(* Basic_1: was: pr3_t *)
(* Basic_1: includes: pr1_t *)
theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
-#G #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
+normalize /2 width=3 by trans_TC/ qed-.
(* Basic_1: was: pr3_confluence *)
(* Basic_1: includes: pr1_confluence *)
theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
-#G #L @TC_confluent2 /2 width=3 by cpr_conf/ qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
+normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -V1 /2 width=1/
+#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=5 by cprs_trans, cprs_bind_dx/
qed.
(* Basic_1: was: pr3_flat *)
theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
-#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /2 width=1/
+#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
qed.
theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1/
-#W #W2 #_ #HW2 #IHW1
-@(cprs_trans … IHW1) -IHW1 /3 width=1/
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
+#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
+@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
qed.
theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1/
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /3 width=1/
+@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
qed.
theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
-#W #W2 #_ #HW2 #IHW1
-@(cprs_trans … IHW1) /2 width=1/
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cprs_ind … H) -W2
+/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
qed.
theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
-#V1 #V0 #HV10 #_ #IHV0
-@(cprs_trans … IHV0) /2 width=1/
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1
+/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
| ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
-#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 [ /3 width=5/ ]
+#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
#U #U2 #_ #HU2 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
elim (cpr_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
| #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
- lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2 /2 width=1/ #HT2
- @or3_intro1 @(ex2_3_intro … HT10) -HT10 /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
+ lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+ /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_abst, ex2_3_intro, or3_intro1/
| #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
+ /5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/
]
-| /4 width=9/
-| /4 width=11/
+| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
]
qed-.
(* Basic_1: was just: pr3_pr2_pr2_t *)
(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma lpr_cpr_trans: ∀G. s_r_trans … (cpr G) (lpr G).
+lemma lpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lpr G).
#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3/
+[ /2 width=3 by/
| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (lpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
- lapply (IHV02 … HK12) -K2 #HV02
- lapply (cprs_strap2 … HV10 … HV02) -V0 /2 width=6/
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓑ{I}V1) ?) -IHT12 /2 width=1/ /3 width=1/
-|4,6: /3 width=1/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- lapply (IHT1 (L1.ⓓV2) ?) -IHT1 /2 width=1/ /2 width=3/
-| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓛW1) ?) -IHT12 /2 width=1/ /3 width=1/
-| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓓW1) ?) -IHT12 /2 width=1/ /3 width=3/
+ /4 width=6 by cprs_strap2, cprs_delta/
+|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
+|4,6: /3 width=1 by cprs_flat, cprs_eps/
+|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
]
qed-.
lemma cpr_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 (lpr_cpr_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
-qed.
+/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
(* Advanced properties ******************************************************)
(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lpr_cprs_trans: ∀G. s_rs_trans … (cpr G) (lpr G).
-/3 width=5 by s_r_trans_TC1, lpr_cpr_trans/ qed-.
+lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G).
+#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+qed-.
(* Basic_1: was: pr3_strip *)
(* Basic_1: includes: pr1_strip *)
lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
-#G #L @TC_strip1 /2 width=3 by cpr_conf/ qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
+normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
lemma cprs_lpr_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 #H elim H -T1
-[ #T1 #HT01 #L1 #HL01
- elim (lpr_cpr_conf_dx … HT01 … HL01) -L0 /3 width=3/
-| #T #T1 #_ #HT1 #IHT0 #L1 #HL01
- elim (IHT0 … HL01) #T2 #HT2 #HT02
- elim (lpr_cpr_conf_dx … HT1 … HL01) -L0 #T3 #HT3 #HT13
- elim (cprs_strip … HT2 … HT3) -T #T #HT2 #HT3
- lapply (cprs_strap2 … HT13 … HT3) -T3
- lapply (cprs_strap1 … HT02 … HT2) -T2 /2 width=3/
-]
+#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/
+#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01)
+#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0
+#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T
+/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
qed-.
lemma cprs_lpr_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 (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1
-lapply (lpr_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
+#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lpr_cprs_trans, ex2_intro/
qed-.
lemma cprs_bind2_dx: ∀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 (lpr_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
-qed.
+/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.