(* *)
(**************************************************************************)
-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_preserve_cpcs.ma".
include "basic_2/dynamic/nta.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
#H destruct
qed.
+(* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
+lemma nta_cpr_conf_lpr (a) (h) (G):
+ ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[a,h] U.
+#a #h #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
+/3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
+qed-.
+
+(* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
+lemma nta_cpr_conf (a) (h) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
+#a #h #G #L #T1 #U #H #T2 #HT12
+/3 width=6 by cnv_cpm_trans, cpm_cast/
+qed-.
+
+(* Note: this is the preservation property *)
+(* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
+lemma nta_cprs_conf (a) (h) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
+#a #h #G #L #T1 #U #H #T2 #HT12
+/3 width=6 by cnv_cpms_trans, cpms_cast/
+qed-.
+
+(* Basic_1: uses: ty3_cred_pr2 *)
+lemma nta_lpr_conf (a) (h) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
+#a #h #G #L1 #T #U #HTU #L2 #HL12
+/2 width=3 by cnv_lpr_trans/
+qed-.
+
+(* Basic_1: uses: ty3_cred_pr3 *)
+lemma nta_lprs_conf (a) (h) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
+#a #h #G #L1 #T #U #HTU #L2 #HL12
+/2 width=3 by cnv_lprs_trans/
+qed-.
+
(* Inversion lemmas based on preservation ***********************************)
lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
∃∃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].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
-elim (cnv_inv_appl … H1) * [ | #n ] #p #W #U #Hn #HV #HT #HVW #HTU
-[ lapply (cnv_cpms_trans … HT … HTU) #H
- elim (cnv_inv_bind … H) -H #_ #HU
- elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
- 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/
-qed-.
-(*
- (ltc_ind
- :∀A: Type \sub 0
- .(A→A→A)
- →∀B: Type \sub 0
- .relation3 A B B
- →∀Q_:∀x_3:A.∀x_2:B.∀x_1:B.ltc A __6 B __4 x_3 x_2 x_1→Prop
- .(∀a:A
- .∀b1:B
- .∀b2:B.∀x_5:__5 a b1 b2.Q_ a b1 b2 (ltc_rc A __8 B __6 a b1 b2 x_5))
- →(∀a1:A
- .∀a2:A
- .∀b1:B
- .∀b:B
- .∀b2:B
- .∀x_7:ltc A __10 B __8 a1 b1 b
- .∀x_6:ltc A __11 B __9 a2 b b2
- .Q_ a1 b1 b x_7
- →Q_ a2 b b2 x_6
- →Q_ (__14 a1 a2) b1 b2
- (ltc_trans A __14 B __12 a1 a2 b1 b b2 x_7 x_6))
- →∀x_3:A
- .∀x_2:B.∀x_1:B.∀x_4:ltc A __9 B __7 x_3 x_2 x_1.Q_ x_3 x_2 x_1 x_4)
+elim (cnv_inv_appl_true … H1) #p #W #U #HV #HT #HVW #HTU
+/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,T0,U0. ⦃G,L⦄ ⊢ V :*[h] W & ⦃G,L⦄ ⊢ ⓛ{p}W.T0 :*[h] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ T ➡*[h] ⓛ{p}W.T0 & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U0 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h]
+ ∨∨ ∃∃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) -H1 * [| #n ] #p #W0 #T0 #_ #HV #HT #HW0 #HT0
-lapply (cnv_cpms_trans … HT … HT0) #H
-elim (cnv_inv_bind … H) -H #_ #H1T0
-[ elim (cpms_inv_appl_sn_decompose … H) -H #U #HTU #HUX1
-
-
- [ #V0 #U0 #HV0 #HU0 #H destruct
- elim (cnv_cpms_conf … HT … HT0 … HU0)
- <minus_O_n <minus_n_O #X #H #HU0X
- elim (cpms_inv_abst_sn … H) -H #W1 #U1 #HW01 #HU01 #H destruct
- @or_introl
- @(ex5_4_intro … U1 … HT0 … HX2) -HX2
- [ /2 width=1 by cnv_cpms_nta/
- | @nta_bind_cnv /2 width=4 by cnv_cpms_trans/ /2 width=3 by cnv_cpms_nta/
- | @(cpcs_cprs_div … HX21) -HX21
- @(cprs_div … (ⓐV0.ⓛ{p}W1.U1))
- /3 width=1 by cpms_appl, cpms_appl_dx, cpms_bind/
- ]
-*)
+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 *)
lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] X2 →
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 *)
elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
[ #U1 #U0 #HTU1 #HTU0 #H destruct
- elim (cnv_cpms_conf … HT0 … HT0X … HTU0) -HT0 -HT0X -HTU0
- <minus_n_n #X0 #HX0 #HUX0
- lapply (cprs_trans … HT1X … HX0) -X #HT1X0
- /5 width=7 by cnv_cpms_nta, cpcs_cprs_div, cprs_div, cpms_cast, ex4_intro/
+ 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_cpms_conf … HT0 … HT0X … HTX0) -HT0 -HT0X -HTX0
- <minus_n_n #X1 #HX1 #HX01
elim (cnv_nta_sn … HT1) -HT1 #U1 #HTU1
- lapply (cprs_trans … HT1X … HX1) -X #HTX1
- lapply (cprs_trans … HX20 … HX01) -X0 #HX21
- /4 width=5 by cprs_div, cpms_eps, ex4_intro/
+ 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-.
+(* Basic_1: uses: ty3_gen_lift *)
+(* Note: "⦃G,L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G,L⦄ ⊢ X2 ➡*[h] U2" *)
+lemma nta_inv_lifts_sn (a) (h) (G):
+ ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 →
+ ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 →
+ ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[a,h] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
+#a #h #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
+elim (cnv_inv_cast … H) -H #U2 #HX2 #HT2 #HXU2 #HTU2
+lapply (cnv_inv_lifts … HT2 … HLK … HT12) -HT2 #HT1
+elim (cpms_inv_lifts_sn … HTU2 … HLK … HT12) -T2 -HLK #U1 #HU12 #HTU1
+/3 width=5 by cnv_cpms_nta, cpcs_cprs_sn, ex4_2_intro/
+qed-.
+
(* Forward lemmas based on preservation *************************************)
(* Basic_1: was: ty3_unique *)
#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-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: uses: ty3_sconv_pc3 *)
+lemma nta_cpcs_bi (a) (h) (G) (L):
+ ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[a,h] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[a,h] U2 →
+ ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
+#a #h #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
+/3 width=6 by nta_mono, nta_cprs_conf/
qed-.
projects / helm.git / blobdiff