∀T1,T2. T1 ≛ T2 → ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T2] L2 → ❪G,L1❫ ⊢ ⬈[T1] L2.
/2 width=5 by teqx_rex_div/ qed-.
-lemma cpx_teqx_conf_rex (G): R_confluent2_rex … (cpx G) cdeq (cpx G) cdeq.
-#G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #s1 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (teqx_inv_sort1 … H0) -H0 #s2 #H destruct
- /3 width=3 by teqx_sort, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(teqx_inv_lref1 … H0) -H0
- elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
- elim (reqx_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
- elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(teqx_inv_lref1 … H0) -H0
- elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
- elim (reqx_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
- elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (reqx_inv_bind … H2) -H2 #HL02 #H2
- lapply (reqx_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -T0
- /3 width=5 by cpx_bind, teqx_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (reqx_inv_flat … H2) -H2 #HL02 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=5 by cpx_flat, teqx_pair, ex2_intro/
-| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (reqx_inv_bind … H2) -H2 #HL02 #H2
- lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
- lapply (reqx_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
+lemma cpx_teqx_repl_reqx (G) (L0) (T0):
+ ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 → ∀T3. T1 ≛ T3 →
+ ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T2 ⬈ T3.
+#G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1
+[ * #x0 #G #L0 #X2 #HX2 #X3 #HX3 #L2 #_
+ [ elim (teqx_inv_sort1 … HX2) -HX2 #x2 #H destruct
+ elim (teqx_inv_sort1 … HX3) -HX3 #x3 #H destruct //
+ | lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ lapply (teqx_inv_lref1 … HX3) -HX3 #H destruct //
+ | lapply (teqx_inv_gref1 … HX2) -HX2 #H destruct
+ lapply (teqx_inv_gref1 … HX3) -HX3 #H destruct //
+ ]
+| #G #L0 #s0 #s1 #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_sort1 … HX2) -HX2 #s2 #H destruct
+ elim (teqx_inv_sort1 … HX3) -HX3 #s3 #H destruct //
+| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+ lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ elim (reqx_inv_zero_pair_sn … HL2) -HL2 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+ /3 width=3 by cpx_delta/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+ lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ elim (reqx_inv_lref_bind_sn … HL2) -HL2 #I2 #K2 #HK02 #H destruct
+ elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+ /3 width=3 by cpx_lref/
+| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+ elim (reqx_inv_bind … HL02) -HL02 #HV0 #HT0
+ lapply (reqx_bind_repl_dx … HT0 (BPair I V2) ?) -HT0
+ /2 width=1 by ext2_pair/ #HT0
+ /3 width=1 by cpx_bind/
+| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT0
+ /3 width=1 by cpx_flat/
+| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #U2 #HV02 #HU02 #H destruct
+ elim (reqx_inv_bind … HL02) -HL02 #HV0 #HU0
+ lapply (reqx_inv_lifts_bi … HU0 (Ⓣ) … HTU0) -HU0
+ [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #HT0
elim (teqx_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
- elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (reqx_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=3 by cpx_eps, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (reqx_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
- /3 width=3 by cpx_ee, ex2_intro/
-| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (teqx_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (reqx_inv_flat … H2) -H2 #H2LV0 #H2
- elim (reqx_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (reqx_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- /4 width=7 by cpx_beta, teqx_pair, ex2_intro/ (* note: 2 teqx_pair *)
-| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (teqx_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (reqx_inv_flat … H2) -H2 #H2LV0 #H2
- elim (reqx_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (reqx_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- elim (teqx_lifts_sn … HV1 … HVU1) -V1
- /4 width=9 by cpx_theta, teqx_pair, ex2_intro/ (* note: 2 teqx_pair *)
+ /3 width=3 by cpx_zeta/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #_ #HT02 #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT0
+ /3 width=1 by cpx_eps/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #_ #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT1
+ /3 width=1 by cpx_ee/
+| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #X #HV02 #HX #H destruct
+ elim (teqx_inv_pair1 … HX) -HX #W2 #T2 #HW02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #X #T3 #HX #HT13 #H destruct
+ elim (teqx_inv_pair1 … HX) -HX #W3 #V3 #HW13 #HV13 #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HL02
+ elim (reqx_inv_bind … HL02) -HL02 #HW0 #HT0
+ lapply (reqx_bind_repl_dx … HT0 (BPair Abst W2) ?) -HT0
+ /2 width=1 by ext2_pair/ #H2T0
+ /3 width=1 by cpx_beta/
+| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #X #HV02 #HX #H destruct
+ elim (teqx_inv_pair1 … HX) -HX #W2 #T2 #HW02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #W3 #X #HW13 #HX #H destruct
+ elim (teqx_inv_pair1 … HX) -HX #U3 #T3 #HU13 #HT13 #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HL02
+ elim (reqx_inv_bind … HL02) -HL02 #HW0 #HT0
+ lapply (reqx_bind_repl_dx … HT0 (BPair Abbr W2) ?) -HT0
+ /2 width=1 by ext2_pair/ #HT0
+ elim (teqx_inv_lifts_sn … HU13 … HVU1) -U1 #V3 #HVU3 #HV13
+ /3 width=3 by cpx_theta/
]
qed-.
lemma cpx_teqx_conf (G) (L):
- ∀T0:term. ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 →
- ∃∃T. T1 ≛ T & ❪G,L❫ ⊢ T2 ⬈ T.
-#G #L #T0 #T1 #HT01 #T2 #HT02
-elim (cpx_teqx_conf_rex … HT01 … HT02 L … L) -HT01 -HT02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
+ ∀T0:term. ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 → ❪G,L❫ ⊢ T2 ⬈ T1.
+/2 width=7 by cpx_teqx_repl_reqx/ qed-.
lemma teqx_cpx_trans (G) (L):
- ∀T2. ∀T0:term. T2 ≛ T0 → ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 →
- ∃∃T. ❪G,L❫ ⊢ T2 ⬈ T & T ≛ T1.
-#G #L #T2 #T0 #HT20 #T1 #HT01
-elim (cpx_teqx_conf … HT01 T2) -HT01 /3 width=3 by teqx_sym, ex2_intro/
+ ∀T2. ∀T0:term. T2 ≛ T0 → ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ❪G,L❫ ⊢ T2 ⬈ T1.
+/3 width=3 by cpx_teqx_conf, teqx_sym/
qed-.
+lemma teqx_cpx (G) (L):
+ ∀T1,T2:term. T1 ≛ T2 → ❪G,L❫ ⊢ T1 ⬈ T2.
+/2 width=3 by teqx_cpx_trans/ qed.
+
(* Basic_2A1: uses: cpx_lleq_conf *)
lemma cpx_reqx_conf (G):
- ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀L2. L0 ≛[T0] L2 →
- ∃∃T. ❪G,L2❫ ⊢ T0 ⬈ T & T1 ≛ T.
-#G #L0 #T0 #T1 #HT01 #L2 #HL02
-elim (cpx_teqx_conf_rex … HT01 T0 … L0 … HL02) -HT01 -HL02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
+ ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T0 ⬈ T1.
+/2 width=7 by cpx_teqx_repl_reqx/ qed-.
(* Basic_2A1: uses: lleq_cpx_trans *)
lemma reqx_cpx_trans (G):
- ∀L2,L0,T0. L2 ≛[T0] L0 → ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 →
- ∃∃T. ❪G,L2❫ ⊢ T0 ⬈ T & T ≛ T1.
-#G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_reqx_conf … HT01 L2) -HT01
-/3 width=3 by reqx_sym, teqx_sym, ex2_intro/
+ ∀L2,L0,T0. L2 ≛[T0] L0 → ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ❪G,L2❫ ⊢ T0 ⬈ T1.
+/3 width=3 by cpx_reqx_conf, reqx_sym/
qed-.
-lemma rpx_reqx_conf (G) (T): confluent2 … (rpx G T) (reqx T).
-/3 width=6 by rpx_fsge_comp, reqx_fsge_comp, cpx_teqx_conf_rex, rex_conf/ qed-.
+lemma rpx_reqx_conf (G) (T):
+ confluent1 … (rpx G T) (reqx T).
+/3 width=7 by reqx_fsge_comp, cpx_teqx_repl_reqx, rex_conf1/ qed-.
-lemma reqx_rpx_trans (G) (T):
- ∀L2,K2. ❪G,L2❫ ⊢ ⬈[T] K2 → ∀L1. L1 ≛[T] L2 →
- ∃∃K1. ❪G,L1❫ ⊢ ⬈[T] K1 & K1 ≛[T] K2.
-#G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (rpx_reqx_conf … HLK2 L1)
-/3 width=3 by reqx_sym, ex2_intro/
-qed-.
+lemma reqx_rpx_trans (G) (T) (L):
+ ∀L1. L1 ≛[T] L → ∀L2. ❪G,L❫ ⊢ ⬈[T] L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
+/3 width=3 by rpx_reqx_conf, reqx_sym/ qed-.
+
+lemma reqx_rpx (G) (T):
+ ∀L1,L2. L1 ≛[T] L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
+/2 width=3 by reqx_rpx_trans/ qed.
projects / helm.git / blobdiff