(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_3.ma".
include "ground_2/steps/rtc_ist_shift.ma".
include "ground_2/steps/rtc_ist_plus.ma".
include "ground_2/steps/rtc_ist_max.ma".
/3 width=3 by cpg_ee, ist_succ, ex2_intro/
qed.
-(* Basic properties *********************************************************)
-
lemma cpt_refl (h) (G) (L): reflexive … (cpt h G L 0).
/3 width=3 by cpg_refl, ex2_intro/ qed.
#h #G #L * //
#n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/
qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpt_inv_atom_sn (h) (n) (J) (G) (L):
+ ∀X2. ⦃G,L⦄ ⊢ ⓪{J} ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = ⓪{J} & n = 0
+ | ∃∃s. X2 = ⋆(⫯[h]s) & J = Sort s & n =1
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,n] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓓV1 & J = LRef 0
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,m] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬆[h,n] T & ⇧*[1] T ≘ X2 & L = K.ⓘ{I} & J = LRef (↑i).
+#h #n #J #G #L #X2 * #c #Hc #H
+elim (cpg_inv_atom1 … H) -H *
+[ #H1 #H2 destruct /3 width=1 by or5_intro0, conj/
+| #s #H1 #H2 #H3 destruct /3 width=3 by or5_intro1, ex3_intro/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
+ /4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
+ elim (ist_inv_plus_SO_dx … H3) -H3 [| // ] #m #Hc #H destruct
+ /4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
+| #I #K #V2 #i #HV2 #HVT2 #H1 #H2 destruct
+ /4 width=8 by or5_intro4, ex4_4_intro, ex2_intro/
+]
+qed-.
+
+lemma cpt_inv_sort_sn (h) (n) (G) (L) (s):
+ ∀X2. ⦃G,L⦄ ⊢ ⋆s ⬆[h,n] X2 →
+ ∧∧ X2 = ⋆(((next h)^n) s) & n ≤ 1.
+#h #n #G #L #s #X2 * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
+/2 width=1 by conj/
+qed-.
+
+lemma cpt_inv_zero_sn (h) (n) (G) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #0 ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #0 & n = 0
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,n] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓓV1
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,m] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓛV1 & n = ↑m.
+#h #n #G #L #X2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ elim (ist_inv_plus_SO_dx … H2) -H2 // #m #Hc #H destruct
+ /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
+]
+qed-.
+
+lemma cpt_inv_zero_sn_unit (h) (n) (I) (K) (G):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ⬆[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+#h #n #I #G #K #X2 #H
+elim (cpt_inv_zero_sn … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpt_inv_lref_sn (h) (n) (G) (L) (i):
+ ∀X2. ⦃G,L⦄ ⊢ #↑i ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #(↑i) & n = 0
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬆[h,n] T & ⇧*[1] T ≘ X2 & L = K.ⓘ{I}.
+#h #n #G #L #i #X2 * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
+| #I #K #V2 #HV2 #HVT2 #H destruct
+ /4 width=6 by ex3_3_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma cpt_inv_lref_sn_ctop (n) (h) (G) (i):
+ ∀X2. ⦃G,⋆⦄ ⊢ #i ⬆[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G * [| #i ] #X2 #H
+[ elim (cpt_inv_zero_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpt_inv_lref_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cpt_inv_gref_sn (h) (n) (G) (L) (l):
+ ∀X2. ⦃G,L⦄ ⊢ §l ⬆[h,n] X2 → ∧∧ X2 = §l & n = 0.
+#h #n #G #L #l #X2 * #c #Hc #H elim (cpg_inv_gref1 … H) -H
+#H1 #H2 destruct /2 width=1 by conj/
+qed-.
+
+lemma cpt_inv_bind_sn (h) (n) (p) (I) (G) (L) (V1) (T1):
+ ∀X2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬆[h,n] X2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬆[h,0] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬆[h,n] T2
+ & X2 = ⓑ{p,I}V2.T2.
+#h #n #p #I #G #L #V1 #T1 #X2 * #c #Hc #H
+elim (cpg_inv_bind1 … H) -H *
+[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (ist_inv_max … H2) -H2 #nV #nT #HcV #HcT #H destruct
+ elim (ist_inv_shift … HcV) -HcV #HcV #H destruct
+ /3 width=5 by ex3_2_intro, ex2_intro/
+| #cT #T2 #_ #_ #_ #_ #H destruct
+ elim (ist_inv_plus_10_dx … H)
+]
+qed-.
+
+lemma cpt_inv_appl_sn (h) (n) (G) (L) (V1) (T1):
+ ∀X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬆[h,n] X2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬆[h,0] V2 & ⦃G,L⦄ ⊢ T1 ⬆[h,n] T2 & X2 = ⓐV2.T2.
+#h #n #G #L #V1 #T1 #X2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
+[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (ist_inv_max … H2) -H2 #nV #nT #HcV #HcT #H destruct
+ elim (ist_inv_shift … HcV) -HcV #HcV #H destruct
+ /3 width=5 by ex3_2_intro, ex2_intro/
+| #cV #cW #cU #p #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #_ #H destruct
+ elim (ist_inv_plus_10_dx … H)
+| #cV #cW #cU #p #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #_ #_ #H destruct
+ elim (ist_inv_plus_10_dx … H)
+]
+qed-.
+
+lemma cpt_inv_cast_sn (h) (n) (G) (L) (V1) (T1):
+ ∀X2. ⦃G,L⦄ ⊢ ⓝV1.T1 ⬆[h,n] X2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬆[h,n] V2 & ⦃G,L⦄ ⊢ T1 ⬆[h,n] T2 & X2 = ⓝV2.T2
+ | ∃∃m. ⦃G,L⦄ ⊢ V1 ⬆[h,m] X2 & n = ↑m.
+#h #n #G #L #V1 #T1 #X2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
+[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
+ elim (ist_inv_max … H2) -H2 #nV #nT #HcV #HcT #H destruct
+ <idempotent_max
+ /4 width=5 by or_introl, ex3_2_intro, ex2_intro/
+| #cT #_ #H destruct
+ elim (ist_inv_plus_10_dx … H)
+| #cV #H12 #H destruct
+ elim (ist_inv_plus_SO_dx … H) -H [| // ] #m #Hm #H destruct
+ /4 width=3 by ex2_intro, or_intror/
+]
+qed-.
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