(* Advanced properties ******************************************************)
lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
- â\86\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V1 â\86\92 K â\8a¢ V1 [0, |L| - i - 1] â\89«* W1 →
- â\86\91[0, i + 1] W1 â\89¡ W2 â\86\92 L â\8a¢ #i â\87\92 W2.
+ â\87©[0, i] L â\89¡ K. â\93\93V1 â\86\92 K â\8a¢ V1 [0, |L| - i - 1] â\96¶* W1 →
+ â\87§[0, i + 1] W1 â\89¡ W2 â\86\92 L â\8a¢ #i â\9e¡ W2.
#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
-@ex2_1_intro [2: // | skip | @tpss_subst /2 width=6/ ] (**) (* /4 width=6/ is too slow *)
+lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+@ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_1: was: pr2_gen_lref *)
-lemma cpr_inv_lref1: â\88\80L,T2,i. L â\8a¢ #i â\87\92 T2 →
+lemma cpr_inv_lref1: â\88\80L,T2,i. L â\8a¢ #i â\9e¡ T2 →
T2 = #i ∨
- â\88\83â\88\83K,V1,T1. â\86\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V1 &
- K â\8a¢ V1 [0, |L| - i - 1] â\89«* T1 &
- â\86\91[0, i + 1] T1 ≡ T2 &
+ â\88\83â\88\83K,V1,T1. â\87©[0, i] L â\89¡ K. â\93\93V1 &
+ K â\8a¢ V1 [0, |L| - i - 1] â\96¶* T1 &
+ â\87§[0, i + 1] T1 ≡ T2 &
i < |L|.
#L #T2 #i * #X #H
>(tpr_inv_atom1 … H) -H #H
-elim (tpss_inv_lref1 … H) -H /2/
+elim (tpss_inv_lref1 … H) -H /2 width=1/
* /3 width=6/
-qed.
+qed-.
(* Basic_1: was: pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀V1,T1,U2. 𝕔{Abst} V1. T1 ⇒ U2 →
- ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2.
-/2/ qed.
+lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
+#L #V1 #T1 #Y * #X #H1 #H2 #I #W
+elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+lapply (tpss_lsubs_conf … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Basic_1: was pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+ U0 = ⓛW. T1 &
+ U2 = ⓓV2. T2
+ | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓW1. T1 &
+ U2 = ⓓW2. ⓐV. T2.
+#L #V1 #U0 #Y * #X #H1 #H2
+elim (tpr_inv_appl1 … H1) -H1 *
+[ #V #U #HV1 #HU0 #H destruct
+ elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
+| #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
+| #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
+ elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
+ elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒[T1] →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
(* Relocation properties ****************************************************)
(* Basic_1: was: pr2_lift *)
-lemma cpr_lift: â\88\80L,K,d,e. â\86\93[d, e] L ≡ K →
- â\88\80T1,U1. â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\88\80T2,U2. â\86\91[d, e] T2 ≡ U2 →
- K â\8a¢ T1 â\87\92 T2 â\86\92 L â\8a¢ U1 â\87\92 U2.
+lemma cpr_lift: â\88\80L,K,d,e. â\87©[d, e] L ≡ K →
+ â\88\80T1,U1. â\87§[d, e] T1 â\89¡ U1 â\86\92 â\88\80T2,U2. â\87§[d, e] T2 ≡ U2 →
+ K â\8a¢ T1 â\9e¡ T2 â\86\92 L â\8a¢ U1 â\9e¡ U2.
#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
elim (lift_total T d e) #U #HTU
lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
elim (lt_or_ge (|K|) d) #HKd
-[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 T HLK [ /2/ | /3/ ]
-| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 T HLK // /3/
+[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
+| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
]
qed.
(* Basic_1: was: pr2_gen_lift *)
-lemma cpr_inv_lift: â\88\80L,K,d,e. â\86\93[d, e] L ≡ K →
- â\88\80T1,U1. â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\88\80U2. L â\8a¢ U1 â\87\92 U2 →
- â\88\83â\88\83T2. â\86\91[d, e] T2 â\89¡ U2 & K â\8a¢ T1 â\87\92 T2.
+lemma cpr_inv_lift: â\88\80L,K,d,e. â\87©[d, e] L ≡ K →
+ â\88\80T1,U1. â\87§[d, e] T1 â\89¡ U1 â\86\92 â\88\80U2. L â\8a¢ U1 â\9e¡ U2 →
+ â\88\83â\88\83T2. â\87§[d, e] T2 â\89¡ U2 & K â\8a¢ T1 â\9e¡ T2.
#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
elim (lt_or_ge (|L|) d) #HLd
-[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U HLK [ /5/ | /2/ ]
+[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
| elim (lt_or_ge (|L|) (d + e)) #HLde
- [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U HLK // [ /5/ | /2/ ]
- | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U HLK // /5/
+ [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=2/ ]
+ | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/
]
]
qed.
projects / helm.git / blobdiff