(* *)
(**************************************************************************)
-include "basic_2/static/sta_sta.ma".
include "basic_2/unfold/lstas_lift.ma".
(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
(* Main properties **********************************************************)
-theorem lstas_trans: ∀h,G,L. ltransitive … (lstas h G L).
-/2 width=3 by lstar_ltransitive/ qed-.
+theorem lstas_trans: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*[h, d1+d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ #G #L #d1 #k #X #d2 #H >(lstas_inv_sort1 … H) -X
+ <iter_plus /2 width=1 by lstas_sort/
+| #G #L #K #V1 #V #U #i #d1 #HLK #_ #HVU #IHV1 #U2 #d2 #HU2
+ lapply (drop_fwd_drop2 … HLK) #H0
+ elim (lstas_inv_lift1 … HU2 … H0 … HVU)
+ /3 width=6 by lstas_ldef/
+| //
+| #G #L #K #W1 #W #U #i #d1 #HLK #_ #HWU #IHW1 #U2 #d2 #HU2
+ lapply (drop_fwd_drop2 … HLK) #H0
+ elim (lstas_inv_lift1 … HU2 … H0 … HWU)
+ /3 width=6 by lstas_succ/
+| #a #I #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
+ elim (lstas_inv_bind1 … H) -H #T2 #HT2 #H destruct
+ /3 width=1 by lstas_bind/
+| #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
+ elim (lstas_inv_appl1 … H) -H #T2 #HT2 #H destruct
+ /3 width=1 by lstas_appl/
+| /3 width=1 by lstas_cast/
+]
+qed-.
-theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h G L l).
-/3 width=7 by sta_mono, lstar_singlevalued/ qed-.
+(* Note: apparently this was missing in basic_1 *)
+theorem lstas_mono: ∀h,G,L,d. singlevalued … (lstas h d G L).
+#h #G #L #d #T #T1 #H elim H -G -L -T -T1 -d
+[ #G #L #d #k #X #H >(lstas_inv_sort1 … H) -X //
+| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H
+ elim (lstas_inv_lref1 … H) -H *
+ #K0 #V0 #W0 [3: #d0 ] #HLK0
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+ #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct
+ /2 width=5 by lift_mono/
+| #G #L #K #W #W1 #i #HLK #_ #_ #X #H
+ elim (lstas_inv_lref1_O … H) -H *
+ #K0 #V0 #W0 #HLK0
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct //
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H
+ elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+ #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct
+ /2 width=5 by lift_mono/
+| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H
+ elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #V #T #U1 #d #_ #IHTU1 #X #H
+ elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H
+ lapply (lstas_inv_cast1 … H) -H /2 width=1 by/
+]
+qed-.
-theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
- ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
- ⦃G, L⦄ ⊢ U1 •*[h, l2-l1] U2.
-#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
->(plus_minus_m_m … Hl12) in ⊢ (%→?); -Hl12 >commutative_plus #H
-elim (lstas_split … H) -H #U #HTU
->(lstas_mono … HTU … HTU1) -T //
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was just: sty0_correct *)
+lemma lstas_correct: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+ ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2
+ elim (IHV1 d2) -IHV1 #V2
+ elim (lift_total V2 0 (i+1))
+ lapply (drop_fwd_drop2 … HLK) -HLK
+ /3 width=11 by ex_intro, lstas_lift/
+| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2
+ @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+ #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2
+ lapply (lstas_trans … HW1 … HW2) -W
+ elim (lift_total W2 0 (i+1))
+ /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2
+ elim (IHW1 d2) -IHW1 #W2
+ elim (lift_total W2 0 (i+1))
+ lapply (drop_fwd_drop2 … HLK) -HLK
+ /3 width=11 by ex_intro, lstas_lift/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /2 width=2 by ex_intro/
+]
qed-.
-(* Advanced properties ******************************************************)
+(* more main properties *****************************************************)
-lemma lstas_sta_conf_pos: ∀h,G,L,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 →
- ∀U2,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U2 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2.
-#h #G #L #T #U1 #HTU1 #U2 #l #HTU2
-lapply (lstas_conf_le … T U1 1 … HTU2) -HTU2 /2 width=1 by sta_lstas/
+theorem lstas_conf_le: ∀h,G,L,T,U1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] U1 →
+ ∀U2,d2. d1 ≤ d2 → ⦃G, L⦄ ⊢ T •*[h, d2] U2 →
+ ⦃G, L⦄ ⊢ U1 •*[h, d2-d1] U2.
+#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12
+>(plus_minus_m_m … Hd12) in ⊢ (%→?); -Hd12 >commutative_plus #H
+elim (lstas_split … H) -H #U #HTU
+>(lstas_mono … HTU … HTU1) -T //
qed-.
-lemma lstas_strip_pos: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
- ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
- ∃∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2 & ⦃G, L⦄ ⊢ U1 •*[h, l+1] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12
-elim (lstas_fwd_correct … HTU1 … HT12)
-lapply (lstas_sta_conf_pos … HTU1 … HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/
+theorem lstas_conf: ∀h,G,L,T0,T1,d1. ⦃G, L⦄ ⊢ T0 •*[h, d1] T1 →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T0 •*[h, d2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T2 •*[h, d1] T.
+#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02
+elim (lstas_lstas … HT01 (d1+d2)) #T #HT0
+lapply (lstas_conf_le … HT01 … HT0) // -HT01 <minus_plus_m_m_commutative
+lapply (lstas_conf_le … HT02 … HT0) // -T0 <minus_plus_m_m
+/2 width=3 by ex2_intro/
qed-.
projects / helm.git / blobdiff