lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
Q 0 T2 →
- (â\88\80n1,n2,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
- â\88\80n,T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 → Q n T1.
-#h #G #L #T2 #R @ltc_ind_sn_refl //
+ (â\88\80n1,n2,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n1,h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
+ â\88\80n,T1. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2 → Q n T1.
+#h #G #L #T2 #Q @ltc_ind_sn_refl //
qed-.
lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
Q 0 T1 →
- (â\88\80n1,n2,T,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1, h] T â\86\92 Q n1 T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2 → Q (n1+n2) T2) →
- â\88\80n,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 → Q n T2.
-#h #G #L #T1 #R @ltc_ind_dx_refl //
+ (â\88\80n1,n2,T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1,h] T â\86\92 Q n1 T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[n2,h] T2 → Q (n1+n2) T2) →
+ â\88\80n,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2 → Q n T2.
+#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic properties *********************************************************)
(* Basic_1: includes: pr1_pr0 *)
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): â\88\80n,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2.
+lemma cpm_cpms (h) (G) (L): â\88\80n,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2.
/2 width=1 by ltc_rc/ qed.
-lemma cpms_step_sn (h) (G) (L): â\88\80n1,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T →
- â\88\80n2,T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[n2, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1+n2, h] T2.
+lemma cpms_step_sn (h) (G) (L): â\88\80n1,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n1,h] T →
+ â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n2,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1+n2,h] T2.
/2 width=3 by ltc_sn/ qed-.
-lemma cpms_step_dx (h) (G) (L): â\88\80n1,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1, h] T →
- â\88\80n2,T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1+n2, h] T2.
+lemma cpms_step_dx (h) (G) (L): â\88\80n1,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1,h] T →
+ â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[n2,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1+n2,h] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
lemma cpms_bind_dx (n) (h) (G) (L):
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â\88\80p. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡*[n, h] â\93\91{p,I}V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80I,T1,T2. â\9dªG,L.â\93\91[I]V1â\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡*[n,h] â\93\91[p,I]V2.T2.
#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
+lemma cpms_appl_dx (n) (h) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+ ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
+#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
+qed.
+
+lemma cpms_zeta (n) (h) (G) (L):
+ ∀T1,T. ⇧*[1] T ≘ T1 →
+ ∀V,T2. ❪G,L❫ ⊢ T ➡*[n,h] T2 → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[n,h] T2.
+#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
+/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
+qed.
+
+(* Basic_2A1: uses: cprs_zeta *)
+lemma cpms_zeta_dx (n) (h) (G) (L):
+ ∀T2,T. ⇧*[1] T2 ≘ T →
+ ∀V,T1. ❪G,L.ⓓV❫ ⊢ T1 ➡*[n,h] T → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[n,h] T2.
+#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
+qed.
+
+(* Basic_2A1: uses: cprs_eps *)
+lemma cpms_eps (n) (h) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀V. ❪G,L❫ ⊢ ⓝV.T1 ➡*[n,h] T2.
+#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
+qed.
+
+lemma cpms_ee (n) (h) (G) (L):
+ ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[n,h] U2 →
+ ∀T. ❪G,L❫ ⊢ ⓝU1.T ➡*[↑n,h] U2.
+#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
+[ /3 width=1 by cpm_cpms, cpm_ee/
+| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
+ /3 width=3 by cpms_step_sn, cpm_ee/
+]
+qed.
+
+(* Basic_2A1: uses: cprs_beta_dx *)
+lemma cpms_beta_dx (n) (h) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
+#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
+qed.
+
+(* Basic_2A1: uses: cprs_theta_dx *)
+lemma cpms_theta_dx (n) (h) (G) (L):
+ ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V →
+ ∀V2. ⇧*[1] V ≘ V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
+#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
+qed.
+
(* Basic properties with r-transition ***************************************)
(* Basic_1: was: pr3_refl *)
lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
/2 width=1 by cpm_cpms/ qed.
-(* Basic_2A1: removed theorems 4:
+(* Advanced properties ******************************************************)
+
+lemma cpms_sort (h) (G) (L) (n):
+ ∀s. ❪G,L❫ ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
+#h #G #L #n elim n -n [ // ]
+#n #IH #s <plus_SO_dx
+/3 width=3 by cpms_step_dx, cpm_sort/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ❪G,L❫ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
+#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
+#n1 #n2 #X #X2 #_ #IH #HX2 destruct
+elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
+qed-.
+
+lemma cpms_inv_lref1_ctop (n) (h) (G):
+ ∀X2,i. ❪G,⋆❫ ⊢ #i ➡*[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 #i #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_lref1_ctop … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_zero1_unit (n) (h) (I) (K) (G):
+ ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡*[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+#n #h #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_zero1_unit … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_gref1 (n) (h) (G) (L):
+ ∀X2,l. ❪G,L❫ ⊢ §l ➡*[n,h] X2 → ∧∧ X2 = §l & n = 0.
+#n #h #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_gref1 … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_cast1 (h) (n) (G) (L):
+ ∀W1,T1,X2. ❪G,L❫ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃W2,T2. ❪G,L❫ ⊢ W1 ➡*[n,h] W2 & ❪G,L❫ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+ | ❪G,L❫ ⊢ T1 ➡*[n,h] X2
+ | ∃∃m. ❪G,L❫ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+| #n1 #n2 #X #X2 #_ * [ * || * ]
+ [ #W #T #HW1 #HT1 #H #HX2 destruct
+ elim (cpm_inv_cast1 … HX2) -HX2 [ * || * ]
+ [ #W2 #T2 #HW2 #HT2 #H destruct
+ /4 width=5 by cpms_step_dx, ex3_2_intro, or3_intro0/
+ | #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+ | #m #HX2 #H destruct <plus_n_Sm
+ /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+ ]
+ | #HX #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+ | #m #HX #H #HX2 destruct
+ /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+ ]
+]
+qed-.
+
+(* Basic_2A1: removed theorems 5:
sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs
+ scpds_inv_lstas_eq
*)
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