(* Advanced properties ******************************************************)
-lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 →
- ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[c, h] T2.
-#c #h #G #K #V #V2 #i elim i -i
+lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 →
+ ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2.
+#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct
]
qed.
-lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 →
- ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[c+𝟘𝟙, h] T2.
-#c #h #G #K #V #V2 #i elim i -i
+lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 →
+ ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2.
+#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[c, h] T2 →
+lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 →
∨∨ T2 = #i ∧ c = 𝟘𝟘
- | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & c = cV
- | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & c = cV + 𝟘𝟙.
-#c #h #G #i elim i -i
+#Rt #c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
/4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
| #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1_drops: ∀c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[c, h] T2 →
+lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 →
∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
| ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
- | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV
- | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV + 𝟘𝟙.
-#c #h * #n #G #L #T2 #H
+#Rt #c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
/3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
| elim (cpg_inv_lref1_drops … H) -H *
(* Properties with generic slicing for local environments *******************)
(* Note: it should use drops_split_trans_pair2 *)
-lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G).
-#c #h #G #K #T generalize in match c; -c
+lemma cpg_lifts: ∀Rt. reflexive … Rt → ∀c,h,G. d_liftable2 (cpg Rt h c G).
+#Rt #HRt #c #h #G #K #T generalize in match c; -c
@(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
[ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
lapply (lifts_inv_sort1 … H1) -H1 #H destruct
/2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
| #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
elim (cpg_inv_lref1_drops … H2) -H2 *
- [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
+ [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
#cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
elim (lifts_split_trans … HXU2 f (𝐔❴⫯O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
/3 width=5 by cpg_zeta, ex2_intro/
]
-| #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+| * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (cpg_inv_flat1 … H2) -H2 *
- [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
- elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
- elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
- /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
- | #cT #HT12 #H1 #H2 destruct
- elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
- /3 width=3 by cpg_eps, ex2_intro/
- | #cV #HV12 #H1 #H2 destruct
- elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
- /3 width=3 by cpg_ee, ex2_intro/
- | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
- elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
- elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
- elim (IH … HY12 … HLK … HYZ1) -HY12 //
- elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
- /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
- | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
- elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
- elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
- elim (IH … HY12 … HLK … HYZ1) -HY12 //
- elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
- elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
- lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
- lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/
- /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ [ elim (cpg_inv_appl1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
+ lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
+ lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ ]
+ | elim (cpg_inv_cast1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
+ | #cT #HT12 #H destruct
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cV #HV12 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ ]
]
]
qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma cpg_inv_lifts1: ∀c,h,G. d_deliftable2_sn (cpg h c G).
-#c #h #G #L #U generalize in match c; -c
+lemma cpg_inv_lifts1: ∀Rt. reflexive … Rt → ∀c,h,G. d_deliftable2_sn (cpg Rt h c G).
+#Rt #HRt #c #h #G #L #U generalize in match c; -c
@(fqup_wf_ind_eq … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
[ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
lapply (lifts_inv_sort2 … H1) -H1 #H destruct
/2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
| #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
elim (cpg_inv_lref1_drops … H2) -H2 *
- [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
+ [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
#cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #T2 #HTU2 #HT12
elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/
]
-| #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+| * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (cpg_inv_flat1 … H2) -H2 *
- [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
- elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
- elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
- /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
- | #cU #HU12 #H1 #H2 destruct
- elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
- /3 width=3 by cpg_eps, ex2_intro/
- | #cW #HW12 #H1 #H2 destruct
- elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
- /3 width=3 by cpg_ee, ex2_intro/
- | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
- elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
- elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
- elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
- elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
- /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
- | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
- elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
- elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
- elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
- elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
- lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
- elim (lifts_split_trans … H ? (↑f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
- /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ [ elim (cpg_inv_appl1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
+ elim (lifts_split_trans … H ? (↑f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ ]
+ | elim (cpg_inv_cast1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
+ | #cU #HU12 #H destruct
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cW #HW12 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ ]
]
]
qed-.
projects / helm.git / blobdiff