on sn parallel reduction for local environments ***************)
-
-fact lstas_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lstas_cpr_lpr h o G1 L1 T1.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #s #_ #_ #_ #_ #d1 #d2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
- >(lstas_inv_sort1 … H2) -X2
- >(cpr_inv_sort1 … H3) -X3 /2 width=3 by ex2_intro/
-| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
- elim (da_inv_lref … Hd1) -Hd1 * #K1 [ #V1 | #W1 #d ] #HLK1 [ #HVd1 | #HWd1 #H destruct ]
- lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
- elim (lstas_inv_lref1 … H2) -H2 * #K0 #V0 #X0 [3,6: #d0 ] #HK0 #HVX0 [1,2: #HX02 #H |3,5: #HX02 |4,6: #H1 #H2 ] destruct
- lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
- [ lapply (le_plus_to_le_c … Hd21) -Hd21 #Hd21 |3: -Hd21 ]
- lapply (fqup_lref … G1 … HLK1) #HKV1
- elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #W2 | #V2 ] #HK12 [ #HW12 | #HW12 | #HV12 ] #H destruct
- lapply (drop_fwd_drop2 … HLK2) #H2
- elim (cpr_inv_lref1 … H3) -H3
- [1,3,5: #H destruct -HLK1
- |2,4,6: * #K #V #X #H #HVX #HX3
- lapply (drop_mono … H … HLK1) -H -HLK1 #H destruct
- ]
- [ lapply (IH2 … HWd1 … HW12 … HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
- elim (da_lstas … H d0) -H
- elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
- elim (lift_total V2 0 (i+1))
- /3 width=12 by cpcs_lift, lstas_succ, ex2_intro/
- | elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0
- /3 width=5 by fqup_fpbg, lstas_zero, ex2_intro/
- | elim (IH1 … HVd1 … HVX0 … HV12 … HK12) -IH1 -HVd1 -HVX0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
- elim (lift_total W2 0 (i+1))
- /4 width=12 by cpcs_lift, lstas_ldef, ex2_intro/
- | elim (IH1 … HVd1 … HVX0 … HVX … HK12) -IH1 -HVd1 -HVX0 -HVX -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -d1 #W2 #HXW2 #HW02
- elim (lift_total W2 0 (i+1))
- /3 width=12 by cpcs_lift, lstas_lift, ex2_intro/
- ]
-| #p #_ #_ #HT0 #H1 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #_ #HT1
- lapply (da_inv_bind … Hd1) -Hd1 #Hd1
- elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
- elim (cpr_inv_bind1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH1 … Hd1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hd1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
- /4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
- | #T3 #HT13 #HXT3 #H1 #H2 destruct
- elim (IH1 … Hd1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hd1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
- elim (lstas_inv_lift1 … HTU3 L2 … HXT3) -T3
- /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, drop_drop, ex2_intro/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat … Hd1) -Hd1 #Hd1
- elim (lstas_inv_appl1 … H2) -H2 #X #HT1U #H destruct
- elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -d0 -W1 -U1 -HV1 -IH4 -IH3 -IH2
- elim (IH1 … Hd1 … HT1U … HT12 … HL12) -IH1 -Hd1 -HT1U
- /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_appl, ex2_intro/
- | #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
- lapply (da_inv_bind … Hd1) -Hd1 #Hd1
- elim (lstas_inv_bind1 … HT1U) -HT1U #U #HT2U #H destruct
- elim (scpds_inv_abst1 … HTU1) -HTU1 #W0 #U0 #HW20 #_ #H destruct -U0 -d0
- elim (snv_fwd_da … HW2) #d0 #HW2d
- lapply (cprs_scpds_div … HW20 … HW2d … HVW1) -W0 #H21
- elim (snv_fwd_da … HV1) #d #HV1d
- elim (da_scpes_aux … IH4 IH3 IH2 … HW2d … HV1d … H21) /2 width=1 by fqup_fpbg/ #_ #H
- <minus_n_O #H0 destruct >(plus_minus_k_k d 1) in HV1d; // -H #HV1d
- lapply (scpes_cpr_lpr_aux … IH2 IH1 … H21 … HW23 … HV12 … HL12) -H21 /2 width=1 by fqup_fpbg/ #H32
- lapply (IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
- lapply (IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH2 … HW2d … HW23 … HL12) /2 width=1 by fqup_fpbg/ -HW2 -HW2d #HW3d
- lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV1d #HV2d
- elim (IH1 … Hd1 … HT2U … HT23 (L2.ⓛW3)) -HT2U /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
- elim (lsubsv_lstas_trans … o … HTU3 … Hd21 … (L2.ⓓⓝW3.V2)) -HTU3
- [ #U4 #HT3U4 #HU43 -IH1 -IH2 -IH3 -IH4 -d -d1 -HW3 -HV2 -HT2
- @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
- @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
- @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
- /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
- | -U3
- @(lsubsv_beta … (d-1)) /3 width=7 by fqup_fpbg/
- @shnv_cast [1,2: // ] #d0 #Hd0
- lapply (scpes_le_aux … IH4 IH3 IH2 IH1 … HW3d … HV2d … d0 … H32) -IH4 -IH3 -IH2 -IH1 -HW3d -HV2d -H32
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
- | -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbg, lpr_pair/
- ]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -d0 -W1 -HV1 -IH4 -IH3 -IH2
- elim (snv_inv_bind … HT1) -HT1 #_ #HT0
- lapply (da_inv_bind … Hd1) -Hd1 #Hd1
- elim (lstas_inv_bind1 … HT1U) -HT1U #U0 #HTU0 #H destruct
- elim (IH1 … Hd1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hd1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
- lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
- lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
- lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
- lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
- lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
- /4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #_ #HT1 #_ #_ -U1
- lapply (da_inv_flat … Hd1) -Hd1 #Hd1
- lapply (lstas_inv_cast1 … H2) -H2 #HTU1
- elim (cpr_inv_cast1 … H3) -H3
- [ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 … Hd1 … HTU1 … HT12 … HL12) -IH1 -Hd1 -HTU1 -HL12
- /3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
- | #HT1X3 elim (IH1 … Hd1 … HTU1 … HT1X3 … HL12) -IH1 -Hd1 -HTU1 -HL12
- /2 width=3 by fqup_fpbg, ex2_intro/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fsb_aaa.ma".
-include "basic_2/dynamic/snv_da_lpr.ma".
-include "basic_2/dynamic/snv_lstas.ma".
-include "basic_2/dynamic/snv_lstas_lpr.ma".
-include "basic_2/dynamic/snv_lpr.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Main preservation properties *********************************************)
-
-lemma snv_preserve: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
- ∧∧ IH_da_cpr_lpr h o G L T
- & IH_snv_cpr_lpr h o G L T
- & IH_snv_lstas h o G L T
- & IH_lstas_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_fwd_aaa … HT) -HT
-#A #HT @(aaa_ind_fpbg h o … HT) -G -L -T -A
-#G #L #T #A #_ #IH -A @and4_intro
-[ letin aux ≝ da_cpr_lpr_aux | letin aux ≝ snv_cpr_lpr_aux
-| letin aux ≝ snv_lstas_aux | letin aux ≝ lstas_cpr_lpr_aux
-]
-@(aux … G L T) // #G0 #L0 #T0 #H elim (IH … H) -IH -H //
-qed-.
-
-theorem da_cpr_lpr: ∀h,o,G,L,T. IH_da_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
-qed-.
-
-theorem snv_cpr_lpr: ∀h,o,G,L,T. IH_snv_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
-qed-.
-
-theorem snv_lstas: ∀h,o,G,L,T. IH_snv_lstas h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/
-qed-.
-
-theorem lstas_cpr_lpr: ∀h,o,G,L,T. IH_lstas_cpr_lpr h o G L T.
-#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/
-qed-.
-
-(* Advanced preservation properties *****************************************)
-
-lemma snv_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G #L1 #T1 #HT1 #T2 #H
-@(cprs_ind … H) -T2 /3 width=5 by snv_cpr_lpr/
-qed-.
-
-lemma da_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
-#h #o #G #L1 #T1 #HT1 #d #Hd #T2 #H
-@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/
-qed-.
-
-lemma lstas_cprs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #o #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ]
-#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
-elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (lstas_cpr_lpr … o … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2
-[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1
-/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
-qed-.
-
-lemma lstas_cpcs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, o] →
- ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, o] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 →
- ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #o #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1
-elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2
-lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fpbg_fpbs.ma".
-include "basic_2/equivalence/scpes_cpcs.ma".
-include "basic_2/equivalence/scpes_scpes.ma".
-include "basic_2/dynamic/snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Inductive premises for the preservation results **************************)
-
-definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
-
-definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
-
-definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-
-definition IH_snv_lstas: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
- ∀d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
- ∀U. ⦃G, L⦄ ⊢ T •*[h, d2] U → ⦃G, L⦄ ⊢ U ¡[h, o].
-
-(* Properties for the preservation results **********************************)
-
-fact snv_cprs_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o].
-#h #o #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
-@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
-qed-.
-
-fact da_cprs_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
-#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H
-@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
-qed-.
-
-fact da_scpds_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d1. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
- ∀T2,d2. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d2] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- d2 ≤ d1 ∧ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d1-d2.
-#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12
-lapply (da_mono … H … Hd1) -H #H destruct
-lapply (lstas_da_conf … HT1 … Hd1) #Hd12
-lapply (da_cprs_lpr_aux … IH2 IH1 … Hd12 … HT2 … HL12) -IH2 -IH1 -HT2 -HL12
-/3 width=8 by fpbg_fpbs_trans, lstas_fpbs, conj/
-qed-.
-
-fact da_scpes_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] →
- ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 →
- ∀d21,d22. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 →
- ∧∧ d21 ≤ d11 & d22 ≤ d12 & d11 - d21 = d12 - d22.
-#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2
-elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd11 … HT1 … L) -Hd11 -HT1 //
-elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd12 … HT2 … L) -Hd12 -HT2 //
-/3 width=7 by da_mono, and3_intro/
-qed-.
-
-fact lstas_cprs_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
-#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
-elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (IH1 … Hd21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
-[2: /3 width=12 by da_cprs_lpr_aux/
-|3: /3 width=10 by snv_cprs_lpr_aux/
-|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
-] -G0 -L0 -T0 -T1 -T -d1
-/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
-qed-.
-
-fact scpds_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀U1,d. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, o, d] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
-#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
-lapply (IH2 … H01 … HTd1 … HT12 … HL12) -L0 -T0 // -T1
-lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
-lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/
-qed-.
-
-fact scpes_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, o] →
- ∀d1,d2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ U1 •*⬌*[h, o, d1, d2] U2.
-#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
-elim (scpds_cpr_lpr_aux … IH2 IH1 … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
-elim (scpds_cpr_lpr_aux … IH2 IH1 … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
-elim (cprs_conf … H1 … H2) -T0
-/3 width=5 by scpds_div, scpds_cprs_trans/
-qed-.
-
-fact lstas_scpds_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G,L,T. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, o] →
- ∀d,d1. d1 ≤ d → ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀T1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 →
- ∀T2,d2. ⦃G, L⦄ ⊢ T •*➡*[h, o, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d2-d1, d1-d2] T2.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2
-lapply (da_mono … H … HTd) -H #H destruct
-lapply (lstas_da_conf … HT1 … HTd) #HTd1
-lapply (lstas_da_conf … HTX … HTd) #HXd
-lapply (da_cprs_lpr_aux … IH3 IH2 … HXd … HXT2 L ?)
-[1,2,3: /3 width=8 by fpbg_fpbs_trans, lstas_fpbs/ ] #HTd2
-elim (le_or_ge d1 d2) #Hd12 >(eq_minus_O … Hd12)
-[ elim (da_lstas … HTd2 0) #X2 #HTX2 #_ -IH4 -IH3 -IH2 -IH1 -H0 -HT -HTd -HXd
- /5 width=6 by lstas_scpds, scpds_div, cprs_strap1, lstas_cpr, lstas_conf_le, monotonic_le_minus_l, ex4_2_intro/
-| elim (da_lstas … HTd1 0) #X1 #HTX1 #_
- lapply (lstas_conf_le … HTX … HT1) // #HXT1 -HT1
- elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … HXd … HXT1 … HXT2 L) -IH3 -IH2 -IH1 -HXd -HXT1 -HXT2
- /4 width=8 by cpcs_scpes, cpcs_cpr_conf, fpbg_fpbs_trans, lstas_fpbs, lstas_cpr, monotonic_le_minus_l/
-]
-qed-.
-
-fact scpes_le_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] →
- ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] →
- ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 →
- ∀d21,d22,d. d21 + d ≤ d11 → d22 + d ≤ d12 →
- ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21+d, d22+d] T2.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20
-elim (da_lstas … Hd11 (d21+d)) #X1 #HTX1 #_
-elim (da_lstas … Hd12 (d22+d)) #X2 #HTX2 #_
-lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd11 … HTX1 … HT10) -HT10
-[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_k_k_commutative #HX1 ]
-lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd12 … HTX2 … HT20) -HT20
-[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_k_k_commutative #HX2 ]
-lapply (lstas_scpes_trans … Hd11 … HTX1 … HX1) [ // ] -Hd11 -HTX1 -HX1 -H1 #H1
-lapply (lstas_scpes_trans … Hd12 … HTX2 … HX2) [ // ] -Hd12 -HTX2 -HX2 -H2 #H2
-/2 width=4 by scpes_canc_dx/ (**) (* full auto fails *)
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma snv_cast_scpes: ∀h,o,G,L,U,T. ⦃G, L⦄ ⊢ U ¡[h, o] → ⦃G, L⦄ ⊢ T ¡[h, o] →
- ⦃G, L⦄ ⊢ U •*⬌*[h, o, 0, 1] T → ⦃G, L⦄ ⊢ ⓝU.T ¡[h, o].
-#h #o #G #L #U #T #HU #HT * /2 width=3 by snv_cast/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/tdpos.ma".
-include "basic_2/dynamic/cnv_cpm_tdeq.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-(* Forward lemmas with positive rt-transition for terms *********************)
-
-lemma cpm_tdeq_fwd_tdpos_sn (a) (h) (o) (n) (G):
- ∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 → 𝐏[h,o]⦃T1⦄ →
- ∧∧ T1 = T2 & 0 = n.
-#a #h #o #n #G #L #T1 #H0 #T2 #H1 #H2
-@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
-[ /2 width=1 by conj/
-| #L #s #_ #H1 #H
- elim (tdpos_inv_sort … H) -H #d #H2
- lapply (deg_mono … H2 H1) -H1 -H2 #H destruct
-| #p #I #L #V #T1 #_ #_ #T2 #_ #_ #IH #H
- elim (tdpos_inv_pair … H) -H #_ #HT1
- elim IH // -IH -HT1 #H1 #H2 destruct
- /2 width=1 by conj/
-| #m #_ #L #V #_ #W #_ #q #T1 #U1 #_ #_ #T2 #_ #_ #IH #H -a -m -q -G -L -W -U1
- elim (tdpos_inv_pair … H) -H #_ #HT1
- elim IH // -IH -HT1 #H1 #H2 destruct
- /2 width=1 by conj/
-| #L #U0 #U1 #T1 #_ #_ #U2 #_ #_ #_ #T2 #_ #_ #_ #IHU #IHT #H
- elim (tdpos_inv_pair … H) -H #HU1 #HT1
- elim IHU // -IHU -HU1 #H1 #H2 destruct
- elim IHT // -IHT -HT1 #H1 #H2 destruct
- /2 width=1 by conj/
-]
-qed-.
-
-lemma cpm_fwd_tdpos_sn (a) (h) (o) (n) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → 𝐏[h,o]⦃T1⦄ →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
- ∨∨ ∧∧ T1 = T2 & 0 = n | (T1 ≛[h,o] T2 → ⊥).
-#a #h #o #n #G #L #T1 #H1T1 #H2T1 #T2 #HT12
-elim (tdeq_dec h o T1 T2) #H
-/3 width=9 by cpm_tdeq_fwd_tdpos_sn, or_intror, or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/cnv_cpm_tdpos.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-(* Forward lemmas with positive rt-computation for terms ********************)
-
-lemma cpms_fwd_tdpos_sn (a) (h) (o) (n) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → 𝐏[h,o]⦃T1⦄ →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
- ∨∨ ∧∧ T1 = T2 & 0 = n
- | ∃∃n1,n2,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & (T1 ≛[h,o] T → ⊥) & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 & n1+n2=n.
-#a #h #o #n #G #L #T1 #H1T1 #H2T1 #T2 #H
-@(cpms_ind_dx … H) -n -T2
-[ /3 width=1 by or_introl, conj/
-| #n1 #n2 #T #T2 #HT1 * *
- [ #H1 #H2 #HT2 destruct -HT1
- elim (cpm_fwd_tdpos_sn … H1T1 H2T1 … HT2) -H1T1 -H2T1
- [ * #H1 #H2 destruct -HT2 /3 width=1 by or_introl, conj/
- | #HnT2 >commutative_plus in ⊢ (??%); /4 width=7 by ex4_3_intro, or_intror/
- ]
- | #m1 #m2 #T0 #H1T10 #H2T10 #HT0 #H #HT2 destruct -H1T1 -H2T1 -HT1
- >associative_plus in ⊢ (??%); /4 width=7 by cpms_step_dx, ex4_3_intro, or_intror/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( 𝐏 [ term 46 h, term 46 o ] ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'Positive $h $o $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/notation/relations/positive_3.ma".
-include "static_2/syntax/item_sd.ma".
-include "static_2/syntax/term.ma".
-
-(* DEGREE POSITIVITY ON TERMS ***********************************************)
-
-inductive tdpos (h) (o): predicate term ≝
-| tdpos_sort: ∀s,d. deg h o s (↑d) → tdpos h o (⋆s)
-| tdpos_lref: ∀i. tdpos h o (#i)
-| tdpos_gref: ∀l. tdpos h o (§l)
-| tdpos_pair: ∀I,V,T. tdpos h o V → tdpos h o T → tdpos h o (②{I}V.T)
-.
-
-interpretation
- "context-free degree positivity (term)"
- 'Positive h o T = (tdpos h o T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tdpos_inv_sort_aux (h) (o):
- ∀X. 𝐏[h,o]⦃X⦄ → ∀s. X = ⋆s → ∃d. deg h o s (↑d).
-#h #o #H *
-[ #s #d #Hsd #x #H destruct /2 width=2 by ex_intro/
-| #i #x #H destruct
-| #l #x #H destruct
-| #I #V #T #_ #_ #x #H destruct
-]
-qed-.
-
-lemma tdpos_inv_sort (h) (o): ∀s. 𝐏[h,o]⦃⋆s⦄ → ∃d. deg h o s (↑d).
-/2 width=3 by tdpos_inv_sort_aux/ qed-.
-
-fact tdpos_inv_pair_aux (h) (o): ∀X. 𝐏[h,o]⦃X⦄ → ∀I,V,T. X = ②{I}V.T →
- ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
-#h #o #H *
-[ #s #d #_ #Z #X1 #X2 #H destruct
-| #i #Z #X1 #X2 #H destruct
-| #l #Z #X1 #X2 #H destruct
-| #I #V #T #HV #HT #Z #X1 #X2 #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma tdpos_inv_pair (h) (o): ∀I,V,T. 𝐏[h,o]⦃②{I}V.T⦄ →
- ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
-/2 width=4 by tdpos_inv_pair_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-axiom tdpos_total (h): ∀T. ∃o. 𝐏[h,o]⦃T⦄.
<table name="basic_2_sum"/>
<subsection name="B">Stage "B"</subsection>
+<!--
<news class="alpha" date="Ongoing.">
Context-sensitive subject equivalence
for native type assignment.
</news>
-
- <subsection name="A2">Stage "A2": "Extending the Applicability Condition"</subsection>
-
+-->
+ <news class="alpha" date="2018 September 21.">
+ λδ-2A completed with
+ confluence of rt-computation and
+ preservation of validity for rt-computation.
+ </news>
<news class="alpha" date="2018 June 8.">
Behavioral component rt_computation completed.
</news>
Relocation with reference transforming maps (rtmap).
</news>
<news class="alpha" date="2015 October 9.">
- λδ version 2A2 is started.
+ λδ-2B is started.
</news>
- <subsection name="A1">Stage "A1": "Extending the Applicability Condition"</subsection>
+ <subsection name="A">Stage "A" </subsection>
<news class="delta" date="2015 August 27.">
- λδ version 2A1 appears too complex and is dismissed.
+ λδ-2A appears too complex and is dismissed.
</news>
<news class="gamma" date="2014 October 28.">
- λδ version 2A1 is released.
+ λδ version 2A is released.
</news>
<news class="beta" date="2014 September 9.">
Iterated static type assignment defined (more elegantly)
Confluence for context-free parallel reduction on terms.
</news>
<news class="alpha" date="2011 April 17.">
- λδ version 2 is started.
+ λδ-2A is started.
</news>
<section4 name="structure">Logical Structure of the Specification</section4>
*)
[ { "context-sensitive native validity" * } {
[ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ]
- [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_preserve_sub" + "cnv_preserve" * ]
+ [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_preserve_sub" + "cnv_preserve" * ]
}
]
}
projects / helm.git / commitdiff