(* *)
(**************************************************************************)
-include "basic_2/rt_equivalence/cpcs_cprs.ma".
-include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/rt_equivalence/cpcs_cpcs.ma".
+include "basic_2/dynamic/cnv_cpcs.ma".
include "basic_2/dynamic/nta.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
/4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
qed-.
+(* Basic_1: was: ty3_typecheck *)
+lemma nta_typecheck (a) (h) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] T0.
+/3 width=1 by cnv_cast, cnv_nta_sn/ qed-.
+
+(* Basic_1: was: ty3_correct *)
+(* Basic_2A1: was: ntaa_fwd_correct *)
+lemma nta_fwd_correct (a) (h) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ U :[a,h] T0.
+/3 width=2 by nta_fwd_cnv_dx, cnv_nta_sn/ qed-.
+
lemma nta_pure_cnv (h) (G) (L):
∀T,U. ⦃G,L⦄ ⊢ T :*[h] U →
∀V. ⦃G,L⦄ ⊢ ⓐV.U !*[h] → ⦃G,L⦄ ⊢ ⓐV.T :*[h] ⓐV.U.
(* Inversion lemmas based on preservation ***********************************)
+lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] X2 →
+ ∃∃W,U. ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![a,h].
+#a #h #G #Y #X #X2 #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
+elim (cpms_inv_delta_sn … H2) -H2 *
+[ #_ #H destruct
+| #W #HVW #HWX1
+ /3 width=5 by cnv_cpms_nta, cpcs_cprs_sn, ex4_2_intro/
+]
+qed-.
+
+lemma nta_inv_lref_sn (a) (h) (G) (L):
+ ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[a,h] X2 →
+ ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[a,h] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![a,h] & L = K.ⓘ{I}.
+#a #h #G #L #X2 #i #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
+elim (cpms_inv_lref_sn … H2) -H2 *
+[ #_ #H destruct
+| #X #HX #HX1
+ /3 width=9 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro/
+]
+qed-.
+
+lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L):
+ ∀X2, i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
+ ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]
+ | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
+#a #h #G #L #X2 #i #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
+elim (cpms_inv_lref1_drops … H2) -H2 *
+[ #_ #H destruct
+| #Y #X #W #H #HVW #HUX1
+ lapply (drops_mono … H … HLK) -H #H destruct
+ /4 width=8 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro, or_introl/
+| #n #Y #X #U #H #HVU #HUX1 #H0 destruct
+ lapply (drops_mono … H … HLK) -H #H destruct
+ elim (lifts_total V (𝐔❴↑i❵)) #W #HVW
+ lapply (cpms_lifts_bi … HVU (Ⓣ) … L … HVW … HUX1) -U
+ [ /2 width=2 by drops_isuni_fwd_drop2/ ] #HWX1
+ /4 width=9 by cprs_div, ex5_3_intro, or_intror/
+]
+qed-.
+
lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] X2 →
∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![a,h].
lapply (cpms_step_dx … HTU 1 (ⓛ{p}W.U0) ?) -HTU [ /2 width=1 by cpm_bind/ ] #HTU
| lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Hn #H destruct
]
-lapply (cpms_appl_dx … V V … HTU) [1,3: // ] #HVTU
-elim (cnv_cpms_conf … H1 … H2 … HVTU) -H1 -H2 -HVTU <minus_n_n #X0 #HX0 #HUX0
-@ex4_3_intro [6,13: |*: /2 width=5 by cnv_cpms_nta/ ]
-/3 width=5 by cprs_div, cprs_trans/
+/5 width=11 by cnv_cpms_nta, cnv_cpms_conf_eq, cpcs_cprs_div, cpms_appl_dx, ex4_3_intro/
+qed-.
+
+(* Basic_2A1: uses: nta_fwd_pure1 *)
+lemma nta_inv_pure_sn_cnv (h) (G) (L) (X2):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :*[h] X2 →
+ ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :*[h] W & ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h]
+ | ∃∃U. ⦃G,L⦄ ⊢ T :*[h] U & ⦃G,L⦄ ⊢ ⓐV.U !*[h] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h].
+#h #G #L #X2 #V #T #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H
+elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
+[ lapply (cnv_cpms_trans … HT … HT0) #H0
+ elim (cnv_inv_bind … H0) -H0 #_ #HU
+ elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
+ lapply (cpms_step_dx … HT0 1 (ⓛ{p}W0.U0) ?) -HT0 [ /2 width=1 by cpm_bind/ ] #HT0
+ lapply (cpms_appl_dx … V V … HT0) [ // ] #HTU0
+ lapply (cnv_cpms_conf_eq … H1 … HTU0 … H) -H1 -H -HTU0 #HU0X1
+ /4 width=8 by cnv_cpms_nta, cpcs_cprs_div, ex4_3_intro, or_introl/
+| elim (cnv_cpms_fwd_appl_sn_decompose … H1 … H) -H1 -H #X0 #_ #H #HX01
+ elim (cpms_inv_plus … 1 n … HT0) #U #HTU #HUT0
+ lapply (cnv_cpms_trans … HT … HTU) #HU
+ lapply (cnv_cpms_conf_eq … HT … HTU … H) -H #HUX0
+ @or_intror @(ex4_intro … U … HX2) -HX2
+ [ /2 width=1 by cnv_cpms_nta/
+ | /4 width=7 by cnv_appl, lt_to_le/
+ | /4 width=3 by cpcs_trans, cpcs_cprs_div, cpcs_flat/
+ ]
+]
qed-.
(* Basic_2A1: uses: nta_inv_cast1 *)
elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
[ #U0 #T0 #HU0 #HT0 #H destruct -HU -HU0
- elim (cnv_cpms_conf … HT … HTX … HT0) -HT -HTX -HT0
- <minus_n_n #T1 #HXT1 #HT01
- @and3_intro // @(cprs_div … T1) /3 width=4 by cprs_trans, cpms_eps/ (**) (* full auto too slow *)
+ lapply (cnv_cpms_conf_eq … HT … HTX … HT0) -HT -HT0 -HTX #HXT0
+ lapply (cprs_step_dx … HX20 T0 ?) -HX20 [ /2 width=1 by cpm_eps/ ] #HX20
| #HTX0 -HU
- elim (cnv_cpms_conf … HT … HTX … HTX0) -HT -HTX -HTX0
- <minus_n_n #T1 #HXT1 #HXT01
- @and3_intro // @(cprs_div … T1) /2 width=3 by cprs_trans/ (**) (* full auto too slow *)
+ lapply (cnv_cpms_conf_eq … HT … HTX … HTX0) -HT -HTX -HTX0 #HX0
| #m #HUX0 #H destruct -HT -HTX
- elim (cnv_cpms_conf … HU … HUX … HUX0) -HU -HUX0
- <minus_n_n #U1 #HXU1 #HXU01
- @and3_intro // @(cprs_div … U1) /2 width=3 by cprs_trans/ (**) (* full auto too slow *)
+ lapply (cnv_cpms_conf_eq … HU … HUX … HUX0) -HU -HUX0 #HX0
+]
+/4 width=3 by cpcs_cprs_div, cpcs_cprs_step_sn, and3_intro/
+qed-.
+
+(* Basic_1: uses: ty3_gen_cast *)
+lemma nta_inv_cast_sn_old (a) (h) (G) (L) (X2):
+ ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] X2 →
+ ∃∃T2. ⦃G,L⦄ ⊢ T0 :[a,h] T1 & ⦃G,L⦄ ⊢ T1 :[a,h] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
+#a #h #G #L #X2 #T0 #T1 #H
+elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
+elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
+elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
+[ #U1 #U0 #HTU1 #HTU0 #H destruct
+ lapply (cnv_cpms_conf_eq … HT0 … HT0X … HTU0) -HT0 -HT0X -HTU0 #HXU0
+ /5 width=5 by cnv_cpms_nta, cpcs_cprs_div, cpcs_cprs_step_sn, cpcs_flat, ex4_intro/
+| #HTX0
+ elim (cnv_nta_sn … HT1) -HT1 #U1 #HTU1
+ lapply (cnv_cpms_conf_eq … HT0 … HT0X … HTX0) -HT0 -HT0X -HTX0 #HX0
+ lapply (cprs_step_sn … (ⓝU1.T1) … HT1X) -HT1X [ /2 width=1 by cpm_eps/ ] #HT1X
+ /4 width=5 by cpcs_cprs_div, cpcs_cprs_step_sn, ex4_intro/
+| #n #HT1X0 #H destruct -X -HT0
+ elim (cnv_nta_sn … HT1) -HT1 #U1 #HTU1
+ /4 width=5 by cprs_div, cpms_eps, ex4_intro/
]
qed-.
#a #h #G #L #T #U1 #H1 #U2 #H2
elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
-elim (cnv_cpms_conf … HT … HTX1 … HTX2) -T <minus_n_n #X #HX1 #HX2
-/3 width=5 by cprs_div, cprs_trans/
+lapply (cnv_cpms_conf_eq … HT … HTX1 … HTX2) -T #HX12
+/3 width=3 by cpcs_cprs_div, cpcs_cprs_step_sn/
qed-.
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