L1 = K1.ⓑ{I}V1.
/3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
+(* Advanced eliminators *****************************************************)
+
+lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
+ R (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
+ R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpr_cprs_trans/ qed-.
+
(* Properties on context-sensitive parallel computation for terms ***********)
-lemma lprs_cpr_trans: ∀G. s_r_trans … (cpr G) (lprs G).
-/3 width=5 by s_r_trans_TC2, lpr_cprs_trans/ qed-.
+lemma lprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lprs G).
+/3 width=5 by s_r_trans_LTC2, lpr_cprs_trans/ qed-.
(* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma lprs_cprs_trans: ∀G. s_rs_trans … (cpr G) (lprs G).
-/3 width=5 by s_r_trans_TC1, lprs_cpr_trans/ qed-.
+(* Note: alternative proof /3 width=5 by s_r_trans_LTC1, lprs_cpr_trans/ *)
+lemma lprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lprs G).
+#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
+qed-.
lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #H elim H -L1
-[ #L1 #HL01
- elim (cprs_lpr_conf_dx … HT01 … HL01) -L0 /2 width=3/
-| #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
- elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2 #HT12
- elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3 #HT03
- elim (cprs_conf … HT2 … HT3) -T #T #HT2 #HT3
- lapply (cprs_trans … HT03 … HT3) -T3
- lapply (cprs_trans … HT12 … HT2) -T2 /2 width=3/
-]
+#G #L0 #T0 #T1 #HT01 #L1 #H @(lprs_ind … H) -L1 /2 width=3 by ex2_intro/
+#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
+elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2
+elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3
+elim (cprs_conf … HT2 … HT3) -T
+/3 width=5 by cprs_trans, ex2_intro/
qed-.
lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
/3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
+(* Note: this can be proved on its own using lprs_ind_dx *)
lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 #T #HT1
-lapply (lprs_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
+elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lprs_cprs_trans, ex2_intro/
qed-.
lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
-lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
-qed.
+/4 width=5 by lprs_cprs_trans, lprs_pair, cprs_bind/ qed.
(* Inversion lemmas on context-sensitive parallel computation for terms *****)
lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
U2 = ⓛ{a}W2.T2.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
-lapply (cprs_strap1 … HV10 … HV02) -V0
-lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5/
+lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro/
qed-.
lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
-#a #G #L #W1 #W2 #T1 #T2 #H
-elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1/
+#a #G #L #W1 #W2 #T1 #T2 #H elim (cprs_inv_abst1 … H) -H
+#W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
qed-.
(* Basic_1: was pr3_gen_abbr *)
U2 = ⓓ{a}V2.T2
) ∨
∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
#U0 #U2 #_ #HU02 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
elim (cpr_inv_abbr1 … HU02) -HU02 *
[ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) /2 width=1/ -HT02 #HT02
- lapply (cprs_strap1 … HV10 … HV02) -V0
- lapply (cprs_trans … HT10 … HT02) -T0 /3 width=5/
+ lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
| #T2 #HT02 #HUT2
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1/ -V0 #HT02
- lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/
+ lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by lprs_pair, cprs_trans, ex3_intro, or_intror/
]
-| #U1 #HTU1 #HU01
- elim (lift_total U2 0 1) #U #HU2
- lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0 /2 width=1/ /4 width=3/
+| #U1 #HTU1 #HU01 elim (lift_total U2 0 1)
+ #U #HU2 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
+ /4 width=3 by cprs_strap1, drop_drop, ex3_intro, or_intror/
]
qed-.
projects / helm.git / blobdiff