(* Extended validy (basic_2B) vs. restricted validity (basic_1A) ************)
(* Note: extended validity of a closure, height of cnv_appl > 1 *)
-lemma cnv_extended (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 !*[h].
+lemma cnv_extended (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 !*[h].
#h #p #G #L #s
@(cnv_appl … 2 p … (⋆s) … (⋆s))
[ //
qed.
(* Note: restricted validity of the η-expanded closure, height of cnv_appl = 1 **)
-lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h].
+lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h].
#h #p #G #L #s
@(cnv_appl … 1 p … (⋆s) … (ⓐ#0.#2))
[ /2 width=1 by ylt_inj/
qed-.
lemma cpr_inv_Delta_sn (h) (G) (L) (s):
- ∀X. ⦃G, L⦄ ⊢ Delta s ➡[h] X → Delta s = X.
+ ∀X. ⦃G,L⦄ ⊢ Delta s ➡[h] X → Delta s = X.
#h #G #L #s #X #H
elim (cpm_inv_abst1 … H) -H #X1 #X2 #H1 #H2 #H destruct
lapply (cpr_inv_sort1 … H1) -H1 #H destruct
(* Main properties **********************************************************)
-theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega1 s ➡[h] Omega2 s.
+theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega1 s ➡[h] Omega2 s.
/2 width=1 by cpm_beta/ qed.
-theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega3 s.
+theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega3 s.
/5 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, Delta_lifts/ qed.
-theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega3 s ➡[h] Omega1 s.
+theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega3 s ➡[h] Omega1 s.
/4 width=3 by cpm_zeta, Delta_lifts, lifts_flat/ qed.
(* Main inversion properties ************************************************)
theorem cpr_inv_Omega1_sn (h) (G) (L) (s):
- ∀X. ⦃G, L⦄ ⊢ Omega1 s ➡[h] X →
+ ∀X. ⦃G,L⦄ ⊢ Omega1 s ➡[h] X →
∨∨ Omega1 s = X | Omega2 s = X.
#h #G #L #s #X #H elim (cpm_inv_appl1 … H) -H *
[ #W2 #T2 #HW2 #HT2 #H destruct
]
qed-.
-theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥.
+theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥.
#h #G #L #s #H elim (cpm_inv_bind1 … H) -H *
[ #W #T #_ #_ whd in ⊢ (??%?→?); #H destruct
| #X #H #_ #_ #_
⬆*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
/5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
-lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s.
+lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s.
/2 width=1 by cpm_beta/ qed.
-lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s.
+lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s.
/6 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, ApplDelta_lifts/ qed.
-lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s.
+lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s.
/4 width=3 by cpm_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
-lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s.
+lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s.
/5 width=4 by cpxs_strap1, cpm_fwd_cpx/ qed.
lemma fqup_ApplOmega_41 (G) (L) (s0) (s): ⦃G,L,ApplOmega4 s0 s⦄ ⊐+ ⦃G,L,ApplOmega1 s0 s⦄.
d2 ≤ d1 → ↑[d2,h2]↑[d1,h1]T = ↑[h2+d1,h1]↑[d2,h2]T.
/3 width=1 by flifts_comp, basic_swap/ qed-.
(*
-lemma flift_join: ∀e1,e2,T. ⬆[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T.
+lemma flift_join: ∀e1,e2,T. ⬆[e1,e2] ↑[0,e1] T ≡ ↑[0,e1 + e2] T.
#e1 #e2 #T
lapply (flift_lift T 0 (e1+e2)) #H
elim (lift_split … H e1 e1) -H // #U #H
(* Properties with relocation ***********************************************)
-lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 →
+lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 →
∀T,L,l. ⬇*[l] L ≘ K.ⓓV1 →
- ∀gv,lv. ⦃G, L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T.
+ ∀gv,lv. ⦃G,L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T.
#h #G #K #V1 #V2 #HV #T elim T -T * //
[ #i #L #l #HKL #gv #lv
>mf_lref >mf_lref
(* DENOTATIONAL EQUIVALENCE ************************************************)
definition deq (M): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, lv].
+ λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,lv].
interpretation "denotational equivalence (model)"
'RingEq M G L T1 T2 = (deq M G L T1 T2).
(* Forward lemmas with context-sensitive parallel reduction for terms *******)
lemma cpr_fwd_deq (h) (M): is_model M → is_extensional M →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ≗{M} T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ≗{M} T2.
#h #M #H1M #H2M #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ /2 width=2 by deq_refl/
| #G #K #V1 #V2 #W2 #_ #IH #HVW2 #gv #v #H
inductive li (M) (gv): relation2 lenv (evaluation M) ≝
| li_atom: ∀lv. li M gv (⋆) lv
-| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv, lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv)
+| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv,lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv)
| li_abst: ∀lv,d,L,W. li M gv L lv → li M gv (L.ⓛW) (⫯[0←d]lv)
| li_unit: ∀lv,d,I,L. li M gv L lv → li M gv (L.ⓤ{I}) (⫯[0←d]lv)
| li_veq : ∀lv1,lv2,L. li M gv L lv1 → lv1 ≗ lv2 → li M gv L lv2
(* Note: application: compatibility *)
mp: compatible_3 … (ap M) (sq M) (sq M) (sq M);
(* Note: interpretation: sort *)
- ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv, lv] ≗ sv M s;
+ ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv,lv] ≗ sv M s;
(* Note: interpretation: local reference *)
- ml: ∀gv,lv,i. ⟦#i⟧{M}[gv, lv] ≗ lv i;
+ ml: ∀gv,lv,i. ⟦#i⟧{M}[gv,lv] ≗ lv i;
(* Note: interpretation: global reference *)
- mg: ∀gv,lv,l. ⟦§l⟧{M}[gv, lv] ≗ gv l;
+ mg: ∀gv,lv,l. ⟦§l⟧{M}[gv,lv] ≗ gv l;
(* Note: interpretation: intensional binder *)
- mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1, lv1] ≗ ⟦W⟧{M}[gv2, lv2] →
- (∀d. ⟦T⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2, ⫯[0←d]lv2]) →
- ⟦ⓛ{p}W.T⟧[gv1, lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2, lv2];
+ mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1,lv1] ≗ ⟦W⟧{M}[gv2,lv2] →
+ (∀d. ⟦T⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2,⫯[0←d]lv2]) →
+ ⟦ⓛ{p}W.T⟧[gv1,lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2,lv2];
(* Note: interpretation: abbreviation *)
- md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] ⊕[p] ⟦T⟧[gv, ⫯[0←⟦V⟧[gv, lv]]lv];
+ md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] ⊕[p] ⟦T⟧[gv,⫯[0←⟦V⟧[gv,lv]]lv];
(* Note: interpretation: application *)
- ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] @ ⟦T⟧[gv, lv];
+ ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] @ ⟦T⟧[gv,lv];
(* Note: interpretation: ζ-equivalence *)
mz: ∀d1,d2. d1 ⊕{M}[Ⓣ] d2 ≗ d2;
(* Note: interpretation: ϵ-equivalence *)
- me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, lv];
+ me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv,lv] ≗ ⟦T⟧[gv,lv];
(* Note: interpretation: β-requivalence *)
- mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv, lv] ≗ d ⊕[p] ⟦T⟧[gv, ⫯[0←d]lv];
+ mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv,lv] ≗ d ⊕[p] ⟦T⟧[gv,⫯[0←d]lv];
(* Note: interpretation: θ-requivalence *)
mh: ∀p,d1,d2,d3. d1 @ (d2 ⊕{M}[p] d3) ≗ d2 ⊕[p] (d1 @ d3)
}.
record is_extensional (M): Prop ≝ {
(* Note: interpretation: extensional abstraction *)
- mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1, lv1] ≗ ⟦W2⟧{M}[gv2, lv2] →
- (∀d. ⟦T1⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2, ⫯[0←d]lv2]) →
- ⟦ⓛ{p}W1.T1⟧[gv1, lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2, lv2]
+ mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1,lv1] ≗ ⟦W2⟧{M}[gv2,lv2] →
+ (∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2]) →
+ ⟦ⓛ{p}W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2,lv2]
}.
record is_injective (M): Prop ≝ {
definition tm_dd ≝ term.
-definition tm_sq (h) (T1) (T2) ≝ ⦃⋆, ⋆⦄ ⊢ T1 ⬌*[h] T2.
+definition tm_sq (h) (T1) (T2) ≝ ⦃⋆,⋆⦄ ⊢ T1 ⬌*[h] T2.
definition tm_sv (s) ≝ ⋆s.
lemma ti_comp (M): is_model M →
∀T,gv1,gv2. gv1 ≗ gv2 → ∀lv1,lv2. lv1 ≗ lv2 →
- ⟦T⟧[gv1, lv1] ≗{M} ⟦T⟧[gv2, lv2].
+ ⟦T⟧[gv1,lv1] ≗{M} ⟦T⟧[gv2,lv2].
#M #HM #T elim T -T * [||| #p * | * ]
[ /4 width=5 by seq_trans, seq_sym, ms/
| /4 width=5 by seq_sym, ml, mq/
fact lifts_fwd_vpush_aux (M): is_model M → is_extensional M →
∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀m. 𝐁❴m,1❵ = f →
- ∀gv,lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[m←d]lv].
+ ∀gv,lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[m←d]lv].
#M #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
[ #f #s #m #Hf #gv #lv #d
@(mq … H1M) [4,5: /3 width=2 by seq_sym, ms/ |1,2: skip ]
lemma lifts_SO_fwd_vpush (M) (gv): is_model M → is_extensional M →
∀T1,T2. ⬆*[1] T1 ≘ T2 →
- ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[0←d]lv].
+ ∀lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[0←d]lv].
/2 width=3 by lifts_fwd_vpush_aux/ qed-.
inductive vpushs (M) (gv) (lv): relation2 lenv (evaluation M) ≝
| vpushs_atom: vpushs M gv lv (⋆) lv
-| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv, v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v)
+| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv,v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v)
| vpushs_abst: ∀v,d,K,V. vpushs M gv lv K v → vpushs M gv lv (K.ⓛV) (⫯[0←d]v)
| vpushs_unit: ∀v,d,I,K. vpushs M gv lv K v → vpushs M gv lv (K.ⓤ{I}) (⫯[0←d]v)
| vpushs_repl: ∀v1,v2,L. vpushs M gv lv L v1 → v1 ≗ v2 → vpushs M gv lv L v2
fact vpushs_inv_abbr_aux (M) (gv) (lv): is_model M →
∀y,L. L ⨁{M}[gv] lv ≘ y →
∀K,V. K.ⓓV = L →
- ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y.
+ ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y.
#M #gv #lv #HM #y #L #H elim H -y -L
[ #Y #X #H destruct
| #v #d #K #V #Hv #Hd #_ #Y #X #H destruct
lemma vpushs_inv_abbr (M) (gv) (lv): is_model M →
∀y,K,V. K.ⓓV ⨁{M}[gv] lv ≘ y →
- ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y.
+ ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y.
/2 width=3 by vpushs_inv_abbr_aux/ qed-.
fact vpushs_inv_abst_aux (M) (gv) (lv): is_model M →
lemma vpushs_fold (M): is_model M → is_extensional M →
∀L,T1,T2,gv,lv.
- (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v]) →
- ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv].
+ (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v]) →
+ ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv].
#M #H1M #H2M #L elim L -L [| #K * [| * ]]
[ #T1 #T2 #gv #lv #H12
>fold_atom >fold_atom
(* Inversion lemmas with fold for restricted closures ***********************)
lemma vpushs_inv_fold (M): is_model M → is_injective M →
- ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv] →
- ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v].
+ ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv] →
+ ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v].
#M #H1M #H2M #L elim L -L [| #K * [| * ]]
[ #T1 #T2 #gv #lv
>fold_atom >fold_atom #H12 #v #H
| cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
| cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
| cnv_appl: ∀n,p,G,L,V,W0,T,U0. yinj n < a → cnv a h G L V → cnv a h G L T →
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
| cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
- ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T)
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv a h G L (ⓝU.T)
.
interpretation "context-sensitive native validity (term)"
(* Basic inversion lemmas ***************************************************)
-fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → X = #0 →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
#a #h #G #L #X * -G -L -X
[ #G #L #s #H destruct
| #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
]
qed-.
-lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G,L⦄ ⊢ #0 ![a,h] →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
/2 width=3 by cnv_inv_zero_aux/ qed-.
-fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀i. X = #(↑i) →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
#a #h #G #L #X * -G -L -X
[ #G #L #s #j #H destruct
| #I #G #K #V #_ #j #H destruct
]
qed-.
-lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![a,h] →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
/2 width=3 by cnv_inv_lref_aux/ qed-.
-fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
+fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀l. X = §l → ⊥.
#a #h #G #L #X * -G -L -X
[ #G #L #s #l #H destruct
| #I #G #K #V #_ #l #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_gref *)
-lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
+lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![a,h] → ⊥.
/2 width=8 by cnv_inv_gref_aux/ qed-.
-fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
+fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] →
∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+ ∧∧ ⦃G,L⦄ ⊢ V ![a,h]
+ & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
#a #h #G #L #X * -G -L -X
[ #G #L #s #q #Z #X1 #X2 #H destruct
| #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_bind *)
-lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![a,h]
+ & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
/2 width=4 by cnv_inv_bind_aux/ qed-.
-fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
- ∃∃n,p,W0,U0. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀V,T. X = ⓐV.T →
+ ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
#a #h #G #L #X * -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
- ∃∃n,p,W0,U0. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+ ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
/2 width=3 by cnv_inv_appl_aux/ qed-.
-fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀U,T. X = ⓝU.T →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
#a #h #G #L #X * -G -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
/2 width=3 by cnv_inv_cast_aux/ qed-.
(* Basic forward lemmas *****************************************************)
lemma cnv_fwd_flat (a) (h) (I) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓕ{I}V.T ![a,h] →
- ∧∧ ⦃G, L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ T ![a,h].
+ ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![a,h] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h].
#a #h * #G #L #V #T #H
[ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
| elim (cnv_inv_cast … H) #U #HV #HT #_ #_
(* Forward lemmas on atomic arity assignment for terms **********************)
(* Basic_2A1: uses: snv_fwd_aaa *)
-lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
+lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A.
#a #h #G #L #T #H elim H -G -L -T
[ /2 width=2 by aaa_sort, ex_intro/
| #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/
(* Forward lemmas with t_bound rt_transition for terms **********************)
lemma cnv_fwd_cpm_SO (a) (h) (G) (L):
- ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
#a #h #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by aaa_cpm_SO/
(* Forward lemmas with t_bound rt_computation for terms *********************)
lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L):
- ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
#a #h #n #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by cpms_total_aaa/
(* Advanced inversion lemmas ************************************************)
lemma cnv_inv_appl_pred (a) (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![yinj a, h] →
- ∃∃p,W0,U0. ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[↓a, h] ⓛ{p}W0.U0.
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![yinj a,h] →
+ ∃∃p,W0,U0. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[↓a,h] ⓛ{p}W0.U0.
#a #h #G #L #V #T #H
elim (cnv_inv_appl … H) -H #n #p #W #U #Ha #HV #HT #HVW #HTU
lapply (ylt_inv_inj … Ha) -Ha #Ha
lemma cnv_appl_cpes (a) (h) (G) (L):
∀n. yinj n < a →
- ∀V. ⦃G, L⦄ ⊢ V ![a, h] → ∀T. ⦃G, L⦄ ⊢ T ![a, h] →
- ∀W. ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W →
- ∀p,U. ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U → ⦃G, L⦄ ⊢ ⓐV.T ![a, h].
+ ∀V. ⦃G,L⦄ ⊢ V ![a,h] → ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
+ ∀W. ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W →
+ ∀p,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![a,h].
#a #h #G #L #n #Hn #V #HV #T #HT #W *
/4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
qed.
lemma cnv_cast_cpes (a) (h) (G) (L):
- ∀U. ⦃G, L⦄ ⊢ U ![a, h] →
- ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G, L⦄ ⊢ ⓝU.T ![a, h].
+ ∀U. ⦃G,L⦄ ⊢ U ![a,h] →
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G,L⦄ ⊢ ⓝU.T ![a,h].
#a #h #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/
qed.
(* Inversion lemmas with t-bound rt-equivalence for terms *******************)
lemma cnv_inv_appl_cpes (a) (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
- ∃∃n,p,W,U. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U.
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+ ∃∃n,p,W,U. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U.
#a #h #G #L #V #T #H
elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
/3 width=7 by cpms_div, ex5_4_intro/
qed-.
lemma cnv_inv_appl_pred_cpes (a) (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![yinj a, h] →
- ∃∃p,W,U. ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[↓a, h] ⓛ{p}W.U.
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![yinj a,h] →
+ ∃∃p,W,U. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[↓a,h] ⓛ{p}W.U.
#a #h #G #L #V #T #H
elim (cnv_inv_appl_pred … H) -H #p #W #U #HV #HT #HVW #HTU
/3 width=7 by cpms_div, ex4_3_intro/
qed-.
lemma cnv_inv_cast_cpes (a) (h) (G) (L):
- ∀U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
- ∧∧ ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T.
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
+ ∧∧ ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T.
#a #h #G #L #U #T #H
elim (cnv_inv_cast … H) -H
/3 width=3 by cpms_div, and3_intro/
(* Sub diamond propery with t-bound rt-transition for terms *****************)
fact cnv_cpm_conf_lpr_atom_atom_aux (h) (G) (L1) (L2) (I):
- ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
/2 width=3 by ex2_intro/ qed-.
fact cnv_cpm_conf_lpr_atom_ess_aux (h) (G) (L1) (L2) (s):
fact cnv_cpm_conf_lpr_aux (a) (h):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
#a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
(* Inversion lemmas with restricted rt-transition for terms *****************)
lemma cnv_cpr_tdeq_fwd_refl (a) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 →
- ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 →
+ ⦃G,L⦄ ⊢ T1 ![a,h] → T1 = T2.
#a #h #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
qed-.
lemma cpm_tdeq_inv_bind_sn (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
lemma cpm_tdeq_inv_appl_sn (a) (h) (n) (G) (L):
∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![a,h] →
∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
- ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ V ➡*[1,h] W & ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
- & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
+ ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
+ & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
#a #h #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
qed-.
lemma cpm_tdeq_inv_cast_sn (a) (h) (n) (G) (L):
- ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
+ ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
- & ⦃G, L⦄ ⊢ U1 ![a,h] & ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
- & ⦃G, L⦄ ⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
+ & ⦃G,L⦄ ⊢ U1 ![a,h] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
+ & ⦃G,L⦄ ⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
#a #h #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
qed-.
lemma cpm_tdeq_inv_bind_dx (a) (h) (n) (p) (I) (G) (L):
- ∀X. ⦃G, L⦄ ⊢ X ![a,h] →
- ∀V,T2. ⦃G, L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
- ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
+ ∀X. ⦃G,L⦄ ⊢ X ![a,h] →
+ ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
+ ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
#a #h #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
) →
(∀m. yinj m < a →
- ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G, L⦄ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
+ ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
) →
(∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ⦃G, L⦄ ⊢ U1 ![a,h] → ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ![a,h] → ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ![a,h] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ![a,h] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
) →
∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
(* Advanced properties with restricted rt-transition for terms **************)
lemma cpm_tdeq_free (a) (h) (n) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- ∀F,K. ⦃F, K⦄ ⊢ T1 ➡[n,h] T2.
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
#a #h #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
(* Advanced inversion lemmas with restricted rt-transition for terms ********)
lemma cpm_tdeq_inv_bind_sn_void (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_tdeq_free/
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
definition IH_cnv_cpm_tdeq_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
- ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
(* Diamond propery with restricted rt-transition for terms ******************)
fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (G0) (L1) (L2) (I):
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0, L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0,L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
#h #G0 #L1 #L2 #I
/2 width=5 by ex4_intro/
qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
definition IH_cnv_cpm_tdeq_cpm_trans (a) (h): relation3 genv lenv term ≝
- λG,L,T1. ⦃G, L⦄ ⊢ T1 ![a,h] →
+ λG,L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛ T →
∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡[n1,h] T2 & T0 ≛ T2.
(* Transitive properties restricted rt-transition for terms *****************)
fact cnv_cpm_tdeq_cpm_trans_sub (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0, L0, T0⦄ >[h] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
(∀G,L,T. ⦃G0,L0,T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_cpm_trans a h G L T) →
∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans a h G L T1.
#a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
qed-.
fact cnv_cpm_tdeq_cpm_trans_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0, L0, T0⦄ >[h] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
IH_cnv_cpm_tdeq_cpm_trans a h G0 L0 T0.
#a #h #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #IH0
fact cnv_cpm_trans_lpr_aux (a) (h):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr a h G1 L1 T1.
#a #h #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
[ #s #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #_ destruct -IH2 -IH1 -H1
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X02 … L0 ?) // #HX2
elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … H1X01 … H1X02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
-cut (⦃G0,L0,T0⦄ >[h] ⦃G0,L0,X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
+cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ20 … L0 ?) // #HZ0
elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX01 … HX02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
-cut (⦃G0,L0,T0⦄ >[h] ⦃G0,L0,X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
+cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ10 … L0 ?) // #HZ0
elim (IH1 … HXT1 … HXZ10 … L1 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT1 -HXZ10 #Z1 #HTZ1 #HZ01
(* Properties with restricted rt-computation for terms **********************)
fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (G) (L) (T1):
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛ T2 → ⊥) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → (T1 ≛ T2 → ⊥) →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- ∃∃n1,n2,T0. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G, L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
+ ∃∃n1,n2,T0. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
#a #h #n #G #L #T1 #T2 #H
@(cpms_ind_sn … H) -n -T1
[ #_ #H2T2 elim H2T2 -H2T2 //
qed-.
fact cpms_tdeq_ind_sn (a) (h) (G) (L) (T2) (Q:relation2 …):
- (⦃G, L⦄ ⊢ T2 ![a,h] → Q 0 T2) →
- (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛ T → ⦃G, L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G, L⦄ ⊢ T ![a,h] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛ T2 →
+ (⦃G,L⦄ ⊢ T2 ![a,h] → Q 0 T2) →
+ (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T ![a,h] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T2 →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
(∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
Q n T1.
(* Advanced dproperties *****************************************************)
(* Basic_2A1: uses: snv_lref *)
-lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G, K⦄ ⊢ V ![a, h] →
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ #i ![a, h].
+lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G,K⦄ ⊢ V ![a,h] →
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ #i ![a,h].
#a #h #G #I #K #V #i elim i -i
[ #L #HV #H
lapply (drops_fwd_isid … H ?) -H // #H destruct
(* Basic_2A1: uses: snv_inv_lref *)
lemma cnv_inv_lref_drops (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ![a, h].
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ![a,h].
#a #h #G #i elim i -i
[ #L #H
elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct
qed-.
lemma cnv_inv_lref_pair (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ![a, h].
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ![a,h].
#a #h #G #i #L #H #I #K #V #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX
lapply (drops_mono … HLY … HLK) -L #H destruct //
qed-.
lemma cnv_inv_lref_atom (a) (h) (b) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥.
#a #h #b #G #i #L #H #Hi
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
qed-.
lemma cnv_inv_lref_unit (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
∀I,K. ⬇*[i] L ≘ K.ⓤ{I} → ⊥.
#a #h #G #i #L #H #I #K #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
(* Properties with supclosure ***********************************************)
(* Basic_2A1: uses: snv_fqu_conf *)
-lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
+lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I1 #G1 #L1 #V1 #H
elim (cnv_inv_zero … H) -H #I2 #L2 #V2 #HV2 #H destruct //
qed-.
(* Basic_2A1: uses: snv_fquq_conf *)
-lemma cnv_fquq_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
+lemma cnv_fquq_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*]
/2 width=5 by cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqup_conf *)
-lemma cnv_fqup_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
+lemma cnv_fqup_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=5 by fqup_strap1, cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqus_conf *)
-lemma cnv_fqus_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
+lemma cnv_fqus_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*]
/2 width=5 by cnv_fqup_conf/
qed-.
(* Forward lemmas with strongly rst-normalizing closures ********************)
(* Basic_2A1: uses: snv_fwd_fsb *)
-lemma cnv_fwd_fsb (a) (h): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ≥[h] 𝐒⦃G, L, T⦄.
+lemma cnv_fwd_fsb (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ≥[h] 𝐒⦃G,L,T⦄.
#a #h #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
qed-.
(* Forward lemmas with strongly rt-normalizing terms ************************)
-lemma cnv_fwd_csx (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma cnv_fwd_csx (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#a #h #G #L #T #H
/3 width=2 by cnv_fwd_fsb, fsb_inv_csx/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
lemma cnv_fpbg_refl_false (a) (h) (G) (L) (T):
- ⦃G, L⦄ ⊢ T ![a,h] → ⦃G, L, T⦄ >[h] ⦃G, L, T⦄ → ⊥.
+ ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
/3 width=7 by cnv_fwd_fsb, fsb_fpbg_refl_false/ qed-.
(* Inductive premises for the preservation results **************************)
definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
+ ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
+ ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡*[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
(* Auxiliary properties for preservation ************************************)
fact cnv_cpms_trans_lpr_sub (a) (h):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
#a #h #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2
/3 width=7 by fpbg_cpms_trans/
fact cnv_cpm_conf_lpr_sub (a) (h):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
fact cnv_cpms_strip_lpr_sub (a) (h):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
inductive lsubv (a) (h) (G): relation lenv ≝
| lsubv_atom: lsubv a h G (⋆) (⋆)
| lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubv_beta: ∀L1,L2,W,V. ⦃G, L1⦄ ⊢ ⓝW.V ![a,h] →
+| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![a,h] →
lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW)
.
fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
∀I,K1. L1 = K1.ⓘ{I} →
∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
#a #h #G #L1 #L2 * -L1 -L2
(* Basic_2A1: uses: lsubsv_inv_pair1 *)
lemma lsubv_inv_bind_sn (a) (h) (G): ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 →
∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsubv_inv_bind_sn_aux/ qed-.
fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
∀I,K2. L2 = K2.ⓘ{I} →
∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
#a #h #G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
(* Basic_2A1: uses: lsubsv_inv_pair2 *)
lemma lsubv_inv_bind_dx (a) (h) (G): ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} →
∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsubv_inv_bind_dx_aux/ qed-.
(* Basic_2A1: uses: lsubsv_snv_trans *)
lemma lsubv_cnv_trans (a) (h) (G):
- ∀L2,T. ⦃G, L2⦄ ⊢ T ![a,h] →
- ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G, L1⦄ ⊢ T ![a,h].
+ ∀L2,T. ⦃G,L2⦄ ⊢ T ![a,h] →
+ ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G,L1⦄ ⊢ T ![a,h].
#a #h #G #L2 #T #H elim H -G -L2 -T //
[ #I #G #K2 #V #HV #IH #L1 #H
elim (lsubv_inv_bind_dx … H) -H * /3 width=1 by cnv_zero/
qed-.
lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L):
- ∀X2, i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
+ ∀X2,i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]
| ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
#a #h #G #L #X2 #i #H
qed-.
(* Basic_1: uses: ty3_gen_lift *)
-(* Note: "⦃G,L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G,L⦄ ⊢ X2 ➡*[h] U2" *)
+(* Note: "⦃G, L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G, L⦄ ⊢ X2 ➡*[h] U2" *)
lemma nta_inv_lifts_sn (a) (h) (G):
∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 →
∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 →
(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
definition ntas (a) (h) (n) (G) (L): relation term ≝ λT,U.
- ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[n,h] U0.
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[n,h] U0.
interpretation "iterated native type assignment (term)"
'Colon a h n G L T U = (ntas a h n G L T U).
(* Advanced properties on native type assignment for terms ******************)
-lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U →
- ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U.
+lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U →
+ ∀V. ⦃h,L⦄ ⊢ V : W → ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U.
#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
qed.
-axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U →
- ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z.
+axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U →
+ ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z.
(* REQUIRES SUBJECT CONVERSION
#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
[ #U #HYU
(* Advanced inversion lemmas on native type assignment for terms ************)
-fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
- ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
+fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
#h #L #Z #U #H elim H -L -Z -U
[ #L #k #X #Y #H destruct
| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
qed.
(* Basic_1: was only: ty3_gen_appl *)
-lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U →
- ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
+lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
/2 width=3/ qed-.
-axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U →
- ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W &
+axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U →
+ ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W &
L ⊢ ⓐZ.ⓛY.W ⬌* U.
(* REQUIRES SUBJECT REDUCTION
#h #L #Z #Y #X #U #H
(* Basic eliminators ********************************************************)
axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) →
- ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1.
+ (∀T1,T. ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → R T → R T1) →
+ ∀T1. ⦃h,L⦄ ⊢ T1 :* T2 → R T1.
(*
#h #L #T2 #R #HT2 #IHT2 #T1 #HT12
@(star_ind_dx … HT2 IHT2 … HT12) //
(* Basic properties *********************************************************)
lemma ntas_strap1: ∀h,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+ ⦃h,L⦄ ⊢ T1 :* T → ⦃h,L⦄ ⊢ T : T2 → ⦃h,L⦄ ⊢ T1 :* T2.
/2 width=3/ qed.
lemma ntas_strap2: ∀h,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+ ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → ⦃h,L⦄ ⊢ T1 :* T2.
/2 width=3/ qed.
*)
(* \ / *)
(* \ / This file is distributed under the terms of the *)
(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⥲* [ break term 46 h ] 𝐍 ⦃ break term 46 T2 ⦄ )"
(* Basic_2A1: uses: cpre *)
definition cpme (h) (n) (G) (L): relation2 term term ≝
- λT1,T2. ∧∧ ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T2⦄.
+ λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T2⦄.
interpretation "evaluation for t-bound context-sensitive parallel rt-transition (term)"
'PRedEval h n G L T1 T2 = (cpme h n G L T1 T2).
lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
Q 0 T2 →
- (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
+ (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T1.
#h #G #L #T2 #Q @ltc_ind_sn_refl //
qed-.
lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
Q 0 T1 →
- (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
- ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
+ (∀n1,n2,T,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → Q n1 T → ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → Q (n1+n2) T2) →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T2.
#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic_1: includes: pr1_pr0 *)
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
/2 width=1 by ltc_rc/ qed.
-lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_sn/ qed-.
-lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
lemma cpms_bind_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
lemma cpms_appl_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
qed.
lemma cpms_zeta (n) (h) (G) (L):
∀T1,T. ⬆*[1] T ≘ T1 →
- ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+ ∀V,T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_zeta *)
lemma cpms_zeta_dx (n) (h) (G) (L):
∀T2,T. ⬆*[1] T2 ≘ T →
- ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+ ∀V,T1. ⦃G,L.ⓓV⦄ ⊢ T1 ➡*[n,h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_eps *)
lemma cpms_eps (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
qed.
lemma cpms_ee (n) (h) (G) (L):
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2.
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ∀T. ⦃G,L⦄ ⊢ ⓝU1.T ➡*[↑n,h] U2.
#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=1 by cpm_cpms, cpm_ee/
| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
(* Basic_2A1: uses: cprs_beta_dx *)
lemma cpms_beta_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
qed.
(* Basic_2A1: uses: cprs_theta_dx *)
lemma cpms_theta_dx (n) (h) (G) (L):
- ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V →
+ ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V →
∀V2. ⬆*[1] V ≘ V2 →
- ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
+lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
#n1 #n2 #X #X2 #_ #IH #HX2 destruct
elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
qed-.
lemma cpms_inv_cast1 (h) (n) (G) (L):
- ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2
- | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+ ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ➡*[n,h] X2
+ | ∃∃m. ⦃G,L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #X #X2 #_ * [ * || * ]
/3 width=5 by cpms_fwd_cpxs, cpxs_aaa_conf/ qed-.
lemma cpms_total_aaa (h) (G) (L) (n) (A):
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
#h #G #L #n elim n -n
[ /2 width=3 by ex_intro/
| #n #IH #A #T1 #HT1 <plus_SO
(* Basic_2A1: includes: cprs_bind *)
theorem cpms_bind (n) (h) (G) (L):
- ∀I,V1,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀I,V1,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
#n #h #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_bind_dx/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
qed.
theorem cpms_appl (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
#n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_appl_dx/
| #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_beta_rc *)
theorem cpms_beta_rc (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=1 by cpms_beta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_beta *)
theorem cpms_beta (n) (h) (G) (L):
- ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_beta_rc/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_theta_rc *)
theorem cpms_theta_rc (n) (h) (G) (L):
- ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 →
- ∀W1,T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 →
+ ∀W1,T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=3 by cpms_theta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_theta *)
theorem cpms_theta (n) (h) (G) (L):
- ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V1. ⦃G, L⦄ ⊢ V1 ➡*[h] V →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V1. ⦃G,L⦄ ⊢ V1 ➡*[h] V →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
[ /2 width=3 by cpms_theta_rc/
| #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
(* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *)
theorem cpms_trans (h) (G) (L):
- ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+ ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_trans/ qed-.
(* Basic_2A1: uses: scpds_cprs_trans *)
theorem cpms_cprs_trans (n) (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
/2 width=3 by cpms_trans/ qed-.
(* Advanced inversion lemmas ************************************************)
lemma cpms_inv_appl_sn (n) (h) (G) (L):
- ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] X2 →
+ ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
∨∨ ∃∃V2,T2.
- ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 &
+ ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 &
X2 = ⓐV2.T2
| ∃∃n1,n2,p,W,T.
- ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2, h] X2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2,h] X2 &
n1 + n2 = n
| ∃∃n1,n2,p,V0,V2,V,T.
- ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2, h] X2 &
+ ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2,h] X2 &
n1 + n2 = n.
#n #h #G #L #V1 #T1 #U2 #H
@(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
]
qed-.
-lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T & ⦃G, L⦄ ⊢ T ➡*[n2, h] T2.
+lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2.
#h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/
#n1 #IH #n2 #T1 #T2 <plus_S1 #H
elim (cpms_inv_succ_sn … H) -H #T0 #HT10 #HT02
(* Advanced main properties *************************************************)
theorem cpms_cast (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡*[n, h] ⓝU2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
[ /3 width=3 by cpms_cast_sn/
| #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H
(* Forward lemmas with unbound context-sensitive rt-computation for terms ***)
(* Basic_2A1: includes: scpds_fwd_cpxs cprs_cpxs *)
-lemma cpms_fwd_cpxs (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpms_fwd_cpxs (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_dx … H) -T2
/3 width=4 by cpxs_strap1, cpm_fwd_cpx/
qed-.
(* Advanced properties ******************************************************)
-lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡*[n, h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡*[n, h] W2.
+lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡*[n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_delta/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
]
qed.
-lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡*[n, h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡*[↑n, h] W2.
+lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡*[↑n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_ell/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
]
qed.
-lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ⦃G, K⦄ ⊢ #i ➡*[n, h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡*[n, h] U.
+lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ⦃G,K⦄ ⊢ #i ➡*[n,h] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡*[n,h] U.
#n #h #I #G #K #T #i #H @(cpms_ind_dx … H) -T
[ /3 width=3 by cpm_cpms, cpm_lref/
| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
qed.
lemma cpms_cast_sn (n) (h) (G) (L):
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡*[n, h] ⓝU2.T2.
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=3 by cpm_cpms, cpm_cast/
| #n1 #n2 #U1 #U #HU1 #_ #IH #T1 #T2 #H
(* Basic_2A1: uses: cprs_delta *)
lemma cpms_delta_drops (n) (h) (G):
∀L,K,V,i. ⬇*[i] L ≘ K.ⓓV →
- ∀V2. ⦃G, K⦄ ⊢ V ➡*[n, h] V2 →
- ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡*[n, h] W2.
+ ∀V2. ⦃G,K⦄ ⊢ V ➡*[n,h] V2 →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡*[n,h] W2.
#n #h #G #L #K #V #i #HLK #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=6 by cpm_cpms, cpm_delta_drops/
| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
lemma cpms_ell_drops (n) (h) (G):
∀L,K,W,i. ⬇*[i] L ≘ K.ⓛW →
- ∀W2. ⦃G, K⦄ ⊢ W ➡*[n, h] W2 →
- ∀V2. ⬆*[↑i] W2 ≘ V2 → ⦃G, L⦄ ⊢ #i ➡*[↑n, h] V2.
+ ∀W2. ⦃G,K⦄ ⊢ W ➡*[n,h] W2 →
+ ∀V2. ⬆*[↑i] W2 ≘ V2 → ⦃G,L⦄ ⊢ #i ➡*[↑n,h] V2.
#n #h #G #L #K #W #i #HLK #W2 #H @(cpms_ind_dx … H) -W2
[ /3 width=6 by cpm_cpms, cpm_ell_drops/
| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
(* Advanced inversion lemmas ************************************************)
lemma cpms_inv_lref1_drops (n) (h) (G):
- ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[n, h] T2 →
+ ∀L,T2,i. ⦃G,L⦄ ⊢ #i ➡*[n,h] T2 →
∨∨ ∧∧ T2 = #i & n = 0
- | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡*[n, h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡*[n,h] V2 &
⬆*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡*[m, h] V2 &
+ | ∃∃m,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡*[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & n = ↑m.
#n #h #G #L #T2 #i #H @(cpms_ind_dx … H) -T2
[ /3 width=1 by or3_intro0, conj/
qed-.
fact cpms_inv_succ_sn (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[↑n, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[1, h] T & ⦃G, L⦄ ⊢ T ➡*[n, h] T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[↑n,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[1,h] T & ⦃G,L⦄ ⊢ T ➡*[n,h] T2.
#n #h #G #L #T1 #T2
@(insert_eq_0 … (↑n)) #m #H
@(cpms_ind_sn … H) -T1 -m
/3 width=2 by cpms_fwd_cpxs, cpxs_tdneq_fpbg/ qed-.
lemma fpbg_cpms_trans (h) (n):
- ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fpbs_trans, cpms_fwd_fpbs/ qed-.
lemma cpms_fpbg_trans (h) (n):
- ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ➡*[n,h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ➡*[n,h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fpbg_trans, cpms_fwd_fpbs/ qed-.
lemma fqup_cpms_fwd_fpbg (h):
- ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T⦄ →
- ∀n,T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
-/3 width=5 by cpms_fwd_fpbs, fqup_fpbg,fpbg_fpbs_trans/ qed-.
+ ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T⦄ →
+ ∀n,T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+/3 width=5 by cpms_fwd_fpbs, fqup_fpbg, fpbg_fpbs_trans/ qed-.
lemma cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg (h) (G) (L) (T1):
∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
(* Basic_2A1: uses: cprs_fpbs *)
lemma cpms_fwd_fpbs (n) (h):
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
/3 width=2 by cpms_fwd_cpxs, cpxs_fpbs/ qed-.
(* Properties with parallel rt-transition for full local environments *******)
lemma lpr_cpm_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L2 #T1 #T2 #H @(cpm_ind … H) -n -G -L2 -T1 -T2
[ /2 width=3 by/
| /3 width=2 by cpm_cpms/
qed-.
lemma lpr_cpms_trans (n) (h) (G):
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L1 #L2 #HL12 #T1 #T2 #H @(cpms_ind_sn … H) -n -T1
/3 width=3 by lpr_cpm_trans, cpms_trans/
qed-.
(* Basic_2A1: includes cpr_bind2 *)
lemma cpm_bind2 (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lpr_cpm_trans, cpms_bind_dx, lpr_pair/ qed.
(* Basic_2A1: includes cprs_bind2_dx *)
lemma cpms_bind2_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lpr_cpms_trans, cpms_bind_dx, lpr_pair/ qed.
(* Properties with sort-irrelevant equivalence for local environments *******)
lemma cpms_rdeq_conf_sn (h) (n) (G) (L1) (L2):
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
L1 ≛[T1] L2 → L1 ≛[T2] L2.
/3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_sn/ qed-.
lemma cpms_rdeq_conf_dx (h) (n) (G) (L1) (L2):
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 →
L1 ≛[T1] L2 → L1 ≛[T2] L2.
/3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_dx/ qed-.
(* Basic_1: was: nf2_pr3_confluence *)
theorem cpre_mono (h) (G) (L) (T):
- ∀T1. ⦃G, L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2.
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2.
#h #G #L #T0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20
>(cprs_inv_cnr_sn … HT10 HT1) -T1
(* Basic_1: was just: nf2_sn3 *)
lemma cpre_total_csx (h) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
#h #G #L #T1 #H
@(csx_ind … H) -T1 #T1 #_ #IHT1
elim (cnr_dec_tdeq h G L T1) [ /3 width=3 by ex_intro, conj/ ] *
(* Basic_2A1: was: cprs_ind_dx *)
lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ⦃G,L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T1.
#h #G #L #T2 #Q #IH1 #IH2 #T1
@(insert_eq_0 … 0) #n #H
@(cpms_ind_sn … H) -n -T1 //
(* Basic_2A1: was: cprs_ind *)
lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T → ⦃G,L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T2.
#h #G #L #T1 #Q #IH1 #IH2 #T2
@(insert_eq_0 … 0) #n #H
@(cpms_ind_dx … H) -n -T2 //
(* Basic_1: was: pr3_step *)
(* Basic_2A1: was: cprs_strap2 *)
lemma cprs_step_sn (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_sn/ qed-.
(* Basic_2A1: was: cprs_strap1 *)
lemma cprs_step_dx (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_dx/ qed-.
(* Basic_1: was only: pr3_thin_dx *)
lemma cprs_flat_dx (h) (I) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
lemma cprs_flat_sn (h) (I) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
#h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
+lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
/2 width=4 by cpms_inv_sort1/ qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ➡*[h] X2.
+lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ➡*[h] X2.
#h #G #L #W1 #T1 #X2 #H
elim (cpms_inv_cast1 … H) -H
[ /2 width=1 by or_introl/
(* Basic_1: was: nf2_pr3_unfold *)
(* Basic_2A1: was: cprs_inv_cnr1 *)
lemma cprs_inv_cnr_sn (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ → T1 = T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ → T1 = T2.
#h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T1 //
#T1 #T0 #HT10 #_ #IH #HT1
lapply (HT1 … HT10) -HT10 #H destruct /2 width=1 by/
(* Basic_1: was: pr3_flat *)
theorem cprs_flat (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=3 by cprs_flat_dx/
| /3 width=3 by cpr_pair_sn, cprs_step_dx/
(* Basic_1: was pr3_gen_appl *)
(* Basic_2A1: was: cprs_inv_appl1 *)
lemma cprs_inv_appl_sn (h) (G) (L):
- ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[h] T2 &
+ ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[h] T2 &
X2 = ⓐV2. T2
- | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T &
- ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2
- | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T &
- ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2.
+ | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T &
+ ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2
+ | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T &
+ ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2.
#h #G #L #V1 #T1 #X2 #H elim (cpms_inv_appl_sn … H) -H *
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #p #V2 #T2 #HT12 #HTX2 #H
(* Basic_1: was: pr3_gen_lref *)
(* Basic_2A1: was: cprs_inv_lref1 *)
-lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h] T2 →
+lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G,L⦄ ⊢ #i ➡*[h] T2 →
∨∨ T2 = #i
- | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡*[h] T1 &
+ | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G,K⦄ ⊢ V1 ➡*[h] T1 &
⬆*[↑i] T1 ≘ T2.
#h #G #L #T2 #i #H elim (cpms_inv_lref1_drops … H) -H *
[ /2 width=1 by or_introl/
qed-.
lemma cprs_lpr_conf_dx (h) (G):
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+ ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #H
@(cprs_ind_dx … H) -T1 /2 width=3 by ex2_intro/
#T #T1 #_ #HT1 #IHT0 #L1 #HL01
qed-.
lemma cprs_lpr_conf_sn (h) (G):
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+ ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1 #HT0
/3 width=3 by lpr_cpms_trans, ex2_intro/
(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
definition cpue (h) (G) (L): relation2 term term ≝
- λT1,T2. ∃∃n. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+ λT1,T2. ∃∃n. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
interpretation "evaluation for t-unbound context-sensitive parallel rt-transition (term)"
'PRedITEval h G L T1 T2 = (cpue h G L T1 T2).
(* Properties with strong normalization for unbound rt-transition for terms *)
lemma cpue_total_csx (h) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄.
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄.
#h #G #L #T1 #H
@(csx_ind … H) -T1 #T1 #_ #IHT1
elim (cnu_dec_tdeq h G L T1) [ /3 width=4 by ex2_intro, ex_intro/ ] *
(* Basic eliminators ********************************************************)
lemma cpxs_ind: ∀h,G,L,T1. ∀Q:predicate term. Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T2.
#h #L #G #T1 #Q #HT1 #IHT1 #T2 #HT12
@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
lemma cpxs_ind_dx: ∀h,G,L,T2. ∀Q:predicate term. Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T → ⦃G, L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T → ⦃G,L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T1.
#h #G #L #T2 #Q #HT2 #IHT2 #T1 #HT12
@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
-lemma cpxs_refl: ∀h,G,L,T. ⦃G, L⦄ ⊢ T ⬈*[h] T.
+lemma cpxs_refl: ∀h,G,L,T. ⦃G,L⦄ ⊢ T ⬈*[h] T.
/2 width=1 by inj/ qed.
-lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
/2 width=1 by inj/ qed.
-lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by step/ qed-.
-lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by TC_strap/ qed-.
(* Basic_2A1: was just: cpxs_sort *)
-lemma cpxs_sort: ∀h,G,L,s,n. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
+lemma cpxs_sort: ∀h,G,L,s,n. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
#h #G #L #s #n elim n -n /2 width=1 by cpx_cpxs/
#n >iter_S /2 width=3 by cpxs_strap1/
qed.
-lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
#h #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
qed.
-lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T.
+lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T.
#h #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_pair_sn/
qed.
lemma cpxs_zeta (h) (G) (L) (V):
∀T1,T. ⬆*[1] T ≘ T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
#h #G #L #V #T1 #T #HT1 #T2 #H @(cpxs_ind … H) -T2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_zeta/
qed.
(* Basic_2A1: was: cpxs_zeta *)
lemma cpxs_zeta_dx (h) (G) (L) (V):
∀T2,T. ⬆*[1] T2 ≘ T →
- ∀T1. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
+ ∀T1. ⦃G,L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
#h #G #L #V #T2 #T #HT2 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
qed.
-lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ⬈*[h] T2.
+lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ⬈*[h] T2.
#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
qed.
(* Basic_2A1: was: cpxs_ct *)
-lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ⬈*[h] V2.
+lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ⓝV1.T ⬈*[h] V2.
#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ee/
qed.
lemma cpxs_beta_dx: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
qed.
lemma cpxs_theta_dx: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: wa just: cpxs_inv_sort1 *)
-lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] X2 →
+lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] X2 →
∃n. X2 = ⋆((next h)^n s).
#h #G #L #X2 #s #H @(cpxs_ind … H) -X2 /2 width=2 by ex_intro/
#X #X2 #_ #HX2 * #n #H destruct
@(ex_intro … (↑n)) >iter_S //
qed-.
-lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ⬈*[h] U2
- | ⦃G, L⦄ ⊢ W1 ⬈*[h] U2.
+lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G,L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ⬈*[h] U2
+ | ⦃G,L⦄ ⊢ W1 ⬈*[h] U2.
#h #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
#W #T #HW1 #HT1 #H destruct
(* Inversion lemmas with normal terms ***************************************)
-lemma cpxs_inv_cnx1: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+lemma cpxs_inv_cnx1: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
T1 ≛ T2.
#h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
/5 width=9 by cnx_tdeq_trans, tdeq_trans/
theorem cpxs_trans: ∀h,G,L. Transitive … (cpxs h G L).
normalize /2 width=3 by trans_TC/ qed-.
-theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_bind_dx/
qed.
-theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_flat_dx/
qed.
theorem cpxs_beta_rc: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
/4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
qed.
theorem cpxs_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
/4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
qed.
theorem cpxs_theta_rc: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
/3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
qed.
theorem cpxs_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 &
+lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2
- | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2.
+ | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2
+ | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2.
#h #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
#U #U2 #_ #HU2 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
(* Advanced properties ******************************************************)
-lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2.
+lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2.
#h #I #G #K #V1 #V2 #H @(cpxs_ind … H) -V2
[ /3 width=3 by cpx_cpxs, cpx_delta/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
]
qed.
-lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G, K⦄ ⊢ #i ⬈*[h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U.
+lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G,K⦄ ⊢ #i ⬈*[h] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U.
#h #I #G #K #T #i #H @(cpxs_ind … H) -T
[ /3 width=3 by cpx_cpxs, cpx_lref/
| #T0 #T #_ #HT2 #IH #U #HTU
(* Basic_2A1: was: cpxs_delta *)
lemma cpxs_delta_drops: ∀h,I,G,L,K,V1,V2,i.
- ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
- ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h] W2.
+ ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈*[h] W2.
#h #I #G #L #K #V1 #V2 #i #HLK #H @(cpxs_ind … H) -V2
[ /3 width=7 by cpx_cpxs, cpx_delta_drops/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈*[h] T2 →
+lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈*[h] T2 →
T2 = #0 ∨
- ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 &
+ ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1.
#h #G #L #T2 #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
]
qed-.
-lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈*[h] T2 →
+lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈*[h] T2 →
T2 = #(↑i) ∨
- ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct
qed-.
(* Basic_2A1: was: cpxs_inv_lref1 *)
-lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h] T2 →
+lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈*[h] T2 →
T2 = #i ∨
- ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h] T1 &
+ ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G,K⦄ ⊢ V1 ⬈*[h] T1 &
⬆*[↑i] T1 ≘ T2.
#h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fdeq_cpxs_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛ ⦃G2, L2, T2⦄.
+lemma fdeq_cpxs_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ⬈*[h] T2 →
+ ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
elim (rdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
(* Properties on supclosure *************************************************)
-lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fquq_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
(* Note: a proof based on fqu_cpx_trans_tdneq might exist *)
(* Basic_2A1: uses: fqu_cpxs_trans_neq *)
-lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
#U2 #HVU2 @(ex3_intro … U2)
qed-.
(* Basic_2A1: uses: fquq_cpxs_trans_neq *)
-lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
[ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
qed-.
(* Basic_2A1: uses: fqup_cpxs_trans_neq *)
-lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
qed-.
(* Basic_2A1: uses: fqus_cpxs_trans_neq *)
-lemma fqus_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
[ #H12 elim (fqup_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
(* Advanced properties ******************************************************)
-lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
-lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
(* Properties with plus-iterated structural successor for closures **********)
(* Basic_2A1: uses: lpx_fqup_trans *)
-lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⊐+[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
/3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
(* Properties with star-iterated structural successor for closures **********)
(* Basic_2A1: uses: lpx_fqus_trans *)
-lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⊐*[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H
[ #H12 elim (lpx_fqup_trans … H12 … HKL1) -L1 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: was just: lleq_cpxs_trans *)
-lemma rdeq_cpxs_trans: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+lemma rdeq_cpxs_trans: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 →
∀L2. L2 ≛[T0] L0 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
#h #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
#T0 #T #HT0 #_ #IH #L2 #HL2
elim (rdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2
qed-.
(* Basic_2A1: was just: cpxs_lleq_conf *)
-lemma cpxs_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+lemma cpxs_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 →
∀L2. L0 ≛[T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
/3 width=3 by rdeq_cpxs_trans, rdeq_sym/ qed-.
(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
-lemma cpxs_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 →
+lemma cpxs_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈*[h] T2 →
∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2.
#h #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/
qed-.
(* Basic_2A1: was just: lleq_conf_sn *)
-lemma cpxs_rdeq_conf_sn: ∀h,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 →
+lemma cpxs_rdeq_conf_sn: ∀h,G,L1,T1,T2. ⦃G,L1⦄ ⊢ T1 ⬈*[h] T2 →
∀L2. L1 ≛[T1] L2 → L1 ≛[T2] L2.
/4 width=6 by cpxs_rdeq_conf_dx, rdeq_sym/ qed-.
(* Properties with sort-irrelevant equivalence for terms ********************)
-lemma tdeq_cpxs_trans: ∀h,U1,T1. U1 ≛ T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛ T2.
+lemma tdeq_cpxs_trans: ∀h,U1,T1. U1 ≛ T1 → ∀G,L,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∃∃U2. ⦃G,L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛ T2.
#h #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
#T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1
/3 width=3 by ex2_intro, cpxs_strap1/
(* Note: this requires tdeq to be symmetric *)
(* Nasic_2A1: uses: cpxs_neq_inv_step_sn *)
-lemma cpxs_tdneq_fwd_step_sn: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) →
- ∃∃T,T0. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 ≛ T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T0 & T0 ≛ T2.
+lemma cpxs_tdneq_fwd_step_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) →
+ ∃∃T,T0. ⦃G,L⦄ ⊢ T1 ⬈[h] T & T1 ≛ T → ⊥ & ⦃G,L⦄ ⊢ T ⬈*[h] T0 & T0 ≛ T2.
#h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
[ #H elim H -H //
| #T1 #T0 #HT10 #HT02 #IH #Hn12
(* Forward lemmas with head equivalence for terms ***************************)
-lemma cpxs_fwd_sort: ∀h,G,L,X2,s1. ⦃G, L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2.
+lemma cpxs_fwd_sort: ∀h,G,L,X2,s1. ⦃G,L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2.
#h #G #L #X2 #s1 #H
elim (cpxs_inv_sort1 … H) -H #s2 #H destruct //
qed-.
(* Basic_2A1: was: cpxs_fwd_delta *)
lemma cpxs_fwd_delta_drops: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀X2. ⦃G, L⦄ ⊢ #i ⬈*[h] X2 →
- ∨∨ #i ⩳ X2 | ⦃G, L⦄ ⊢ V2 ⬈*[h] X2.
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬈*[h] X2 →
+ ∨∨ #i ⩳ X2 | ⦃G,L⦄ ⊢ V2 ⬈*[h] X2.
#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #X2 #H
elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
qed-.
(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
- ∨∨ ⓐV.ⓛ{p}W.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ ∨∨ ⓐV.ⓛ{p}W.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] X2.
#h #p #G #L #V #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H *
[ #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
]
qed-.
-lemma cpxs_fwd_theta: ∀h,p,G,L,V1,V,T,X2. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 →
+lemma cpxs_fwd_theta: ∀h,p,G,L,V1,V,T,X2. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 →
∀V2. ⬆*[1] V1 ≘ V2 →
- ∨∨ ⓐV1.ⓓ{p}V.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] X2.
+ ∨∨ ⓐV1.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] X2.
#h #p #G #L #V1 #V #T #X2 #H #V2 #HV12
elim (cpxs_inv_appl1 … H) -H *
[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
]
qed-.
-lemma cpxs_fwd_cast: ∀h,G,L,W,T,X2. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] X2 →
- ∨∨ ⓝW. T ⩳ X2 | ⦃G, L⦄ ⊢ T ⬈*[h] X2 | ⦃G, L⦄ ⊢ W ⬈*[h] X2.
+lemma cpxs_fwd_cast: ∀h,G,L,W,T,X2. ⦃G,L⦄ ⊢ ⓝW.T ⬈*[h] X2 →
+ ∨∨ ⓝW. T ⩳ X2 | ⦃G,L⦄ ⊢ T ⬈*[h] X2 | ⦃G,L⦄ ⊢ W ⬈*[h] X2.
#h #G #L #W #T #X2 #H
elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
#W0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
qed-.
-lemma cpxs_fwd_cnx: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
- ∀X2. ⦃G, L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2.
+lemma cpxs_fwd_cnx: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+ ∀X2. ⦃G,L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2.
/3 width=5 by cpxs_inv_cnx1, tdeq_theq/ qed-.
(* Vector form of forward lemmas with head equivalence for terms ************)
-lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
+lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
#h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
#V #Vs #IHVs #X2 #H
elim (cpxs_inv_appl1 … H) -H *
(* Basic_2A1: was: cpxs_fwd_delta_vector *)
lemma cpxs_fwd_delta_drops_vector: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
- ∨∨ ⒶVs.#i ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
+ ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
#V #Vs #IHVs #X2 #H -K -V1
qed-.
(* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
- ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
#h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
#V0 #Vs #IHVs #V #W #T #X2 #H
elim (cpxs_inv_appl1 … H) -H *
(* Basic_1: was just: pr3_iso_appls_abbr *)
lemma cpxs_fwd_theta_vector: ∀h,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀p,V,T,X2. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
- ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
+ ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
+ ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
#h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
#V1b #V2b #V1a #V2a #HV12a #HV12b #p
generalize in match HV12a; -HV12a
qed-.
(* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
+lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
∨∨ ⒶVs. ⓝW. T ⩳ X2
- | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2
- | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
+ | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2
+ | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
#h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
#V #Vs #IHVs #W #T #X2 #H
elim (cpxs_inv_appl1 … H) -H *
qed-.
(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
- ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
+lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
#h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
#V #Vs #IHVs #X2 #H
elim (cpxs_inv_appl1 … H) -H *
(* Basic eliminators ********************************************************)
lemma csx_ind: ∀h,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) →
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) →
Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
#h #G #L #Q #H0 #T1 #H elim H -T1
/5 width=1 by SN_intro/
qed-.
(* Basic_1: was just: sn3_pr2_intro *)
lemma csx_intro: ∀h,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
/4 width=1 by SN_intro/ qed.
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
- ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ②{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
#h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #V2 #HLV2 #HV2
@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
qed-.
(* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
-fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
- ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+@(IH (ⓑ{p, I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
-fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
- ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
qed-.
(* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
-lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
-lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
(* Basic_1: removed theorems 14:
(* Main properties with atomic arity assignment *****************************)
-theorem aaa_csx: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+theorem aaa_csx: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #T #A #H
@(gcr_aaa … (csx_gcp h) (csx_gcr h) … H)
qed.
(* Advanced eliminators *****************************************************)
fact aaa_ind_csx_aux: ∀h,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T.
#h #G #L #A #Q #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
qed-.
lemma aaa_ind_csx: ∀h,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T.
/5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-.
fact aaa_ind_csx_cpxs_aux: ∀h,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T.
#h #G #L #A #Q #IH #T #H @(csx_ind_cpxs … H) -T /4 width=5 by cpxs_aaa_conf/
qed-.
(* Basic_2A1: was: aaa_ind_csx_alt *)
lemma aaa_ind_csx_cpxs: ∀h,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T.
/5 width=9 by aaa_ind_csx_cpxs_aux, aaa_csx/ qed-.
(* Properties with normal terms for unbound parallel rt-transition **********)
(* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma cnx_csx: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=1 by NF_to_SN/ qed.
(* Advanced properties ******************************************************)
-lemma csx_sort: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃⋆s⦄.
+lemma csx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃⋆s⦄.
/3 width=4 by cnx_csx, cnx_sort/ qed.
(* Properties with normal terms for unbound parallel rt-transition **********)
(* Basic_1: was just: sn3_appls_lref *)
-lemma csx_applv_cnx: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
- ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄.
+lemma csx_applv_cnx: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄.
#h #G #L #T #H1T #H2T #Vs elim Vs -Vs
[ #_ normalize in ⊢ (????%); /2 width=1 by cnx_csx/
| #V #Vs #IHV #H
(* Advanced properties ******************************************************)
-lemma csx_applv_sort: ∀h,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.⋆s⦄.
+lemma csx_applv_sort: ∀h,G,L,s,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.⋆s⦄.
/3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ qed.
(* Basic_1: was just: sn3_intro *)
lemma csx_intro_cpxs: ∀h,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
/2 width=3 by csx_cpx_trans/
qed-.
(* Eliminators with unbound context-sensitive rt-computation for terms ******)
lemma csx_ind_cpxs_tdeq: ∀h,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T0. ⦃G,L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
#h #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
(* Basic_2A1: was: csx_ind_alt *)
lemma csx_ind_cpxs: ∀h,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
#h #G #L #Q #IH #T #HT
@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
-lemma csx_tdeq_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀T2. T1 ≛ T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_tdeq_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
@csx_intro #T1 #HT21 #HnT21 elim (tdeq_cpx_trans … HT2 … HT21) -HT21
/4 width=5 by tdeq_repl/
qed-.
-lemma csx_cpx_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_cpx_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
elim (tdeq_dec T1 T2) /3 width=4 by csx_tdeq_trans/
qed-.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄.
+lemma csx_cast: ∀h,G,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄.
#h #G #L #W #HW @(csx_ind … HW) -W
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
(* Basic_1: was just: sn3_abbr *)
(* Basic_2A1: was: csx_lref_bind *)
lemma csx_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄.
+ ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄.
#h #I #G #L #K #V #i #HLK #HV
@csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
[ #H destruct elim Hi //
(* Basic_1: was: sn3_gen_def *)
(* Basic_2A1: was: csx_inv_lref_bind *)
lemma csx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
#h #I #G #L #K #V #i #HLK #Hi
elim (lifts_total V (𝐔❴↑i❵))
/4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
qed-.
-lemma csx_inv_lref: ∀h,G,L,i. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ →
- ∨∨ ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆
+lemma csx_inv_lref: ∀h,G,L,i. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ →
+ ∨∨ ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆
| ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I}
- | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+ | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
#h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
* * /4 width=9 by csx_inv_lref_pair, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
qed-.
(* Advanced properties ************************************* ****************)
(* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
+lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
#h #p #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
#V0 #Vs #IHV #V #W #T #H1T
lapply (csx_fwd_pair_sn … H1T) #HV0
lemma csx_applv_delta: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
+ ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
[ /4 width=11 by csx_inv_lifts, csx_lref_pair, drops_isuni_fwd_drop2/
| #V #Vs #IHV #H1T
(* Basic_1: was just: sn3_appls_abbr *)
lemma csx_applv_theta: ∀h,p,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
#h #p #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/
#V1b #V2b #V1 #V2 #HV12 #H
generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
qed.
(* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
+lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
#h #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
#V #Vs #IHV #U #H1U #T #H1T
lapply (csx_fwd_pair_sn … H1U) #HV
(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma csx_fdeq_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fdeq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
/3 width=3 by csx_rdeq_conf, csx_tdeq_trans/
qed-.
(* Properties with parallel rst-transition for closures *********************)
(* Basic_2A1: was: csx_fpb_conf *)
-lemma csx_fpbq_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fpbq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
/2 width=6 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_fdeq_conf/
qed-.
(* Properties with extended supclosure **************************************)
-lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by csx_inv_lref_pair, drops_refl/
| /2 width=3 by csx_fwd_pair_sn/
]
qed-.
-lemma csx_fquq_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fquq_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=6 by csx_fqu_conf/
* #HG #HL #HT destruct //
qed-.
-lemma csx_fqup_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fqup_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=6 by csx_fqu_conf/
qed-.
-lemma csx_fqus_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma csx_fqus_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -H
/3 width=6 by csx_fquq_conf/
qed-.
(* Properties with unbound parallel rt-transition on all entries ************)
-lemma csx_lpx_conf: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_lpx_conf: ∀h,G,L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L1 #T #H @(csx_ind_cpxs … H) -T
/4 width=3 by csx_intro, lpx_cpx_trans/
qed-.
(* Advanced properties ******************************************************)
-lemma csx_abst: ∀h,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
- ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄.
+lemma csx_abst: ∀h,p,G,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
+ ∀T. ⦃G,L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄.
#h #p #G #L #W #HW
@(csx_ind … HW) -W #W #_ #IHW #T #HT
@(csx_ind … HT) -T #T #HT #IHT
]
qed.
-lemma csx_abbr: ∀h,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
- ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄.
+lemma csx_abbr: ∀h,p,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
+ ∀T. ⦃G,L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄.
#h #p #G #L #V #HV
@(csx_ind … HV) -V #V #_ #IHV #T #HT
@(csx_ind_cpxs … HT) -T #T #HT #IHT
]
qed.
-fact csx_appl_theta_aux: ∀h,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
- ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+fact csx_appl_theta_aux: ∀h,p,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
+ ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
#h #p #G #L #X #H
@(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
lapply (csx_fwd_pair_sn … HVT) #HV
]
qed-.
-lemma csx_appl_theta: ∀h,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
- ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+lemma csx_appl_theta: ∀h,p,G,L,V,V2,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
+ ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
/2 width=5 by csx_appl_theta_aux/ qed.
(* Properties with unbound parallel rt-computation on all entries ***********)
-lemma csx_lpxs_conf: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_lpxs_conf: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
/3 by lpxs_step_dx, csx_lpx_conf/
qed-.
(* Advanced properties ******************************************************)
-fact csx_appl_beta_aux: ∀h,p,G,L,U1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U1⦄ →
- ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄.
+fact csx_appl_beta_aux: ∀h,p,G,L,U1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U1⦄ →
+ ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄.
#h #p #G #L #X #H @(csx_ind … H) -X
#X #HT1 #IHT1 #V #W #T1 #H1 destruct
@csx_intro #X #H1 #H2
qed-.
(* Basic_1: was just: sn3_beta *)
-lemma csx_appl_beta: ∀h,p,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T⦄.
+lemma csx_appl_beta: ∀h,p,G,L,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T⦄.
/2 width=3 by csx_appl_beta_aux/ qed.
(* Advanced forward lemmas **************************************************)
-fact csx_fwd_bind_dx_unit_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
- ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+fact csx_fwd_bind_dx_unit_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #U #H elim H -H #U0 #_ #IH #p #I #J #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{p,I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2
+@(IH (ⓑ{p, I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
-lemma csx_fwd_bind_dx_unit: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ∀J. ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx_unit: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ∀J. ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=6 by csx_fwd_bind_dx_unit_aux/ qed-.
-lemma csx_fwd_bind_unit: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ∀J. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_bind_unit: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ∀J. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=4 by csx_fwd_pair_sn, csx_fwd_bind_dx_unit, conj/ qed-.
(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: uses: csx_lleq_conf *)
-lemma csx_rdeq_conf: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
- ∀L2. L1 ≛[T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_rdeq_conf: ∀h,G,L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
+ ∀L2. L1 ≛[T] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L1 #T #H
@(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
@csx_intro #T2 #HT12 #HnT12
(* Basic_2A1: uses: csx_lleq_conf *)
lemma csx_rdeq_trans: ∀h,L1,L2,T. L1 ≛[T] L2 →
- ∀G. ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+ ∀G. ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_rdeq_conf, rdeq_sym/ qed-.
(* Properties with simple terms *********************************************)
-lemma csx_appl_simple: ∀h,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
+lemma csx_appl_simple: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
+ 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
#h #G #L #V #H @(csx_ind … H) -V
#V #_ #IHV #T1 #IHT1 #HT1
@csx_intro #X #H1 #H2
(* Basic_1: was just: sn3_appl_appl *)
(* Basic_2A1: was: csx_appl_simple_tsts *)
-lemma csx_appl_simple_theq: ∀h,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
+lemma csx_appl_simple_theq: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
+ 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
#h #G #L #V #H @(csx_ind … H) -V
#V #_ #IHV #T1 #H @(csx_ind … H) -T1
#T1 #H1T1 #IHT1 #H2T1 #H3T1
(* Basic inversion lemmas ***************************************************)
-lemma csxv_inv_cons: ∀h,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⨮Ts⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Ts⦄.
+lemma csxv_inv_cons: ∀h,G,L,T,Ts. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⨮Ts⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Ts⦄.
normalize // qed-.
(* Basic forward lemmas *****************************************************)
-lemma csx_fwd_applv: ∀h,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_fwd_applv: ∀h,G,L,T,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
#V #Vs #IHVs #HVs
lapply (csx_fwd_pair_sn … HVs) #HV
definition fpbg: ∀h. tri_relation genv lenv term ≝
λh,G1,L1,T1,G2,L2,T2.
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄.
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ & ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄.
interpretation "proper parallel rst-computation (closure)"
'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/2 width=5 by ex2_3_intro/ qed.
lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
/3 width=9 by fpbs_strap1, ex2_3_intro/
qed-.
lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
/4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
qed-.
(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
qed-.
(* Basic_2A1: uses: fpbg_fleq_trans *)
-lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-.
(* Properties with t-bound rt-transition for terms **************************)
lemma cpm_tdneq_cpm_fpbg (h) (G) (L):
- ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2,h] T2 → ⦃G, L, T1⦄ >[h] ⦃G, L, T2⦄.
+ ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.
(* Properties with unbound context-sensitive parallel rt-computation ********)
(* Basic_2A1: was: cpxs_fpbg *)
-lemma cpxs_tdneq_fpbg (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- (T1 ≛ T2 → ⊥) → ⦃G, L, T1⦄ >[h] ⦃G, L, T2⦄.
+lemma cpxs_tdneq_fpbg (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ (T1 ≛ T2 → ⊥) → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
#h #G #L #T1 #T2 #H #H0
elim (cpxs_tdneq_fwd_step_sn … H … H0) -H -H0
/4 width=5 by cpxs_tdeq_fpbs, fpb_cpx, ex2_3_intro/
qed.
-lemma cpxs_fpbg_trans (h): ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma cpxs_fpbg_trans (h): ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fpbg_trans, cpxs_fpbs/ qed-.
(* Advanced forward lemmas **************************************************)
lemma fpbg_fwd_fpbs: ∀h,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 *
/3 width=5 by fpbs_strap2, fpb_fpbq/
qed-.
(* Advanced properties with sort-irrelevant equivalence on closures *********)
(* Basic_2A1: uses: fleq_fpbg_trans *)
-lemma fdeq_fpbg_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbg_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
elim (fdeq_fpb_trans … H1 … H0) -G -L -T
/4 width=9 by fpbs_strap2, fpbq_fdeq, ex2_3_intro/
(* Properties with parallel proper rst-reduction on closures ****************)
lemma fpb_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≻[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
(* Properties with parallel rst-reduction on closures ***********************)
lemma fpbq_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
elim (fpbq_inv_fpb … H1) -H1
/2 width=5 by fdeq_fpbg_trans, fpb_fpbg_trans/
(* Properties with parallel rst-compuutation on closures ********************)
-lemma fpbs_fpbg_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpbs_fpbg_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
qed-.
(* Advanced inversion lemmas of parallel rst-computation on closures ********)
(* Basic_2A1: was: fpbs_fpbg *)
-lemma fpbs_inv_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∨∨ ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpbs_inv_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
[ /2 width=1 by or_introl/
| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
(* Advanced properties of parallel rst-computation on closures **************)
-lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h] ⦃G2, L2, U2⦄.
+lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≥[h] ⦃F2,K2,T2⦄ →
+ ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄.
#h #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
[ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
/3 width=5 by fdeq_fpbs, ex2_3_intro/
(* Advanced properties with sort-irrelevant equivalence for terms ***********)
-lemma fpbg_tdeq_div: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T⦄ →
- ∀T2. T2 ≛ T → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fpbg_tdeq_div: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ →
+ ∀T2. T2 ≛ T → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/4 width=5 by fpbg_fdeq_trans, tdeq_fdeq, tdeq_sym/ qed-.
(* Properties with plus-iterated structural successor for closures **********)
(* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄.
+lemma fqup_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
qed.
(* Properties with unbound rt-computation on full local environments ********)
(* Basic_2A1: uses: lpxs_fpbg *)
-lemma lpxs_rdneq_fpbg: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- (L1 ≛[T] L2 → ⊥) → ⦃G, L1, T⦄ >[h] ⦃G, L2, T⦄.
+lemma lpxs_rdneq_fpbg: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ (L1 ≛[T] L2 → ⊥) → ⦃G,L1,T⦄ >[h] ⦃G,L2,T⦄.
#h #G #L1 #L2 #T #H #H0
elim (lpxs_rdneq_inv_step_sn … H … H0) -H -H0
/4 width=7 by fpb_lpx, lpxs_fdeq_fpbs, fdeq_intro_sn, ex2_3_intro/
(* Basic eliminators ********************************************************)
lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
/3 width=8 by tri_TC_star_ind/ qed-.
lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
/3 width=8 by tri_TC_star_ind_dx/ qed-.
(* Basic properties *********************************************************)
lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h).
/2 width=1 by tri_inj/ qed.
-lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
-lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed-.
-lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
-lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=1 by fpbq_fpbs, fpbq_fdeq/ qed.
(* Basic_2A1: uses: fpbs_lleq_trans *)
-lemma fpbs_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=9 by fpbs_strap1, fpbq_fdeq/ qed-.
(* Basic_2A1: uses: lleq_fpbs_trans *)
-lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_strap2, fpbq_fdeq/ qed-.
lemma tdeq_rdeq_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
- ∀G,L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃G, L1, T1⦄ ≥[h] ⦃G, L2, T2⦄.
+ ∀G,L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃G,L1,T1⦄ ≥[h] ⦃G,L2,T2⦄.
/4 width=5 by fdeq_fpbs, fpbs_strap1, fpbq_lpx, fdeq_intro_dx/ qed.
(* Basic_2A1: removed theorems 3:
(* Properties with atomic arity assignment for terms ************************)
-lemma fpbs_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma fpbs_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A
/2 width=8 by fpbq_aaa_conf/
(* Properties with unbound context-sensitive parallel rt-transition *********)
(* Basic_2A1: uses: fpbs_cpx_trans_neq *)
-lemma fpbs_cpx_tdneq_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h] ⦃G2, L2, U2⦄.
+lemma fpbs_cpx_tdneq_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2
elim (fpbs_inv_star … H) -H #G0 #L0 #L3 #T0 #T3 #HT10 #H10 #HL03 #H32
elim (fdeq_cpx_trans … H32 … HTU2) -HTU2 #T4 #HT34 #H42
(* Properties with unbound context-sensitive parallel rt-computation ********)
-lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄.
+lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by fpbq_cpx, fpbs_strap1/
qed.
-lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T2⦄.
+lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T2⦄.
#h #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by fpbs_strap1, fpbq_cpx/
qed-.
-lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1, L1, T⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀T1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀T1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=5 by fpbs_strap2, fpbq_cpx/
qed-.
-lemma cpxs_tdeq_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
+lemma cpxs_tdeq_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
∀T0. T ≛ T0 →
- ∀G2,L2,T2. ⦃G1, L1, T0⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+ ∀G2,L2,T2. ⦃G1,L1,T0⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by cpxs_fpbs_trans, tdeq_fpbs_trans/ qed-.
-lemma cpxs_tdeq_fpbs: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T →
- ∀T2. T ≛ T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄.
+lemma cpxs_tdeq_fpbs: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T →
+ ∀T2. T ≛ T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
/4 width=3 by cpxs_fpbs_trans, fdeq_fpbs, tdeq_fdeq/ qed.
(* Properties with star-iterated structural successor for closures **********)
-lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ ⊐* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
(* Properties with plus-iterated structural successor for closures **********)
-lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
(* Properties with sn for unbound parallel rt-transition for terms **********)
(* Basic_2A1: was: csx_fpbs_conf *)
-lemma fpbs_csx_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+lemma fpbs_csx_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
/2 width=5 by csx_fpbq_conf/
qed-.
(* Properties with proper parallel rst-reduction on closures ****************)
-lemma fpb_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpb_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=1 by fpbq_fpbs, fpb_fpbq/ qed.
(* Advanced properties ******************************************************)
lemma tdeq_fpbs_trans: ∀h,T1,T. T1 ≛ T →
- ∀G1,G2,L1,L2,T2. ⦃G1, L1, T⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+ ∀G1,G2,L1,L2,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fdeq_fpbs_trans, tdeq_fdeq/ qed-.
-lemma fpbs_tdeq_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T⦄ →
- ∀T2. T ≛ T2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_tdeq_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T⦄ →
+ ∀T2. T ≛ T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fdeq_trans, tdeq_fdeq/ qed-.
(* Properties with plus-iterated structural successor for closures **********)
-lemma fqup_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fqup_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
qed.
-lemma fpbs_fqup_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_fqup_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
-lemma fqup_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fqup_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐+ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-.
(* Properties with star-iterated structural successor for closures **********)
-lemma fqus_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fqus_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
/3 width=5 by fpbq_fquq, tri_step/
qed.
-lemma fpbs_fqus_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_fqus_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⊐* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
/3 width=5 by fpbs_strap1, fpbq_fquq/
qed-.
-lemma fqus_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fqus_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐* ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
/3 width=5 by fpbs_strap2, fpbq_fquq/
qed-.
(* Properties with unbound rt-computation on full local environments *******)
-lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → ⦃G, L1, T⦄ ≥[h] ⦃G, L2, T⦄.
+lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → ⦃G,L1,T⦄ ≥[h] ⦃G,L2,T⦄.
#h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
/3 width=5 by fpbq_lpx, fpbs_strap1/
qed.
-lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
/3 width=5 by fpbs_strap1, fpbq_lpx/
qed-.
-lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∀L1. ⦃G1, L1⦄ ⊢ ⬈*[h] L → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1,L,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀L1. ⦃G1,L1⦄ ⊢ ⬈*[h] L → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
/3 width=5 by fpbs_strap2, fpbq_lpx/
qed-.
(* Basic_2A1: uses: lpxs_lleq_fpbs *)
-lemma lpxs_fdeq_fpbs: ∀h,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] L →
- ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma lpxs_fdeq_fpbs: ∀h,G1,L1,L,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] L →
+ ∀G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by lpxs_fpbs_trans, fdeq_fpbs/ qed.
-lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
(* Properties with star-iterated structural successor for closures **********)
-lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
(* Properties with unbound context-sensitive parallel rt-computation ********)
-lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ →
- ∀L2.⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L,T2. ⦃G1,L1,T⦄ ⊐* ⦃G2,L,T2⦄ →
+ ∀L2.⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
-lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1, L1, T1⦄ ≥ [h] ⦃G3, L3, T3⦄ →
- ∀T4. ⦃G3, L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 →
- ∀G2,L4,T2. ⦃G3, L3, T5⦄ ⊐+ ⦃G2, L4, T2⦄ →
- ∀L2. ⦃G2, L4⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥ [h] ⦃G2, L2, T2⦄.
+lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1,L1,T1⦄ ≥ [h] ⦃G3,L3,T3⦄ →
+ ∀T4. ⦃G3,L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 →
+ ∀G2,L4,T2. ⦃G3,L3,T5⦄ ⊐+ ⦃G2,L4,T2⦄ →
+ ∀L2. ⦃G2,L4⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥ [h] ⦃G2,L2,T2⦄.
#h #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
@(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
@(fpbs_fqup_trans … H34) -G2 -L4 -T2
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: fpbs_intro_alt *)
-lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ →
- ∀L0. ⦃G, L⦄ ⊢ ⬈*[h] L0 →
- ∀G2,L2,T2. ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ .
+lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G,L,T0. ⦃G1,L1,T⦄ ⊐* ⦃G,L,T0⦄ →
+ ∀L0. ⦃G,L⦄ ⊢ ⬈*[h] L0 →
+ ∀G2,L2,T2. ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ .
/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_fdeq/ qed.
(* Advanced inversion lemmas *************************************************)
(* Basic_2A1: uses: fpbs_inv_alt *)
-lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ →
- ∃∃G,L,L0,T,T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄
- & ⦃G, L⦄ ⊢ ⬈*[h] L0 & ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄.
+lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∃∃G,L,L0,T,T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,L1,T⦄ ⊐* ⦃G,L,T0⦄
+ & ⦃G,L⦄ ⊢ ⬈*[h] L0 & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
[ /2 width=9 by ex4_5_intro/
| #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
inductive fsb (h): relation3 genv lenv term ≝
| fsb_intro: ∀G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → fsb h G2 L2 T2
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → fsb h G2 L2 T2
) → fsb h G1 L1 T1
.
(* Note: eliminator with shorter ground hypothesis *)
(* Note: to be named fsb_ind when fsb becomes a definition like csx, lfsx ***)
lemma fsb_ind_alt: ∀h. ∀Q: relation3 …. (
- ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → (
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2
) → Q G1 L1 T1
) →
- ∀G,L,T. ≥[h] 𝐒⦃G, L, T⦄ → Q G L T.
+ ∀G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → Q G L T.
#h #Q #IH #G #L #T #H elim H -G -L -T
/4 width=1 by fsb_intro/
qed-.
(* Main properties with atomic arity assignment for terms *******************)
(* Note: this is the "big tree" theorem *)
-theorem aaa_fsb: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ≥[h] 𝐒⦃G, L, T⦄.
+theorem aaa_fsb: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ≥[h] 𝐒⦃G,L,T⦄.
/3 width=2 by aaa_csx, csx_fsb/ qed.
(* Advanced eliminators with atomic arity assignment for terms **************)
fact aaa_ind_fpb_aux: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
#h #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
qed-.
lemma aaa_ind_fpb: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
/4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
fact aaa_ind_fpbg_aux: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
#h #Q #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
qed-.
lemma aaa_ind_fpbg: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
/4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-.
(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄.
+lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
#G0 #L0 #T0 #IHu #H10
]
qed.
-lemma csx_fsb: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G, L, T⦄.
+lemma csx_fsb: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G,L,T⦄.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.
(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fsb_fdeq_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄.
+lemma fsb_fdeq_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #_ #IH #G2 #L2 #T2 #H12
@fsb_intro #G #L #T #H2
(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄.
+lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
elim (fpbs_inv_fpbg … H12) -H12
(* Properties with proper parallel rst-computation for closures *************)
lemma fsb_intro_fpbg: ∀h,G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄
- ) → ≥[h] 𝐒⦃G1, L1, T1⦄.
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄
+ ) → ≥[h] 𝐒⦃G1,L1,T1⦄.
/4 width=1 by fsb_intro, fpb_fpbg/ qed.
(* Eliminators with proper parallel rst-computation for closures ************)
lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#h #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
@IH1 -IH1
qed-.
lemma fsb_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → Q G1 L1 T1.
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → Q G1 L1 T1.
#h #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H
/3 width=1 by/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
lemma fsb_fpbg_refl_false (h) (G) (L) (T):
- ≥[h] 𝐒⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G, L, T⦄ → ⊥.
+ ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
#h #G #L #T #H
@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
/2 width=5 by/
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lprs_pair_refl *)
-lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
+lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lprs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+lemma lprs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
-lemma lprs_refl (h) (G): ∀L. ⦃G, L⦄ ⊢ ➡*[h] L.
+lemma lprs_refl (h) (G): ∀L. ⦃G,L⦄ ⊢ ➡*[h] L.
/2 width=1 by lex_refl/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: lprs_inv_atom1 *)
-lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
+lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
(* Basic_2A1: was: lprs_inv_pair1 *)
lemma lprs_inv_pair_sn (h) (G):
- ∀I,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
+ ∀I,K1,L2,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: uses: lprs_inv_atom2 *)
-lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
+lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lprs_inv_pair2 *)
lemma lprs_inv_pair_dx (h) (G):
- ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
+ ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
lemma lprs_ind (h) (G): ∀Q:relation lenv.
Q (⋆) (⋆) → (
∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 →
Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
) → (
∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 →
Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.
(* Properties with t-bound context-sensitive rt-computarion for terms *******)
lemma lprs_cpms_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L2 #T1 #T2 #HT12 #L1 #H
@(lprs_ind_sn … H) -L1 /2 width=3 by lpr_cpms_trans/
qed-.
lemma lprs_cpm_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
/3 width=3 by lprs_cpms_trans, cpm_cpms/ qed-.
(* Basic_2A1: includes cprs_bind2 *)
lemma cpms_bind_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lprs_cpms_trans, lprs_pair, cpms_bind/ qed.
(* Inversion lemmas with t-bound context-sensitive rt-computarion for terms *)
(* Basic_2A1: includes: cprs_inv_abst1 *)
(* Basic_2A1: uses: scpds_inv_abst1 *)
lemma cpms_inv_abst_sn (n) (h) (G) (L):
- ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 &
+ ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n,h] X2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡*[n,h] T2 &
X2 = ⓛ{p}V2.T2.
#n #h #G #L #p #V1 #T1 #X2 #H
@(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/
(* Basic_2A1: includes: cprs_inv_abst *)
lemma cpms_inv_abst_bi (n) (h) (p1) (p2) (G) (L):
- ∀W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
- ∧∧ p1 = p2 & ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n,h] ⓛ{p2}W2.T2 →
+ ∧∧ p1 = p2 & ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #p1 #p2 #G #L #W1 #W2 #T1 #T2 #H
elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct
/2 width=1 by and3_intro/
(* Basic_1: was pr3_gen_abbr *)
(* Basic_2A1: includes: cprs_inv_abbr1 *)
lemma cpms_inv_abbr_sn_dx (n) (h) (G) (L):
- ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n, h] X2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T2 & X2 = ⓓ{p}V2.T2
- | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
+ ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓓ{p}V2.T2
+ | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
#n #h #G #L #p #V1 #T1 #X2 #H
@(cpms_ind_dx … H) -X2 -n /3 width=5 by ex3_2_intro, or_introl/
#n1 #n2 #X #X2 #_ * *
(* Basic_2A1: uses: scpds_inv_abbr_abst *)
lemma cpms_inv_abbr_abst (n) (h) (G) (L):
- ∀p1,p2,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
+ ∀p1,p2,V1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n,h] ⓛ{p2}W2.T2 →
+ ∃∃T. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
#n #h #G #L #p1 #p2 #V1 #W2 #T1 #T2 #H
elim (cpms_inv_abbr_sn_dx … H) -H *
[ #V #T #_ #_ #H destruct
(* Advanced properties ******************************************************)
(* Basic_2A1: was: lprs_pair2 *)
-lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
/3 width=3 by lprs_pair, lprs_cpms_trans/ qed.
(* Properties on context-sensitive parallel r-computation for terms *********)
-lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #H
@(lprs_ind_dx … H) -L1 /2 width=3 by ex2_intro/
#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
/3 width=5 by cprs_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
/3 width=3 by lprs_cprs_conf_dx, cpm_cpms/ qed-.
(* Note: this can be proved on its own using lprs_ind_sn *)
-lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
/3 width=3 by lprs_cpms_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
/3 width=3 by lprs_cprs_conf_sn, cpm_cpms/ qed-.
(* Properties with contextual transitive closure ****************************)
lemma lprs_CTC (h) (G):
- ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G, L1⦄⊢ ➡*[h] L2.
+ ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G,L1⦄⊢ ➡*[h] L2.
/3 width=3 by cprs_CTC, lex_co/ qed.
(* Inversion lemmas with contextual transitive closure **********************)
lemma lprs_inv_CTC (h) (G):
- ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2.
+ ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2.
/3 width=3 by cprs_inv_CTC, lex_co/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|.
+lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* Basic_2A1: was: lprs_ind_dx *)
lemma lprs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1.
+ (∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L → ⦃G,L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1.
/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
(* Basic_2A1: was: lprs_ind *)
lemma lprs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → Q L → Q L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L2.
+ (∀L,L2. ⦃G,L1⦄ ⊢ ➡*[h] L → ⦃G,L⦄ ⊢ ➡[h] L2 → Q L → Q L2) →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L2.
/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
(* Properties with unbound rt-transition for full local environments ********)
-lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_CTC, lpr_cprs_trans, lex_CTC_inj/ qed.
(* Basic_2A1: was: lprs_strap2 *)
-lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L →
- ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L →
+ ∀L2.⦃G,L⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_sn/ qed-.
(* Basic_2A1: was: lpxs_strap1 *)
-lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡*[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_dx/ qed-.
lemma lprs_strip (h) (G): confluent2 … (lprs h G) (lpr h G).
(* Basic_2A1: was: lprs_lpxs *)
(* Note: original proof uses lpr_fwd_lpx and monotonic_TC *)
-lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by cpms_fwd_cpxs, lex_co/ qed-.
(* Properties with transitive closure ***************************************)
lemma lprs_TC (h) (G):
- ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G, L1⦄⊢ ➡*[h] L2.
+ ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G,L1⦄⊢ ➡*[h] L2.
/4 width=3 by lprs_CTC, lex_CTC, lpr_cprs_trans/ qed.
(* Inversion lemmas with transitive closure *********************************)
lemma lprs_inv_TC (h) (G):
- ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2.
+ ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2.
/3 width=3 by lprs_inv_CTC, lex_inv_CTC/ qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lpxs_pair_refl *)
-lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
+lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
+lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpxs_inv_atom1 *)
-lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
+lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
+lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_pair1 *)
-lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_atom2 *)
-lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lpxs_inv_pair2 *)
-lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
lemma lpxs_ind (h) (G): ∀Q:relation lenv.
Q (⋆) (⋆) → (
∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
+ ⦃G,K1⦄ ⊢ ⬈*[h] K2 →
Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
) → (
∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,K1⦄ ⊢ ⬈*[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 →
Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.
(* Properties with context-sensitive extended rt-computation for terms ******)
(* Basic_2A1: was: cpxs_bind2 *)
-lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
(* Inversion lemmas with context-sensitive ext rt-computation for terms *****)
-lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
+lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓛ{p}V2.T2.
#h #G #p #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
(* Basic_2A1: was: cpxs_inv_abbr1 *)
lemma cpxs_inv_abbr1_dx (h) (p) (G) (L):
- ∀V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
+ ∀V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ.
+ | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ.
#h #p #G #L #V1 #T1 #U2 #H
@(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
#U0 #U2 #_ #HU02 * *
(* Properties with sort-irrelevant equivalence on closures ******************)
-lemma fdeq_lpxs_trans (h): ∀G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L0, T2⦄ →
- ∀L2. ⦃G2, L0⦄ ⊢⬈*[h] L2 →
- ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h] L & ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄.
+lemma fdeq_lpxs_trans (h): ∀G1,G2,L1,L0,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L0,T2⦄ →
+ ∀L2. ⦃G2,L0⦄ ⊢⬈*[h] L2 →
+ ∃∃L. ⦃G1,L1⦄ ⊢⬈*[h] L & ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
elim (fdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct
elim (rdeq_lpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02
(* Forward lemmas with length for local environments ************************)
-lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|.
+lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* Properties with unbound rt-transition for full local environments ********)
-lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_inj/ qed.
(* Basic_2A1: was: lpxs_strap2 *)
-lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ⬈*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ⬈*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_step_sn/ qed-.
(* Basic_2A1: was: lpxs_strap1 *)
-lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈*[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈*[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_step_dx/ qed-.
(* Eliminators with unbound rt-transition for full local environments *******)
(* Basic_2A1: was: lpxs_ind_dx *)
lemma lpxs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L → ⦃G, L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1.
+ (∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L → ⦃G,L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) →
+ ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1.
/3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_sn/ qed-.
(* Basic_2A1: was: lpxs_ind *)
lemma lpxs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L → ⦃G, L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L2.
+ (∀L,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L → ⦃G,L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L2.
/3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_dx/ qed-.
(* Properties with context-sensitive extended rt-transition for terms *******)
(* Advanced properties ******************************************************)
(* Basic_2A1: was: lpxs_pair2 *)
-lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
+lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
(* Basic_2A1: uses: lleq_lpxs_trans *)
lemma rdeq_lpxs_trans (h) (G) (T:term):
- ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 →
+ ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈*[h] K2 →
∀L1. L1 ≛[T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[T] K2.
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[T] K2.
#h #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
#L #L2 #HL2 #_ #IH #L1 #HT
elim (rdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT
(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
lemma lpxs_rdneq_inv_step_sn (h) (G) (T:term):
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) →
- ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[T] L → ⊥ &
- ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[T] L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) →
+ ∃∃L,L0. ⦃G,L1⦄ ⊢ ⬈[h] L & L1 ≛[T] L → ⊥ &
+ ⦃G,L⦄ ⊢ ⬈*[h] L0 & L0 ≛[T] L2.
#h #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
[ #H elim H -H //
| #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec L1 L T) #H
lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
| lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ →
+| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ →
lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
.
(* Basic inversion lemmas ***************************************************)
-fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
L1 = ⋆ → L2 = ⋆.
#h #g #G #L1 #L2 * -g -L1 -L2 //
[ #f #I #K1 #K2 #_ #H destruct
]
qed-.
-lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, g] L2 → L2 = ⋆.
+lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h,g] L2 → L2 = ⋆.
/2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
-fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #_ #H destruct
| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
]
qed-.
-lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, ⫯f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
+lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h,⫯f] L2 →
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
/2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
-fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #_ #H destruct
| #f #I #K1 #K2 #_ #g #J #L1 #H
]
qed-.
-lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, ↑f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h,↑f] L2 →
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
/2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
-fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
- ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+ ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #W #_ #H destruct
| #f #I #K1 #K2 #_ #g #J #L1 #W #H
qed-.
(* Basic_2A1: uses: lcosx_inv_pair *)
-lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, ↑f] L2 →
- ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,↑f] L2 →
+ ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
/2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, g] L2 →
- ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
- | ∃∃f,K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ↑f & L2 = K2.ⓧ.
+lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,g] L2 →
+ ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
+ | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ↑f & L2 = K2.ⓧ.
#h #g #I #G #K1 #L2 #V #H
elim (pn_split g) * #f #Hf destruct
[ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
(* Advanced forward lemmas **************************************************)
-lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, g] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, ⫱g] K2 & L2 = K2.ⓘ{I2}.
+lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h,g] L2 →
+ ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h,⫱g] K2 & L2 = K2.ⓘ{I2}.
#h #g #I1 #G #K1 #L2
elim (pn_split g) * #f #Hf destruct
[ #H elim (lsubsx_inv_push_sn … H) -H
(* Basic properties *********************************************************)
-lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
[ #f #I #L1 #L2 #_ #IH #x #H
elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
]
qed-.
-lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
#h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
qed-.
(* Main properties **********************************************************)
-theorem lsubsx_fix: ∀h,f,G,L1,L. G ⊢ L1 ⊆ⓧ[h, f] L →
- ∀L2. G ⊢ L ⊆ⓧ[h, f] L2 → L = L2.
+theorem lsubsx_fix: ∀h,f,G,L1,L. G ⊢ L1 ⊆ⓧ[h,f] L →
+ ∀L2. G ⊢ L ⊆ⓧ[h,f] L2 → L = L2.
#h #f #G #L1 #L #H elim H -f -L1 -L
[ #f #L2 #H
>(lsubsx_inv_atom_sn … H) -L2 //
(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
lemma rdsx_cpx_trans_lsubsx (h):
- ∀G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ⬈[h] T2 →
- ∀f,L. G ⊢ L0 ⊆ⓧ[h, f] L →
- G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
+ ∀G,L0,T1,T2. ⦃G,L0⦄ ⊢ T1 ⬈[h] T2 →
+ ∀f,L. G ⊢ L0 ⊆ⓧ[h,f] L →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
#h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2 //
[ #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #L #HK0 #HL
elim (lsubsx_inv_pair_sn_gen … HK0) -HK0 *
(* Basic_2A1: uses: lsx_cpx_trans_O *)
lemma rdsx_cpx_trans (h):
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
/3 width=6 by rdsx_cpx_trans_lsubsx, lsubsx_refl/ qed-.
lemma rdsx_cpxs_trans (h):
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
#h #G #L #T1 #T2 #H
@(cpxs_ind_dx ???????? H) -T1 //
/3 width=3 by rdsx_cpx_trans/
(* Basic_2A1: uses: lsx_ind *)
lemma rdsx_ind (h) (G) (T):
∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → Q L.
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
#h #G #T #Q #H0 #L1 #H elim H -L1
/5 width=1 by SN_intro/
qed-.
(* Basic_2A1: uses: lsx_intro *)
lemma rdsx_intro (h) (G) (T):
∀L1.
- (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) →
- G ⊢ ⬈*[h, T] 𝐒⦃L1⦄.
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
/5 width=1 by SN_intro/ qed.
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
lemma rdsx_fwd_pair_sn (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h, ②{I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, V] 𝐒⦃L⦄.
+ ∀I,L,V,T. G ⊢ ⬈*[h,②{I}V.T] 𝐒⦃L⦄ →
+ G ⊢ ⬈*[h,V] 𝐒⦃L⦄.
#h #G #I #L #V #T #H
@(rdsx_ind … H) -L #L1 #_ #IHL1
@rdsx_intro #L2 #HL12 #HnL12
(* Basic_2A1: uses: lsx_fwd_flat_dx *)
lemma rdsx_fwd_flat_dx (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, T] 𝐒⦃L⦄.
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+ G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
#h #G #I #L #V #T #H
@(rdsx_ind … H) -L #L1 #_ #IHL1
@rdsx_intro #L2 #HL12 #HnL12
qed-.
fact rdsx_fwd_pair_aux (h) (G):
- ∀L. G ⊢ ⬈*[h, #0] 𝐒⦃L⦄ →
- ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, V] 𝐒⦃K⦄.
+ ∀L. G ⊢ ⬈*[h,#0] 𝐒⦃L⦄ →
+ ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
#h #G #L #H
@(rdsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
/5 width=5 by lpx_pair, rdsx_intro, rdeq_fwd_zero_pair/
qed-.
lemma rdsx_fwd_pair (h) (G):
- ∀I,K,V. G ⊢ ⬈*[h, #0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h, V] 𝐒⦃K⦄.
+ ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
/2 width=4 by rdsx_fwd_pair_aux/ qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: lsx_inv_flat *)
lemma rdsx_inv_flat (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄ →
- ∧∧ G ⊢ ⬈*[h, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, T] 𝐒⦃L⦄.
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
/3 width=3 by rdsx_fwd_pair_sn, rdsx_fwd_flat_dx, conj/ qed-.
(* Basic_2A1: removed theorems 9:
(* Basic_2A1: uses: lsx_lref_be_lpxs *)
lemma rdsx_pair_lpxs (h) (G):
- ∀K1,V. ⦃G, K1⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
- ∀K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
- ∀I. G ⊢ ⬈*[h, #0] 𝐒⦃K2.ⓑ{I}V⦄.
+ ∀K1,V. ⦃G,K1⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
+ ∀K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → ⦃G,K1⦄ ⊢ ⬈*[h] K2 →
+ ∀I. G ⊢ ⬈*[h,#0] 𝐒⦃K2.ⓑ{I}V⦄.
#h #G #K1 #V #H
@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
@(rdsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
(* Basic_2A1: uses: lsx_lref_be *)
lemma rdsx_lref_pair_drops (h) (G):
- ∀K,V. ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h, V] 𝐒⦃K⦄ →
- ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄.
+ ∀K,V. ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄ →
+ ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
#h #G #K #V #HV #HK #I #i elim i -i
[ #L #H >(drops_fwd_isid … H) -H /2 width=3 by rdsx_pair_lpxs/
| #i #IH #L #H
(* Main properties **********************************************************)
(* Basic_2A1: uses: csx_lsx *)
-theorem csx_rdsx (h): ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → G ⊢ ⬈*[h, T] 𝐒⦃L⦄.
+theorem csx_rdsx (h): ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
#h #G #L #T @(fqup_wf_ind_eq (Ⓕ) … G L T) -G -L -T
#Z #Y #X #IH #G #L * * //
[ #i #HG #HL #HT #H destruct
(* Note: this uses length *)
(* Basic_2A1: uses: lsx_lift_le lsx_lift_ge *)
-lemma rdsx_lifts (h) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄).
+lemma rdsx_lifts (h) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄).
#h #G #K #T #H @(rdsx_ind … H) -K
#K1 #_ #IH #b #f #L1 #HLK1 #Hf #U #HTU @rdsx_intro
#L2 #HL12 #HnL12 elim (lpx_drops_conf … HLK1 … HL12)
(* Inversion lemmas on relocation *******************************************)
(* Basic_2A1: uses: lsx_inv_lift_le lsx_inv_lift_be lsx_inv_lift_ge *)
-lemma rdsx_inv_lifts (h) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄).
+lemma rdsx_inv_lifts (h) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄).
#h #G #L #U #H @(rdsx_ind … H) -L
#L1 #_ #IH #b #f #K1 #HLK1 #Hf #T #HTU @rdsx_intro
#K2 #HK12 #HnK12 elim (drops_lpx_trans … HLK1 … HK12) -HK12
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lsx_lref_free *)
-lemma rdsx_lref_atom (h) (G): ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄.
+lemma rdsx_lref_atom (h) (G): ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
#h #G #L1 #i #HL1
@(rdsx_lifts … (#0) … HL1) -HL1 //
qed.
(* Basic_2A1: uses: lsx_lref_skip *)
-lemma rdsx_lref_unit (h) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄.
+lemma rdsx_lref_unit (h) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
#h #G #I #L1 #K1 #i #HL1
@(rdsx_lifts … (#0) … HL1) -HL1 //
qed.
(* Basic_2A1: uses: lsx_fwd_lref_be *)
lemma rdsx_fwd_lref_pair (h) (G):
- ∀L,i. G ⊢ ⬈*[h, #i] 𝐒⦃L⦄ →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, V] 𝐒⦃K⦄.
+ ∀L,i. G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
#h #G #L #i #HL #I #K #V #HLK
lapply (rdsx_inv_lifts … HL … HLK … (#0) ?) -L
/2 width=2 by rdsx_fwd_pair/
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lsx_atom *)
-lemma lfsx_atom (h) (G) (T): G ⊢ ⬈*[h, T] 𝐒⦃⋆⦄.
+lemma lfsx_atom (h) (G) (T): G ⊢ ⬈*[h,T] 𝐒⦃⋆⦄.
#h #G #T
@rdsx_intro #Y #H #HnT
lapply (lpx_inv_atom_sn … H) -H #H destruct
(* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *)
(* Note: the old proof without the exclusion binder requires lreq *)
lemma rdsx_fwd_bind_dx (h) (G):
- ∀p,I,L,V,T. G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄.
+ ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ →
+ G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄.
#h #G #p #I #L #V #T #H
@(rdsx_ind … H) -L #L1 #_ #IH
@rdsx_intro #Y #H #HT
(* Basic_2A1: uses: lsx_inv_bind *)
lemma rdsx_inv_bind (h) (G):
- ∀p,I,L,V,T. G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
- ∧∧ G ⊢ ⬈*[h, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄.
+ ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄.
/3 width=4 by rdsx_fwd_pair_sn, rdsx_fwd_bind_dx, conj/ qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lsx_sort *)
-lemma rdsx_sort (h) (G): ∀L,s. G ⊢ ⬈*[h, ⋆s] 𝐒⦃L⦄.
+lemma rdsx_sort (h) (G): ∀L,s. G ⊢ ⬈*[h,⋆s] 𝐒⦃L⦄.
#h #G #L1 #s @rdsx_intro #L2 #H #Hs
elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_sort_length/
qed.
(* Basic_2A1: uses: lsx_gref *)
-lemma rdsx_gref (h) (G): ∀L,l. G ⊢ ⬈*[h, §l] 𝐒⦃L⦄.
+lemma rdsx_gref (h) (G): ∀L,l. G ⊢ ⬈*[h,§l] 𝐒⦃L⦄.
#h #G #L1 #s @rdsx_intro #L2 #H #Hs
elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_gref_length/
qed.
-lemma rdsx_unit (h) (G): ∀I,L. G ⊢ ⬈*[h, #0] 𝐒⦃L.ⓤ{I}⦄.
+lemma rdsx_unit (h) (G): ∀I,L. G ⊢ ⬈*[h,#0] 𝐒⦃L.ⓤ{I}⦄.
#h #G #I #L1 @rdsx_intro
#Y #HY #HnY elim HnY -HnY
elim (lpx_inv_unit_sn … HY) -HY #L2 #HL12 #H destruct
(* Basic_2A1: uses: lsx_intro_alt *)
lemma rdsx_intro_lpxs (h) (G):
- ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) →
- G ⊢ ⬈*[h, T] 𝐒⦃L1⦄.
+ ∀L1,T. (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
/4 width=1 by lpx_lpxs, rdsx_intro/ qed-.
(* Basic_2A1: uses: lsx_lpxs_trans *)
lemma rdsx_lpxs_trans (h) (G):
- ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄.
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
/2 width=3 by rdsx_lpx_trans/
qed-.
lemma rdsx_ind_lpxs_rdeq (h) (G):
∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- ∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+ ∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G,L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
#h #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
@IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
(* Basic_2A1: uses: lsx_ind_alt *)
lemma rdsx_ind_lpxs (h) (G):
∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → Q L.
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
#h #G #T #Q #IH #L #HL
@(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
fact rdsx_bind_lpxs_aux (h) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
(* Basic_2A1: uses: lsx_bind *)
lemma rdsx_bind (h) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓑ{I}V⦄ →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓑ{I}V⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_bind_lpxs_aux/ qed.
(* Basic_2A1: uses: lsx_flat_lpxs *)
lemma rdsx_flat_lpxs (h) (G):
- ∀I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀L2,T. G ⊢ ⬈*[h, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L2⦄.
+ ∀I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2,T. G ⊢ ⬈*[h,T] 𝐒⦃L2⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄.
#h #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2
#L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs
(* Basic_2A1: uses: lsx_flat *)
lemma rdsx_flat (h) (G):
- ∀I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄.
+ ∀I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_flat_lpxs/ qed.
fact rdsx_bind_lpxs_void_aux (h) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓧ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
qed-.
lemma rdsx_bind_void (h) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄ →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_bind_lpxs_void_aux/ qed.
(* Basic_2A1: uses: lsx_lleq_trans *)
lemma rdsx_rdeq_trans (h) (G):
- ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L2. L1 ≛[T] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄.
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. L1 ≛[T] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
#h #G #L1 #T #H @(rdsx_ind … H) -L1
#L1 #_ #IHL1 #L2 #HL12 @rdsx_intro
#L #HL2 #HnL2 elim (rdeq_lpx_trans … HL2 … HL12) -HL2
(* Basic_2A1: uses: lsx_lpx_trans *)
lemma rdsx_lpx_trans (h) (G):
- ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄.
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
#h #G #L1 #T #H @(rdsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
elim (rdeq_dec L1 L2 T) /3 width=4 by rdsx_rdeq_trans/
qed-.
(* CONTEXT-SENSITIVE PARALLEL R-CONVERSION FOR TERMS ************************)
definition cpc: sh → relation4 genv lenv term term ≝
- λh,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G, L⦄ ⊢ T2 ➡[h] T1.
+ λh,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G,L⦄ ⊢ T2 ➡[h] T1.
interpretation
"context-sensitive parallel r-conversion (term)"
(* Basic forward lemmas *****************************************************)
-lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h] T & ⦃G, L⦄ ⊢ T2 ➡[h] T.
+lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[h] T & ⦃G,L⦄ ⊢ T2 ➡[h] T.
#h #G #L #T1 #T2 * /2 width=3 by ex2_intro/
qed-.
(* Main properties **********************************************************)
-theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G, L⦄ ⊢ T0 ⬌[h] T1 → ⦃G, L⦄ ⊢ T0 ⬌[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ⬌[h] T & ⦃G, L⦄ ⊢ T2 ⬌[h] T.
+theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G,L⦄ ⊢ T0 ⬌[h] T1 → ⦃G,L⦄ ⊢ T0 ⬌[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ⬌[h] T & ⦃G,L⦄ ⊢ T2 ⬌[h] T.
/3 width=3 by cpc_sym, ex2_intro/ qed-.
lemma lpce_inv_unit_sn (h) (G):
∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
lemma lpce_inv_pair_sn (h) (G):
(* Basic_2A1: was: cpcs_ind_dx *)
lemma cpcs_ind_sn (h) (G) (L) (T2):
∀Q:predicate term. Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T → ⦃G,L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T1.
normalize /3 width=6 by TC_star_ind_dx/
qed-.
(* Basic_2A1: was: cpcs_ind *)
lemma cpcs_ind_dx (h) (G) (L) (T1):
∀Q:predicate term. Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → ⦃G,L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T2.
normalize /3 width=6 by TC_star_ind/
qed-.
/2 width=1 by cpc_sym/
qed-.
-lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/2 width=1 by inj/ qed.
(* Basic_2A1: was: cpcs_strap2 *)
-lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by TC_strap/
qed-.
(* Basic_2A1: was: cpcs_strap1 *)
-lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by step/
qed-.
(* Basic_1: was: pc3_pr2_r *)
-lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_introl/ qed.
(* Basic_1: was: pc3_pr2_x *)
-lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_intror/ qed.
(* Basic_1: was: pc3_pr2_u *)
(* Basic_2A1: was: cpcs_cpr_strap2 *)
-lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_introl/ qed-.
(* Basic_2A1: was: cpcs_cpr_strap1 *)
-lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_introl/ qed-.
-lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_intror/ qed-.
-lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpr_cpcs_dx, cpcs_step_dx, or_intror/ qed-.
(* Basic_1: was: pc3_pr2_u2 *)
-lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_intror/ qed-.
(* Basic_1: removed theorems 9:
(* Main inversion lemmas with atomic arity assignment on terms **************)
(* Note: lemma 1500 *)
-theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
+theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 →
A1 = A2.
#h #G #L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2
elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
(* Advanced properties ******************************************************)
-lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
+lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
-lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
+lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
(* Advanced properties with r-transition for full local environments ********)
(* Basic_1: was: pc3_wcpr0 *)
-lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
qed-.
(* Inversion lemmas with context sensitive r-computation on terms ***********)
-lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[h] T & ⦃G, L⦄ ⊢ T2 ➡*[h] T.
+lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[h] T & ⦃G,L⦄ ⊢ T2 ➡*[h] T.
#h #G #L #T1 #T2 #H @(cpcs_ind_dx … H) -T2
[ /3 width=3 by ex2_intro/
| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
(* Basic_1: was: pc3_gen_sort *)
(* Basic_2A1: was: cpcs_inv_sort *)
-lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2.
+lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G,L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2.
#h #G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H
#T #H1 >(cprs_inv_sort1 … H1) -T #H2
lapply (cprs_inv_sort1 … H2) -L #H destruct //
(* Basic_2A1: was: cpcs_inv_abst1 *)
lemma cpcs_inv_abst_sn (h) (G) (L):
- ∀p,W1,T1,X. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
+ ∀p,W1,T1,X. ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X →
+ ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
#h #G #L #p #W1 #T1 #T #H
elim (cpcs_inv_cprs … H) -H #X #H1 #H2
elim (cpms_inv_abst_sn … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
(* Basic_2A1: was: cpcs_inv_abst2 *)
lemma cpcs_inv_abst_dx (h) (G) (L):
- ∀p,W1,T1,X. ⦃G, L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
+ ∀p,W1,T1,X. ⦃G,L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 →
+ ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
/3 width=1 by cpcs_inv_abst_sn, cpcs_sym/ qed-.
(* Basic_1: was: pc3_gen_sort_abst *)
lemma cpcs_inv_sort_abst (h) (G) (L):
- ∀p,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥.
+ ∀p,W,T,s. ⦃G,L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥.
#h #G #L #p #W #T #s #H
elim (cpcs_inv_cprs … H) -H #X #H1
>(cprs_inv_sort1 … H1) -X #H2
(* Properties with context sensitive r-computation on terms *****************)
(* Basic_1: was: pc3_pr3_r *)
-lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2
/3 width=3 by cpcs_cpr_step_dx, cpcs_step_dx, cpc_cpcs/
qed.
(* Basic_1: was: pc3_pr3_x *)
-lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T2
/3 width=3 by cpcs_cpr_div, cpcs_step_sn, cpcs_cprs_dx/
qed.
(* Basic_2A1: was: cpcs_cprs_strap1 *)
-lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_step_dx/
qed-.
(* Basic_2A1: was: cpcs_cprs_strap2 *)
-lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_sn … H) -T1 /2 width=3 by cpcs_cpr_step_sn/
qed-.
-lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2 /2 width=3 by cpcs_cpr_div/
qed-.
(* Basic_1: was: pc3_pr3_conf *)
-lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_conf/
qed-.
(* Basic_1: was: pc3_pr3_t *)
(* Basic_1: note: pc3_pr3_t should be renamed *)
-lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2
/2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/
qed.
-lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=5 by cpm_cpms, cprs_div/ qed-.
-lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpm_cpms, cprs_div/ qed-.
-lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
/2 width=3 by cpr_cprs_div/
qed-.
-lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T2 ⬌*[h] T1.
+lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h] T1.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
/2 width=3 by cprs_cpr_div/
qed-.
-lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
+lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
#h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12
elim (cpcs_inv_cprs … HV12) -HV12
elim (cpcs_inv_cprs … HT12) -HT12
/3 width=5 by cprs_flat, cprs_div/
qed.
-lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V2 ➡[h] V1 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
+lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V2 ➡[h] V1 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
-lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2.
+lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2.
#h #G #L #I #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
/3 width=5 by cprs_div, cpms_bind/
qed.
-lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T.
+lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T.
#h #G #L #I #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
/3 width=5 by cprs_div, cpms_bind/
qed.
(* Properties with parallel r-computation for full local environments *******)
-lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/4 width=5 by cprs_div, lpr_cpms_trans/
qed-.
-lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2.
+lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/4 width=5 by cprs_div, lprs_cpms_trans/
qed-.
-lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was: pc3_wcpr0_t *)
(* Basic_1: note: pc3_wcpr0_t should be renamed *)
(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
-lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was only: pc3_pr0_pr2_t *)
(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
-lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
/3 width=5 by lpr_cprs_conf, cpm_cpms/ qed-.
(* Advanced inversion lemmas ************************************************)
(* Note: there must be a proof suitable for lfpr *)
-lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
+lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
+ ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
#h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #H
elim (cpcs_inv_cprs … H) -H #T #H1 #H2
elim (cpms_inv_abst_sn … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
qed-.
-lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
+lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
+ ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
#h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
qed-.
(* Basic_2A1: uses: scpes *)
definition cpes (h) (n1) (n2): relation4 genv lenv term term ≝
λG,L,T1,T2.
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G, L⦄ ⊢ T2 ➡*[n2,h] T.
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T.
interpretation "t-bound context-sensitive parallel rt-equivalence (term)"
'PConvStar h n1 n2 G L T1 T2 = (cpes h n1 n2 G L T1 T2).
(* Basic_2A1: uses: scpds_div *)
lemma cpms_div (h) (n1) (n2):
- ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1,h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[n2,h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2.
+ ∀G,L,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2.
/2 width=3 by ex2_intro/ qed.
lemma cpes_refl (h): ∀G,L. reflexive … (cpes h 0 0 G L).
(* Basic_2A1: uses: scpes_sym *)
lemma cpes_sym (h) (n1) (n2):
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → ⦃G, L⦄ ⊢ T2 ⬌*[h,n2,n1] T1.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h,n2,n1] T1.
#h #n1 #n2 #G #L #T1 #T2 * /2 width=3 by cpms_div/
qed-.
(* Basic_2A1: uses: scpes_refl *)
lemma cpes_refl_aaa (h) (n):
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ T ⬌*[h,n,n] T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ T ⬌*[h,n,n] T.
#h #n #G #L #T #A #HA
elim (cpms_total_aaa h … n … HA) #U #HTU
/2 width=3 by cpms_div/
(* Basic_2A1: uses: scpes_aaa_mono *)
theorem cpes_aaa_mono (h) (n1) (n2):
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 →
- ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 → A1 = A2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 →
+ ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 → A1 = A2.
#h #n1 #n2 #G #L #T1 #T2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
lapply (cpms_aaa_conf … HA1 … HT1) -T1 #HA1
lapply (cpms_aaa_conf … HA2 … HT2) -T2 #HA2
(* Basic inversion lemmas ***************************************************)
lemma cnr_inv_abst (h) (p) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}V.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L.ⓛV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}V.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓛV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
#h #p #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct //
(* Basic_2A1: was: cnr_inv_abbr *)
lemma cnr_inv_abbr_neg (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
#h #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct //
qed-.
(* Basic_2A1: was: cnr_inv_eps *)
-lemma cnr_inv_cast (h) (G) (L): ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓝV.T⦄ → ⊥.
+lemma cnr_inv_cast (h) (G) (L): ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓝV.T⦄ → ⊥.
#h #G #L #V #T #H lapply (H T ?) -H
/2 width=4 by cpm_eps, discr_tpair_xy_y/
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: nf2_sort *)
-lemma cnr_sort (h) (G) (L): ∀s. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃⋆s⦄.
+lemma cnr_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃⋆s⦄.
#h #G #L #s #X #H
>(cpr_inv_sort1 … H) //
qed.
-lemma cnr_gref (h) (G) (L): ∀l. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃§l⦄.
+lemma cnr_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃§l⦄.
#h #G #L #l #X #H
>(cpr_inv_gref1 … H) //
qed.
(* Basic_1: was: nf2_abst *)
lemma cnr_abst (h) (p) (G) (L):
- ∀W,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}W.T⦄.
+ ∀W,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}W.T⦄.
#h #p #G #L #W #T #HW #HT #X #H
elim (cpm_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
<(HW … HW0) -W0 <(HT … HT0) -T0 //
qed.
lemma cnr_abbr_neg (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G, L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄.
#h #G #L #V #T #HV #HT #X #H
elim (cpm_inv_abbr1 … H) -H *
[ #V0 #T0 #HV0 #HT0 #H destruct
(* Basic_1: was only: nf2_csort_lref *)
lemma cnr_lref_atom (h) (b) (G) (L):
- ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
#h #b #G #L #i #Hi #X #H
elim (cpr_inv_lref1_drops … H) -H // * #K #V1 #V2 #HLK
lapply (drops_gen b … HLK) -HLK #HLK
(* Basic_1: was: nf2_lref_abst *)
lemma cnr_lref_abst (h) (G) (L):
- ∀K,V,i. ⬇*[i] L ≘ K.ⓛV → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+ ∀K,V,i. ⬇*[i] L ≘ K.ⓛV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
#h #G #L #K #V #i #HLK #X #H
elim (cpr_inv_lref1_drops … H) -H // *
#K0 #V1 #V2 #HLK0 #_ #_
qed.
lemma cnr_lref_unit (h) (I) (G) (L):
- ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
#h #I #G #L #K #i #HLK #X #H
elim (cpr_inv_lref1_drops … H) -H // *
#K0 #V1 #V2 #HLK0 #_ #_
(* Basic_2A1: was: cnr_inv_delta *)
lemma cnr_inv_lref_abbr (h) (G) (L):
- ∀K,V,i. ⬇*[i] L ≘ K.ⓓV → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄ → ⊥.
+ ∀K,V,i. ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄ → ⊥.
#h #G #L #K #V #i #HLK #H
elim (lifts_total V 𝐔❴↑i❵) #W #HVW
lapply (H W ?) -H [ /3 width=6 by cpm_delta_drops/ ] -HLK #H destruct
(* Inversion lemmas with simple terms ***************************************)
lemma cnr_inv_appl (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
#h #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
(* Basic_1: was only: nf2_appl_lref *)
lemma cnr_appl_simple (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄.
#h #G #L #V #T #HV #HT #HS #X #H
elim (cpm_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
<(HV … HV0) -V0 <(HT … HT0) -T0 //
(* Basic_1: was: nf2_dec *)
(* Basic_2A1: uses: cnr_dec *)
lemma cnr_dec_tdeq (h) (G) (L):
- ∀T1. ∨∨ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T1⦄
- | ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥).
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄
+ | ∃∃T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥).
#h #G #L #T1
@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
[ #s #HG #HL #HT destruct -IH
(* Advanced properties ******************************************************)
lemma cnu_atom_drops (h) (b) (G) (L):
- ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
#h #b #G #L #i #Hi #n #X #H
elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #K #V1 #V2 #HLK
lapply (drops_gen b … HLK) -HLK #HLK
qed.
lemma cnu_unit_drops (h) (I) (G) (L):
- ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
#h #I #G #L #K #i #HLK #n #X #H
elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #Y #V1 #V2 #HLY
lapply (drops_mono … HLK … HLY) -L #H destruct
(* Properties with context-free sort-irrelevant equivalence for terms *******)
lemma cnu_dec_tdeq (h) (G) (L):
- ∀T1. ∨∨ ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
- | ∃∃n,T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
+ | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
#h #G #L #T1
@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
[ #s #HG #HL #HT destruct -IH
(* Basic inversion lemmas ***************************************************)
-lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
+lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓛV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
#h #p #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
| #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
qed-.
(* Basic_2A1: was: cnx_inv_abbr *)
-lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
+lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓓV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
#h #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
qed-.
(* Basic_2A1: was: cnx_inv_eps *)
-lemma cnx_inv_cast: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓝV.T⦄ → ⊥.
+lemma cnx_inv_cast: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓝV.T⦄ → ⊥.
#h #G #L #V #T #H lapply (H T ?) -H
/2 width=6 by cpx_eps, tdeq_inv_pair_xy_y/
qed-.
(* Basic properties *********************************************************)
-lemma cnx_sort: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃⋆s⦄.
+lemma cnx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃⋆s⦄.
#h #G #L #s #X #H elim (cpx_inv_sort1 … H) -H
/2 width=1 by tdeq_sort/
qed.
-lemma cnx_abst: ∀h,p,G,L,W,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}W.T⦄.
+lemma cnx_abst: ∀h,p,G,L,W,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}W.T⦄.
#h #p #G #L #W #T #HW #HT #X #H
elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
@tdeq_pair [ @HW | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
(* Advanced properties ******************************************************)
-lemma cnx_tdeq_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
- ∀T2. T1 ≛ T2 → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T2⦄.
+lemma cnx_tdeq_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+ ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T2⦄.
#h #G #L #T1 #HT1 #T2 #HT12 #T #HT2
elim (tdeq_cpx_trans … HT12 … HT2) -HT2 #T0 #HT10 #HT0
lapply (HT1 … HT10) -HT1 -HT10 /2 width=5 by tdeq_repl/ (**) (* full auto fails *)
(* Properties with generic slicing ******************************************)
-lemma cnx_lref_atom: ∀h,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
+lemma cnx_lref_atom: ∀h,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
#h #G #L #i #Hi #X #H elim (cpx_inv_lref1_drops … H) -H // *
#I #K #V1 #V2 #HLK lapply (drops_mono … Hi … HLK) -L #H destruct
qed.
-lemma cnx_lref_unit: ∀h,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
+lemma cnx_lref_unit: ∀h,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
#h #I #G #L #K #i #HLK #X #H elim (cpx_inv_lref1_drops … H) -H // *
#Z #Y #V1 #V2 #HLY lapply (drops_mono … HLK … HLY) -L #H destruct
qed.
(* Inversion lemmas with generic slicing ************************************)
(* Basic_2A1: was: cnx_inv_delta *)
-lemma cnx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄ → ⊥.
+lemma cnx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄ → ⊥.
#h #I #G #L #K #V #i #HLK #H
elim (lifts_total V (𝐔❴↑i❵)) #W #HVW
lapply (H W ?) -H /2 width=7 by cpx_delta_drops/ -HLK
(* Inversion lemmas with simple terms ***************************************)
-lemma cnx_inv_appl: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+lemma cnx_inv_appl: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄ →
+ ∧∧ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
#h #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2
#H elim (tdeq_inv_pair … H) -H //
(* Properties with simple terms *********************************************)
-lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄.
+lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄.
#h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
#V0 #T0 #HV0 #HT0 #H destruct
@tdeq_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
include "ground_2/steps/rtc_max.ma".
include "ground_2/steps/rtc_plus.ma".
include "basic_2/notation/relations/predty_7.ma".
-include "static_2/syntax/item_sh.ma".
+include "static_2/syntax/sort.ma".
include "static_2/syntax/lenv.ma".
include "static_2/syntax/genv.ma".
include "static_2/relocation/lifts.ma".
(* Basic properties *********************************************************)
(* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *)
-lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G, L⦄ ⊢ T ⬈[Rt, 𝟘𝟘, h] T.
+lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G,L⦄ ⊢ T ⬈[Rt,𝟘𝟘,h] T.
#Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/
* /2 width=1 by cpg_appl, cpg_cast/
qed.
(* Basic inversion lemmas ***************************************************)
-fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[Rt, c, h] T2 → ∀J. T1 = ⓪{J} →
+fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
| ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
[ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[Rt, c, h] T2 →
+lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[Rt,c,h] T2 →
∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
| ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
/2 width=3 by cpg_inv_atom1_aux/ qed-.
-lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[Rt, c, h] T2 →
+lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[Rt,c,h] T2 →
∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(next h s) ∧ c = 𝟘𝟙.
#Rt #c #h #G #L #T2 #s #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
]
qed-.
-lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[Rt, c, h] T2 →
+lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[Rt,c,h] T2 →
∨∨ T2 = #0 ∧ c = 𝟘𝟘
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & c = cV+𝟘𝟙.
#Rt #c #h #G #L #T2 #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[Rt, c, h] T2 →
+lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#Rt #c #h #G #L #T2 #i #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s #H destruct
]
qed-.
-lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[Rt, c, h] T2 → T2 = §l ∧ c = 𝟘𝟘.
+lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘.
#Rt #c #h #G #L #T2 #l #H
elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
[ #s #H destruct
]
qed-.
-fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓑ{p,J}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & J = Abbr & c = cT+𝟙𝟘.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #q #J #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓑ{p,I}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & I = Abbr & c = cT+𝟙𝟘.
/2 width=3 by cpg_inv_bind1_aux/ qed-.
-lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓓ{p}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & c = cT+𝟙𝟘.
#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
/3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
qed-.
-lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt, c, h] U2 →
- ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt,c,h] U2 →
+ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓛ{p}V2.T2 & c = ((↕*cV)∨cT).
#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
[ /3 width=8 by ex4_4_intro/
]
qed-.
-fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀V1,U1. U = ⓐV1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
- | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
- | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
/2 width=3 by cpg_inv_appl1_aux/ qed-.
-fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀V1,U1. U = ⓝV1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙.
+ | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #W #U1 #H destruct
| #G #L #s #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙.
+ | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
/2 width=3 by cpg_inv_cast1_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt, c, h] T2 →
+lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt,c,h] T2 →
∨∨ T2 = #0 ∧ c = 𝟘𝟘
- | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
I = Abbr & c = cV
- | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
I = Abst & c = cV+𝟘𝟙.
#Rt #c #h #I #G #K #V1 #T2 #H elim (cpg_inv_zero1 … H) -H /2 width=1 by or3_intro0/
* #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/
qed-.
-lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt, c, h] T2 →
+lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
- | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2.
+ | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2.
#Rt #c #h #I #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/
* #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt, c, h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] ⓑ{p,I}V2.T2 &
+lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt,c,h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#Rt #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/
(* Advanced properties ******************************************************)
-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.
+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
]
qed.
-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.
+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
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,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 ⬈[Rt, 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 ⬈[Rt, cV, h] V2 &
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 &
⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
#Rt #c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, 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 ⬈[Rt, 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 ⬈[Rt, 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.
#Rt #c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
(* Properties with simple terms *********************************************)
(* Note: the main property of simple terms *)
-lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[Rt, c, h] U → 𝐒⦃T1⦄ →
- ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[Rt,c,h] U → 𝐒⦃T1⦄ →
+ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U = ⓐV2.T2 & c = ((↕*cV)∨cT).
#Rt #c #h #G #L #V1 #T1 #U #H #HT1 elim (cpg_inv_appl1 … H) -H *
[ /2 width=8 by ex4_4_intro/
(* Basic_2A1: includes: cpr *)
definition cpm (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2.
+ λT1,T2. ∃∃c. 𝐑𝐓⦃n,c⦄ & ⦃G,L⦄ ⊢ T1 ⬈[eq_t,c,h] T2.
interpretation
"t-bound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
-lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s).
+lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(next h s).
/2 width=3 by cpg_ess, ex2_intro/ qed.
-lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2.
+lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
-lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[↑n, h] W2.
+lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_ell, ex2_intro, isrt_succ/
qed.
-lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
- ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡[n, h] U.
+lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T →
+ ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U.
#n #h #I #G #K #T #U #i *
/3 width=5 by cpg_lref, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_bind *)
lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2.
#n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/
qed.
lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
#n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/
qed.
lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
- ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n, h] ⓝU2.T2.
+ ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
#n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
/4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_zeta *)
lemma cpm_zeta (n) (h) (G) (L):
- ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[n,h] T2 →
- ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
+ ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[n,h] T2.
#n #h #G #L #T1 #T #HT1 #T2 *
/3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_eps *)
-lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
+lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ➡[n,h] T2.
#n #h #G #L #V #T1 #T2 *
/3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
qed.
-lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[↑n, h] V2.
+lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ➡[↑n,h] V2.
#n #h #G #L #V1 #V2 #T *
/3 width=3 by cpg_ee, isrt_succ, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_beta *)
lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
/6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_theta *)
lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
/6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
+lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 →
∨∨ T2 = ⓪{J} ∧ n = 0
| ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & n = ↑m
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
]
qed-.
-lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 →
+lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] T2 →
∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
#n #h #G #L #T2 #s * #c #Hc #H
elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
#H destruct /2 width=1 by conj/
qed-.
-lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 →
+lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
∨∨ T2 = #0 ∧ n = 0
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1
- | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & n = ↑m.
#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
]
qed-.
-lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 →
+lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
∨∨ T2 = #(↑i) ∧ n = 0
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
| #I #K #V2 #HV2 #HVT2 #H destruct
]
qed-.
-lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0.
+lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
#n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
qed-.
(* Basic_2A1: includes: cpr_inv_bind1 *)
-lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓑ{p,I}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 &
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 &
p = true & I = Abbr.
#n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
(* Basic_2A1: includes: cpr_inv_abbr1 *)
-lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & p = true.
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
(* Basic_2A1: includes: cpr_inv_abst1 *)
-lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓛ{p}V2.T2.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
(* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
]
qed-.
-lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
- | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ↑m.
+ | ⦃G,L⦄ ⊢ U1 ➡[n,h] U2
+ | ∃∃m. ⦃G,L⦄ ⊢ V1 ➡[m,h] U2 & n = ↑m.
#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
+lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n,h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex2_intro/
lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
(∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
(∀G,L,s. Q 1 G L (⋆s) (⋆(next h s))) →
- (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+ (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
- ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+ ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
- ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → Q n G K (#i) T →
+ ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
⬆*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
- ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[n, h] T2 →
+ ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
Q n G L T T2 → Q n G L (+ⓓV.T1) T2
- ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
- ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 →
+ ) → (∀n,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 →
Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
- ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → Q n G L T1 T2.
+ ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
* #c #HC #H generalize in match HC; -HC generalize in match n; -n
elim H -c -G -L -T1 -T2
(* Note: one of these U is the inferred type of T *)
lemma aaa_cpm_SO (h) (G) (L) (A):
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
#h #G #L #A #T #H elim H -G -L -T -A
[ /3 width=2 by ex_intro/
| * #G #L #V #B #_ * #V0 #HV0
(* Forward lemmas with unbound context-sensitive rt-transition for terms ****)
(* Basic_2A1: includes: cpr_cpx *)
-lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2.
+lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2.
#n #h #G #L #T1 #T2 * #c #Hc #H elim H -L -T1 -T2
/2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
qed-.
(* Basic_1: includes: pr2_delta1 *)
(* Basic_2A1: includes: cpr_delta *)
lemma cpm_delta_drops: ∀n,h,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2.
+ ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[n,h] W2.
#n #h #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_delta_drops, ex2_intro/
qed.
lemma cpm_ell_drops: ∀n,h,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[↑n, h] W2.
+ ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[↑n,h] W2.
#n #h #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[n, h] T2 →
+lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] T2 →
∨∨ T2 = ⓪{I} ∧ n = 0
| ∃∃s. T2 = ⋆(next h s) & I = Sort s & n = 1
- | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i
- | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
#n #h #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
]
qed-.
-lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 →
+lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[n,h] T2 →
∨∨ T2 = #i ∧ n = 0
- | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & n = ↑m.
#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
(* Advanced forward lemmas **************************************************)
-fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
∀n1,n2. n1+n2 = n →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2.
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
#n #h #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
[ #I #G #L #n1 #n2 #H
elim (plus_inv_O3 … H) -H #H1 #H2 destruct
]
qed-.
-lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n1+n2, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2.
+lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n1+n2,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
/2 width=3 by cpm_fwd_plus_aux/ qed-.
#n #h #G #L1 #T1 #T2 * /3 width=5 by lsubr_cpg_trans, ex2_intro/
qed-.
-lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ➡[n, h] T2 →
- ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2.
/4 width=4 by lsubr_cpm_trans, cpm_bind, lsubr_unit/ qed.
(* Properties with simple terms *********************************************)
(* Basic_2A1: includes: cpr_inv_appl1_simple *)
-lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 &
U = ⓐV2.T2.
#n #h #G #L #V1 #T1 #U * #c #Hc #H #HT1 elim (cpg_inv_appl1_simple … H HT1) -H -HT1
#cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct elim (isrt_inv_max … Hc) -Hc
(* Note: cpr_flat: does not hold in basic_1 *)
(* Basic_1: includes: pr2_thin_dx *)
lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
#h * /2 width=1 by cpm_cast, cpm_appl/
qed.
(* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
+lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
#h * /2 width=1 by cpm_bind, cpr_flat/
qed.
(* Basic inversion properties ***********************************************)
-lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
+lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[h] T2 →
∨∨ T2 = ⓪{J}
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
[2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
qed-.
(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
+lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H //
qed-.
-lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
+lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[h] T2 →
∨∨ T2 = #0
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1.
#h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
#n #K #V1 #V2 #_ #_ #_ #H destruct
qed-.
-lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[h] T2 →
+lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
qed-.
-lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
+lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
#h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
qed-.
(* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ➡[h] U2.
+ | ⦃G,L⦄ ⊢ U1 ➡[h] U2.
#h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
/2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
qed-.
-lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 &
U2 = ⓕ{I}V2.T2
- | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
+ | (⦃G,L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
#h * #G #L #V1 #U1 #U2 #H
lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
(∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
- (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
+ (∀G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
+ ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[h] T2 →
Q G L T T2 → Q G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
+ ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
- ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
@(insert_eq_0 … 0) #n #H
@(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ]
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: includes: cpr_inv_atom1 *)
-lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[h] T2 →
+lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[h] T2 →
∨∨ T2 = ⓪{I}
- | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+ | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i.
#h #I #G #L #T2 #H elim (cpm_inv_atom1_drops … H) -H *
[ /2 width=1 by or_introl/
(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
(* Basic_2A1: includes: cpr_inv_lref1 *)
-lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 →
+lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[h] T2 →
∨∨ T2 = #i
- | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 &
⬆*[↑i] V2 ≘ T2.
#h #G #L #T2 #i #H elim (cpm_inv_lref1_drops … H) -H *
[ /2 width=1 by or_introl/
lemma cpr_subst (h) (G) (L) (U1) (i):
∀K,V. ⬇*[i] L ≘ K.ⓓV →
- ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ➡[h] U2 & ⬆[i,1] T2 ≘ U2.
+ ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ➡[h] U2 & ⬆[i,1] T2 ≘ U2.
#h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
#G0 #L0 #U0 #IH #G #L * *
[ #s #HG #HL #HT #i #K #V #_ destruct -IH
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
definition cpx (h): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
+ λG,L,T1,T2. ∃c. ⦃G,L⦄ ⊢ T1 ⬈[eq_f,c,h] T2.
interpretation
"unbound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
+lemma cpx_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
/2 width=2 by cpg_ess, ex_intro/ qed.
-lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
+lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
#h * #G #K #V1 #V2 #W2 *
/3 width=4 by cpg_delta, cpg_ell, ex_intro/
qed.
-lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
- ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
+lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T →
+ ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
#h #I #G #K #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_bind, ex_intro/
qed.
lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=5 by cpg_appl, cpg_cast, ex_intro/
qed.
lemma cpx_zeta (h) (G) (L):
- ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 →
- ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
+ ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
#h #G #L #T1 #T #HT1 #T2 *
/3 width=4 by cpg_zeta, ex_intro/
qed.
-lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
+lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
#h #G #L #V #T1 #T2 *
/3 width=2 by cpg_eps, ex_intro/
qed.
(* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2.
+lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ⬈[h] V2.
#h #G #L #V1 #V2 #T *
/3 width=2 by cpg_ee, ex_intro/
qed.
lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
/3 width=2 by cpg_beta, ex_intro/
qed.
lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
/3 width=4 by cpg_theta, ex_intro/
qed.
(* Advanced properties ******************************************************)
-lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
+lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
#h * /2 width=2 by cpx_flat, cpx_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
+lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[h] T2 →
∨∨ T2 = ⓪{J}
| ∃∃s. T2 = ⋆(next h s) & J = Sort s
- | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1 & J = LRef 0
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
qed-.
-lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
+lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] T2 →
∨∨ T2 = ⋆s | T2 = ⋆(next h s).
#h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
/2 width=1 by or_introl, or_intror/
qed-.
-lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1.
#h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
/4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
+lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
qed-.
-lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓑ{p,I}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 &
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 &
p = true & I = Abbr.
#h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
/4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & p = true.
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 & p = true.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
/4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓛ{p}V2.T2.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
/4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ⬈[h] U2
- | ⦃G, L⦄ ⊢ V1 ⬈[h] U2.
+ | ⦃G,L⦄ ⊢ U1 ⬈[h] U2
+ | ⦃G,L⦄ ⊢ V1 ⬈[h] U2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
/4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2.
+ | ∃∃V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2.
#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
qed-.
-lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
+ | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓕ{I}V2.T2
- | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
- | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ | (⦃G,L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+ | (⦃G,L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓛ{p}W1.T1 &
U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
#h * #G #L #V1 #U1 #U2 #H
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex_intro/
lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
(∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
(∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
- (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+ (∀I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
+ ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
+ ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
Q G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
+ ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
- ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+ ) → (∀G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
Q G L (ⓝV1.T) V2
- ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
qed-.
(* Basic_2A1: was: cpx_delta *)
lemma cpx_delta_drops: ∀h,I,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈[h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈[h] W2.
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⬈[h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈[h] W2.
#h * #G #L #K #V #V2 #W2 #i #HLK *
/3 width=7 by cpg_ell_drops, cpg_delta_drops, ex_intro/
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: cpx_inv_atom1 *)
-lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] T2 →
+lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ⬈[h] T2 →
∨∨ T2 = ⓪{I}
| ∃∃s. T2 = ⋆(next h s) & I = Sort s
- | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
+ | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i.
#h #I #G #L #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H *
/4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_intro, ex_intro/
qed-.
(* Basic_2A1: was: cpx_inv_lref1 *)
-lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈[h] T2 →
+lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈[h] T2 →
T2 = #i ∨
- ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
+ ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 &
⬆*[↑i] V2 ≘ T2.
#h #G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
lemma cpx_subst (h) (G) (L) (U1) (i):
∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V →
- ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ⬈[h] U2 & ⬆[i,1] T2 ≘ U2.
+ ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ⬈[h] U2 & ⬆[i,1] T2 ≘ U2.
#h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
#G0 #L0 #U0 #IH #G #L * *
[ #s #HG #HL #HT #i #I #K #V #_ destruct -IH
(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fdeq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1, L1, T0⦄ ≛ ⦃G2, L2, T2⦄.
+lemma fdeq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ⬈[h] T2 →
+ ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
elim (rdeq_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02
(* Properties on supclosure *************************************************)
-lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/
[ #I #G #L2 #V2 #X2 #HVX2
]
qed-.
-lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
[ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
/3 width=3 by fqu_fqup, ex2_intro/
]
qed-.
-lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
[ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
#U2 #HVU2 @(ex3_intro … U2)
]
qed-.
-lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
[ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
]
qed-.
-lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
]
qed-.
-lemma fqus_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
[ #H12 elim (fqup_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
#h #G #L1 #T1 #T2 * /3 width=4 by lsubr_cpg_trans, ex_intro/
qed-.
-lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
- ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
+ ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
/4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit/ qed.
/3 width=6 by cpx_rex_conf/ qed-.
(* Basic_2A1: was just: cpx_lleq_conf_dx *)
-lemma cpx_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+lemma cpx_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2.
/4 width=5 by cpx_rdeq_conf_sn, rdeq_sym/ qed-.
qed-.
(*
(* Basic_2A1: was: cpx_lleq_conf *)
-lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L2 ≘[T1] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h] T2.
+lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L2 ≘[T1] L1 → ⦃G,L1⦄ ⊢ T1 ⬈[h] T2.
/3 width=3 by req_cpx_trans, req_sym/ qed-.
*)
(* Basic_2A1: was: cpx_lleq_conf_sn *)
/2 width=5 by cpx_rex_conf/ qed-.
(*
(* Basic_2A1: was: cpx_lleq_conf_dx *)
-lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
∀L1. L1 ≘[T1] L2 → L1 ≘[T2] L2.
/4 width=6 by cpx_req_conf_sn, req_sym/ qed-.
*)
(* Inversion lemmas with simple terms ***************************************)
-lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[h] T2 &
U = ⓐV2.T2.
#h #G #L #V1 #T1 #U * #c #H #HT1 elim (cpg_inv_appl1_simple … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
inductive fpb (h) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h G1 L1 T1 G2 L2 T2
-| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → fpb h G1 L1 T1 G1 L1 T2
-| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T1] L2 → ⊥) → fpb h G1 L1 T1 G1 L2 T1
+| fpb_fqu: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ → fpb h G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → fpb h G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T1] L2 → ⊥) → fpb h G1 L1 T1 G1 L2 T1
.
interpretation
(* Basic properties *********************************************************)
(* Basic_2A1: includes: cpr_fpb *)
-lemma cpm_fpb (n) (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≛ T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h] ⦃G, L, T2⦄.
+lemma cpm_fpb (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → (T1 ≛ T2 → ⊥) →
+ ⦃G,L,T1⦄ ≻[h] ⦃G,L,T2⦄.
/3 width=2 by fpb_cpx, cpm_fwd_cpx/ qed.
-lemma lpr_fpb (h) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → (L1 ≛[T] L2 → ⊥) →
- ⦃G, L1, T⦄ ≻[h] ⦃G, L2, T⦄.
+lemma lpr_fpb (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → (L1 ≛[T] L2 → ⊥) →
+ ⦃G,L1,T⦄ ≻[h] ⦃G,L2,T⦄.
/3 width=1 by fpb_lpx, lpr_fwd_lpx/ qed.
(* Properties with degree-based equivalence for closures ********************)
(* Basic_2A1: uses: fleq_fpb_trans *)
-lemma fdeq_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≛ ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≛ ⦃G2, L2, U2⦄.
+lemma fdeq_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≛ ⦃F2,K2,T2⦄ →
+ ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄.
#h #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
elim (tdeq_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
(* Inversion lemmas with degree-based equivalence for closures **************)
(* Basic_2A1: uses: fpb_inv_fleq *)
-lemma fpb_inv_fdeq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥.
+lemma fpb_inv_fdeq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥.
#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #H elim (fdeq_inv_gen_sn … H) -H
/3 width=11 by rdeq_fwd_length, fqu_inv_tdeq/
(* Properties with sort-irrelevant equivalence for local environments *******)
lemma tdeq_fpb_trans: ∀h,U2,U1. U2 ≛ U1 →
- ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h] ⦃G2, L2, T1⦄ →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∀G1,G2,L1,L2,T1. ⦃G1,L1,U1⦄ ≻[h] ⦃G2,L2,T1⦄ →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ≻[h] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#h #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
[ #G2 #L2 #T1 #H
elim (tdeq_fqu_trans … H … HU21) -H
(* Basic_2A1: was just: lleq_fpb_trans *)
lemma rdeq_fpb_trans: ∀h,F,K1,K2,T. K1 ≛[T] K2 →
- ∀G,L2,U. ⦃F, K2, T⦄ ≻[h] ⦃G, L2, U⦄ →
- ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h] ⦃G, L1, U0⦄ & U0 ≛ U & L1 ≛[U] L2.
+ ∀G,L2,U. ⦃F,K2,T⦄ ≻[h] ⦃G,L2,U⦄ →
+ ∃∃L1,U0. ⦃F,K1,T⦄ ≻[h] ⦃G,L1,U0⦄ & U0 ≛ U & L1 ≛[U] L2.
#h #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
[ #G #L2 #U #H2 elim (rdeq_fqu_trans … H2 … HT) -K2
/3 width=5 by fpb_fqu, ex3_2_intro/
(* Basic_2A1: includes: fleq_fpbq fpbq_lleq *)
inductive fpbq (h) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
-| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h G1 L1 T1 G1 L1 T2
-| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h G1 L1 T1 G1 L2 T1
-| fpbq_fdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → fpbq h G1 L1 T1 G1 L2 T1
+| fpbq_fdeq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
.
interpretation
/2 width=1 by fpbq_cpx/ qed.
(* Basic_2A1: includes: cpr_fpbq *)
-lemma cpm_fpbq (n) (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h] ⦃G, L, T2⦄.
+lemma cpm_fpbq (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L,T1⦄ ≽[h] ⦃G,L,T2⦄.
/3 width=2 by fpbq_cpx, cpm_fwd_cpx/ qed.
-lemma lpr_fpbq (h) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1, T⦄ ≽[h] ⦃G, L2, T⦄.
+lemma lpr_fpbq (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1,T⦄ ≽[h] ⦃G,L2,T⦄.
/3 width=1 by fpbq_lpx, lpr_fwd_lpx/ qed.
(* Basic_2A1: removed theorems 2:
(* Properties with atomic arity assignment for terms ************************)
-lemma fpbq_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma fpbq_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_fdeq_conf, aaa_fquq_conf, ex_intro/
qed-.
(* Properties with proper parallel rst-transition for closures **************)
-lemma fpb_fpbq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄.
+lemma fpb_fpbq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
qed.
(* Basic_2A1: fpb_fpbq_alt *)
-lemma fpb_fpbq_ffdneq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ →
- ∧∧ ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥).
+lemma fpb_fpbq_ffdneq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ∧∧ ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ & (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥).
/3 width=10 by fpb_fpbq, fpb_inv_fdeq, conj/ qed-.
(* Inversrion lemmas with proper parallel rst-transition for closures *******)
(* Basic_2A1: uses: fpbq_ind_alt *)
-lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ →
- ∨∨ ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄.
+lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
/3 width=1 by fpb_fqu, fdeq_intro_sn, or_intror, or_introl/
qed-.
(* Basic_2A1: fpbq_inv_fpb_alt *)
-lemma fpbq_ffdneq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄.
+lemma fpbq_ffdneq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥) → ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄.
#h #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 //
qed-.
(* Basic properties *********************************************************)
-lemma lpr_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
- ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ➡[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
+lemma lpr_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 →
+ ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ➡[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
/2 width=1 by lex_bind/ qed.
(* Note: lemma 250 *)
(* Advanced properties ******************************************************)
-lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
- ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
+lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 →
+ ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h] V2 →
- ∀I. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
+lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡[h] V2 →
+ ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpr_inv_atom1 *)
(* Basic_1: includes: wcpr0_gen_sort *)
-lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
+lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 &
L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpr_inv_atom2 *)
-lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
+lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
-lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 &
L1 = K1.ⓘ{I1}.
/2 width=1 by lex_inv_bind_dx/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
+lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
+ ∃∃K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
(* Basic_2A1: was: lpr_inv_pair1 *)
(* Basic_1: includes: wcpr0_gen_head *)
-lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 &
L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
-lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
- ∃∃K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
+lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
/2 width=1 by lex_inv_unit_dx/ qed-.
(* Basic_2A1: was: lpr_inv_pair2 *)
-lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 &
L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
-lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
+lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ➡[h] L2 & ⦃G,L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
/2 width=1 by lex_inv_pair/ qed-.
(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
(* Properties with extended structural successor for closures ***************)
-lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
]
qed-.
-lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
]
qed-.
-lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_bind_refl_dx, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
(* Note: does not hold in Basic_2A1 because it requires cpm *)
(* Note: L1 = K0.ⓛV0 and T1 = #0 require n = 1 *)
-lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
- ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ * #G #K #V #K1 #H
elim (lpr_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
(* Properties with extended optional structural successor for closures ******)
-lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
[ #H12 elim (fqu_lpr_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
- ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
[ #H12 elim (lpr_fqu_trans … H12 … HKL1) -L1 /3 width=7 by fqu_fquq, ex4_3_intro/
| * #H1 #H2 #H3 destruct /2 width=7 by ex4_3_intro/
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
-lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|.
+lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
definition IH_cpr_conf_lpr (h): relation3 genv lenv term ≝ λG,L,T.
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0.
+ ∀T1. ⦃G,L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G,L1⦄ ⊢ T1 ➡[h] T0 & ⦃G,L2⦄ ⊢ T2 ➡[h] T0.
(* Main properties with context-sensitive parallel reduction for terms ******)
fact cpr_conf_lpr_atom_atom (h):
- ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T.
+ ∀I,G,L1,L2. ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡[h] T.
/2 width=3 by cpr_refl, ex2_intro/ qed-.
fact cpr_conf_lpr_atom_delta (h):
∀G0,L0,i. (
- ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,#i⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V2. ⦃G0, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ #i ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V2. ⦃G0,K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ #i ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
fact cpr_conf_lpr_delta_delta (h):
∀G0,L0,i. (
- ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,#i⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V1. ⦃G0, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
+ ∀V1. ⦃G0,K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
- ∀V2. ⦃G0, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V2. ⦃G0,KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
lapply (drops_mono … H … HLK0) -H #H destruct
fact cpr_conf_lpr_bind_bind (h):
∀p,I,G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
#h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) //
fact cpr_conf_lpr_bind_zeta (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
- ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
+ ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01 [| /3 width=1 by drops_refl, drops_drop/ ] #T #HT1 #HT2
fact cpr_conf_lpr_zeta_zeta (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0, L0⦄ ⊢ T1 ➡[h] X1 →
- ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ X1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T.
+ ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0,L0⦄ ⊢ T1 ➡[h] X1 →
+ ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ X1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #T1 #HT10 #X1 #HTX1
#T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
lapply (lifts_inj … HT20 … HT10) -HT20 #H destruct
fact cpr_conf_lpr_flat_flat (h):
∀I,G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓕ{I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
#h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) //
fact cpr_conf_lpr_flat_eps (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1,T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
#T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
fact cpr_conf_lpr_eps_eps (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #T1 #HT01
#T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
fact cpr_conf_lpr_flat_beta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
*)
fact cpr_conf_lpr_flat_theta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
fact cpr_conf_lpr_beta_beta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
(* Basic_1: was: pr0_upsilon_upsilon *)
fact cpr_conf_lpr_theta_theta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
- ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
+ ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
(* Properties with context-sensitive parallel reduction for terms ***********)
-lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L1⦄ ⊢ T1 ➡[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01
/2 width=3 by ex2_intro/
qed-.
-lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L0⦄ ⊢ T1 ➡[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01
/2 width=3 by ex2_intro/
(* Forward lemmas with unbound parallel rt-transition for ref local envs ****)
(* Basic_2A1: was: lpr_lpx *)
-lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h] L2.
+lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h] L2.
/3 width=3 by cpm_fwd_cpx, lex_co/ qed-.
(* Basic properties *********************************************************)
-lemma lpx_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
+lemma lpx_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 →
+ ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
/2 width=1 by lex_bind/ qed.
lemma lpx_refl (h) (G): reflexive … (lpx h G).
(* Advanced properties ******************************************************)
-lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
+lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 →
+ ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpx_pair (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
- ∀I.⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
+lemma lpx_pair (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I.⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpx_inv_atom1 *)
-lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
+lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 &
L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpx_inv_atom2 *)
-lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
+lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G,L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
-lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 &
L1 = K1.ⓘ{I1}.
/2 width=1 by lex_inv_bind_dx/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
+lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
(* Basic_2A1: was: lpx_inv_pair1 *)
-lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 &
L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
-lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
- ∃∃K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
+lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
/2 width=1 by lex_inv_unit_dx/ qed-.
(* Basic_2A1: was: lpx_inv_pair2 *)
-lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 &
L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
-lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
+lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
/2 width=1 by lex_inv_pair/ qed-.
(* Note: lemma 500 *)
(* Basic_2A1: was: cpx_lpx_aaa_conf *)
-lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G,L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
[ #G #L1 #s #X #H
elim (cpx_inv_sort1 … H) -H #H destruct //
(* Properties with extended structural successor for closures ***************)
-lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #K #V #K1 #H
elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
]
qed-.
-lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
(* Properties with extended optional structural successor for closures ******)
-lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
[ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
[ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
(* Forward lemmas with free variables inclusion for restricted closures *****)
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
-lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄.
/3 width=4 by rpx_cpx_conf_fsge, lpx_rpx/ qed-.
(* Basic_2A1: uses: lpx_frees_trans *)
-lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄.
+lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T0⦄ ⊆ ⦃L0,T0⦄.
/2 width=4 by lpx_cpx_conf_fsge/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|.
+lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: uses: lleq_lpx_trans *)
-lemma rdeq_lpx_trans (h) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 →
+lemma rdeq_lpx_trans (h) (G): ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈[h] K2 →
∀L1. ∀T:term. L1 ≛[T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[T] K2.
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h] K1 & K1 ≛[T] K2.
#h #G #L2 #K2 #HLK2 #L1 #T #HL12
lapply (lpx_rpx … T HLK2) -HLK2 #HLK2
elim (rdeq_rpx_trans … HLK2 … HL12) -L2 #K #H #HK2
(* Basic properties ***********************************************************)
-lemma rpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
+lemma rpx_atom: ∀h,I,G. ⦃G,⋆⦄ ⊢ ⬈[h,⓪{I}] ⋆.
/2 width=1 by rex_atom/ qed.
lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s.
- ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2}.
/2 width=1 by rex_sort/ qed.
lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2.
- ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
+ ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 → ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2.
/2 width=1 by rex_pair/ qed.
lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i.
- ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2}.
/2 width=1 by rex_lref/ qed.
lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l.
- ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2}.
/2 width=1 by rex_gref/ qed.
lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
- ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
+ ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I1} →
+ ∀I2. ⦃G,L1⦄ ⊢ I ⬈[h] I2 →
+ ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I2}.
/2 width=2 by rex_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
+lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G,⋆⦄ ⊢ ⬈[h,T] Y2 → Y2 = ⋆.
/2 width=3 by rex_inv_atom_sn/ qed-.
-lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
+lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G,Y1⦄ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆.
/2 width=3 by rex_inv_atom_dx/ qed-.
-lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
+lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_sort/ qed-.
-lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
+lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_lref/ qed-.
-lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
+lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_gref/ qed-.
-lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
/2 width=2 by rex_inv_bind/ qed-.
-lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=2 by rex_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_sort_bind_sn/ qed-.
-lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_sort_bind_dx/ qed-.
-lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] Y2 →
+ ∃∃L2,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y2 = L2.ⓑ{I}V2.
/2 width=1 by rex_inv_zero_pair_sn/ qed-.
-lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y1 = L1.ⓑ{I}V1.
/2 width=1 by rex_inv_zero_pair_dx/ qed-.
-lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_lref_bind_sn/ qed-.
-lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_lref_bind_dx/ qed-.
-lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_gref_bind_sn/ qed-.
-lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_gref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+ ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,V] L2.
/2 width=3 by rex_fwd_pair_sn/ qed-.
lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
/2 width=2 by rex_fwd_bind_dx/ qed-.
lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=3 by rex_fwd_flat_dx/ qed-.
lemma rpx_refl: ∀h,G,T. reflexive … (rpx h G T).
/2 width=1 by rex_refl/ qed.
-lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
+lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T. ⦃G,L.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L.ⓑ{I}V2.
/2 width=1 by rex_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
-lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ.
/2 width=3 by rex_inv_bind_void/ qed-.
(* Advanced forward lemmas **************************************************)
lemma rpx_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ.
/2 width=4 by rex_fwd_bind_dx_void/ qed-.
(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
-(* Note: This invalidates rpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
-lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+(* Note: This invalidates rpxs_cpx_conf: "∀h, G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
+lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄.
#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
#G #L #T #IH #G0 #L0 * *
[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
lemma rpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (rpx h G).
/2 width=5 by cpx_rex_conf/ qed-.
-lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
+lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T1⦄.
/3 width=5 by rpx_cpx_conf, rpx_fsge_comp/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
+lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → |L1| = |L2|.
/2 width=3 by rex_fwd_length/ qed-.
(* Inversion lemmas with length for local environments **********************)
-lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
+lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
- ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+ | ∃∃I,L1,L2,V1,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 &
+ ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
| ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
/2 width=1 by rex_inv_zero_length/ qed-.
(* Properties with syntactic equivalence for referred local environments ****)
-lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=1 by req_fwd_rex/ qed.
(* Properties with unbound parallel rt-transition for full local envs *******)
-lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=1 by rex_lex/ qed.
(* Inversion lemmas with unbound parallel rt-transition for full local envs *)
-lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
+lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 →
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
/3 width=3 by rpx_fsge_comp, rex_inv_lex_req/ qed-.
(* Properties with sort-irrelevant equivalence for local environments *******)
-lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀I,T.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[V] L2.
+lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → ∀I,T.
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L & L ≛[V] L2.
/3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
-lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀I,V.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[T] L2.
+lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ∀I,V.
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L & L ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
-lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀p.
- ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
+lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2 → ∀p.
+ ∃∃L,V. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
-lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀p,I,V.
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2.
+lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G,K1.ⓧ⦄ ⊢ ⬈[h,T] L2 → ∀p,I,V.
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
lemma rpx_tdeq_conf: ∀h,G. s_r_confluent1 … cdeq (rpx h G).
/2 width=5 by tdeq_rex_conf/ qed-.
lemma rpx_tdeq_div: ∀h,T1,T2. T1 ≛ T2 →
- ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
+ ∀G,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,T2] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T1] L2.
/2 width=5 by tdeq_rex_div/ qed-.
lemma cpx_tdeq_conf_sex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq.
]
qed-.
-lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
∀T2. T0 ≛ T2 →
- ∃∃T. T1 ≛ T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
+ ∃∃T. T1 ≛ T & ⦃G,L⦄ ⊢ T2 ⬈[h] T.
#h #G #L #T0 #T1 #HT01 #T2 #HT02
elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02
/2 width=3 by rex_refl, ex2_intro/
qed-.
lemma tdeq_cpx_trans: ∀h,G,L,T2. ∀T0:term. T2 ≛ T0 →
- ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛ T1.
+ ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G,L⦄ ⊢ T2 ⬈[h] T & T ≛ T1.
#h #G #L #T2 #T0 #HT20 #T1 #HT01
elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
qed-.
(* Basic_2A1: uses: cpx_lleq_conf *)
-lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
∀L2. L0 ≛[T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T.
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T.
#h #G #L0 #T0 #T1 #HT01 #L2 #HL02
elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02
/2 width=3 by rex_refl, ex2_intro/
(* Basic_2A1: uses: lleq_cpx_trans *)
lemma rdeq_cpx_trans: ∀h,G,L2,L0,T0. L2 ≛[T0] L0 →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1.
+ ∀T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1.
#h #G #L2 #L0 #T0 #HL20 #T1 #HT01
elim (cpx_rdeq_conf … HT01 L2) -HT01
/3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
lemma rpx_rdeq_conf: ∀h,G,T. confluent2 … (rpx h G T) (rdeq T).
/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-.
-lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
+lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G,L2⦄ ⊢ ⬈[h,T] K2 →
∀L1. L1 ≛[T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[T] K2.
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h,T] K1 & K1 ≛[T] K2.
#h #G #T #L2 #K2 #HLK2 #L1 #HL12
elim (rpx_rdeq_conf … HLK2 L1)
/3 width=3 by rdeq_sym, ex2_intro/
(* Main properties **********************************************************)
-theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 →
- ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2 →
- ∀p. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L2.
+theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 →
+ ∀I,V2,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V2 →
+ ∀p. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L2.
/2 width=2 by rex_bind/ qed.
-theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀I,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2.
+theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 →
+ ∀I,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2.
/2 width=1 by rex_flat/ qed.
-theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀T. ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ →
- ∀p,I. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2.
+theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 →
+ ∀T. ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ →
+ ∀p,I. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2.
/2 width=1 by rex_bind_void/ qed.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( 𝐁❴ break term 46 l, break term 46 h ❵ )"
+notation "hvbox( 𝐁❴ term 46 l, break term 46 h ❵ )"
non associative with precedence 90
for @{ 'Basic $l $h }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox(f2 ~ \circ break f1)"
+notation "hvbox(f2 ~ \circ break f1)" (**)
right associative with precedence 60
for @{ 'CoCompose $f2 $f1 }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ⫱ * [ term 46 n ] term 46 T )"
+notation "hvbox( ⫱ * [ term 46 n ] break term 46 T )"
non associative with precedence 46
for @{ 'DropPreds $n $T }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ↑ * [ term 46 n ] term 70 T )"
+notation "hvbox( ↑ * [ term 46 n ] break term 70 T )"
non associative with precedence 70
for @{ 'UpArrowStar $n $T }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ⫯ * [ term 46 n ] term 46 T )"
+notation "hvbox( ⫯ * [ term 46 n ] break term 46 T )"
non associative with precedence 46
for @{ 'UpSpoonStar $n $T }.
rec definition pluss (cs:mr2) (i:nat) on cs ≝ match cs with
[ nil2 ⇒ ◊
-| cons2 l m cs ⇒ {l + i, m};pluss cs i
+| cons2 l m cs ⇒ {l + i,m};pluss cs i
].
interpretation "plus (multiple relocation with pairs)"
(* Basic properties *********************************************************)
-lemma pluss_SO2: ∀l,m,cs. ({l, m};cs) + 1 = {↑l, m};cs + 1.
+lemma pluss_SO2: ∀l,m,cs. ({l,m};cs) + 1 = {↑l,m};cs + 1.
normalize // qed.
(* Basic inversion lemmas ***************************************************)
#l #m #cs #H destruct
qed.
-lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l, m};cs2 →
- ∃∃cs1. cs1 + i = cs2 & cs = {l - i, m};cs1.
+lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l,m};cs2 →
+ ∃∃cs1. cs1 + i = cs2 & cs = {l - i,m};cs1.
#i #l #m #cs2 *
[ normalize #H destruct
| #l1 #m1 #cs1 whd in ⊢ (??%?→?); #H destruct
'RAt i1 f i2 = (at f i1 i2).
definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1.
- ∀jf,jg,j. @⦃jf, f2⦄ ≘ j → @⦃jg, g2⦄ ≘ j →
- ∃∃j0. @⦃j0, f1⦄ ≘ jf & @⦃j0, g1⦄ ≘ jg.
+ ∀jf,jg,j. @⦃jf,f2⦄ ≘ j → @⦃jg,g2⦄ ≘ j →
+ ∃∃j0. @⦃j0,f1⦄ ≘ jf & @⦃j0,g1⦄ ≘ jg.
(* Basic inversion lemmas ***************************************************)
-lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2.
+lemma at_inv_ppx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2.
#f #i1 #i2 * -f -i1 -i2 //
[ #f #i1 #i2 #_ #g #j1 #j2 #_ * #_ #x #H destruct
| #f #i1 #i2 #_ #g #j2 * #_ #x #_ #H elim (discr_push_next … H)
]
qed-.
-lemma at_inv_npx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
- ∃∃j2. @⦃j1, g⦄ ≘ j2 & ↑j2 = i2.
+lemma at_inv_npx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
+ ∃∃j2. @⦃j1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 #_ * #_ #x #x1 #H destruct
| #f #i1 #i2 #Hi #g #j1 #j2 * * * #x #x1 #H #Hf >(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xnx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ↑g = f →
- ∃∃j2. @⦃i1, g⦄ ≘ j2 & ↑j2 = i2.
+lemma at_inv_xnx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ↑g = f →
+ ∃∃j2. @⦃i1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
| #f #i1 #i2 #_ #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
(* Advanced inversion lemmas ************************************************)
-lemma at_inv_ppn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_ppn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g,j2. 0 = i1 → ⫯g = f → ↑j2 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j2 #H1 #H <(at_inv_ppx … Hf … H1 H) -f -g -i1 -i2
#H destruct
qed-.
-lemma at_inv_npp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_npp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g,j1. ↑j1 = i1 → ⫯g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j1 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_npn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
- ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1, g⦄ ≘ j2.
+lemma at_inv_npn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1,g⦄ ≘ j2.
#f #i1 #i2 #Hf #g #j1 #j2 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_xnp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_xnp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g. ↑g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_xnn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
- ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1, g⦄ ≘ j2.
+lemma at_inv_xnn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1,g⦄ ≘ j2.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_pxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f.
+lemma at_inv_pxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f.
#f elim (pn_split … f) * /2 width=2 by ex_intro/
#g #H #i1 #i2 #Hf #H1 #H2 cases (at_inv_xnp … Hf … H H2)
qed-.
-lemma at_inv_pxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 →
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_pxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 →
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split … f) *
#g #H #i1 #i2 #Hf #j2 #H1 #H2
[ elim (at_inv_ppn … Hf … H1 H H2)
]
qed-.
-lemma at_inv_nxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_nxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀j1. ↑j1 = i1 → 0 = i2 → ⊥.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j1 #H1 #H2
]
qed-.
-lemma at_inv_nxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
- (∃∃g. @⦃j1, g⦄ ≘ j2 & ⫯g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_nxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
+ (∃∃g. @⦃j1,g⦄ ≘ j2 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
/4 width=7 by at_inv_xnn, at_inv_npn, ex2_intro, or_intror, or_introl/
qed-.
(* Note: the following inversion lemmas must be checked *)
-lemma at_inv_xpx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f →
+lemma at_inv_xpx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f →
(0 = i1 ∧ 0 = i2) ∨
- ∃∃j1,j2. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
+ ∃∃j1,j2. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
#f * [2: #i1 ] #i2 #Hf #g #H
[ elim (at_inv_npx … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/
| >(at_inv_ppx … Hf … H) -f /3 width=1 by conj, or_introl/
]
qed-.
-lemma at_inv_xpp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1.
+lemma at_inv_xpp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1.
#f #i1 #i2 #Hf #g #H elim (at_inv_xpx … Hf … H) -f * //
#j1 #j2 #_ #_ * -i2 #H destruct
qed-.
-lemma at_inv_xpn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
- ∃∃j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1.
+lemma at_inv_xpn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
+ ∃∃j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xpx … Hf … H) -f *
[ #_ * -i2 #H destruct
| #x1 #x2 #Hg #H1 * -i2 #H destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i2 →
+lemma at_inv_xxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i2 →
∃∃g. 0 = i1 & ⫯g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #H2
]
qed-.
-lemma at_inv_xxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. ↑j2 = i2 →
- (∃∃g,j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_xxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. ↑j2 = i2 →
+ (∃∃g,j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j2 #H2
[ elim (at_inv_xpn … Hf … H H2) -i2 /3 width=5 by or_introl, ex3_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma at_increasing: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → i1 ≤ i2.
+lemma at_increasing: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → i1 ≤ i2.
#i2 elim i2 -i2
[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf //
| #i2 #IH * //
]
qed-.
-lemma at_increasing_strict: ∀g,i1,i2. @⦃i1, g⦄ ≘ i2 → ∀f. ↑f = g →
- i1 < i2 ∧ @⦃i1, f⦄ ≘ ↓i2.
+lemma at_increasing_strict: ∀g,i1,i2. @⦃i1,g⦄ ≘ i2 → ∀f. ↑f = g →
+ i1 < i2 ∧ @⦃i1,f⦄ ≘ ↓i2.
#g #i1 #i2 #Hg #f #H elim (at_inv_xnx … Hg … H) -Hg -H
/4 width=2 by conj, at_increasing, le_S_S/
qed-.
-lemma at_fwd_id_ex: ∀f,i. @⦃i, f⦄ ≘ i → ∃g. ⫯g = f.
+lemma at_fwd_id_ex: ∀f,i. @⦃i,f⦄ ≘ i → ∃g. ⫯g = f.
#f elim (pn_split f) * /2 width=2 by ex_intro/
#g #H #i #Hf elim (at_inv_xnx … Hf … H) -Hf -H
#j2 #Hg #H destruct lapply (at_increasing … Hg) -Hg
(* Basic properties *********************************************************)
-corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≘ i2).
+corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 #f1 #H1 cases H1 -f1 -i1 -i2
[ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /2 width=2 by at_refl/
| #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /3 width=7 by at_push/
]
qed-.
-lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1, f⦄ ≘ i2).
+lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 @eq_repl_sym /2 width=3 by at_eq_repl_back/
qed-.
-lemma at_le_ex: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀i1. i1 ≤ i2 →
- ∃∃j1. @⦃i1, f⦄ ≘ j1 & j1 ≤ j2.
+lemma at_le_ex: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀i1. i1 ≤ i2 →
+ ∃∃j1. @⦃i1,f⦄ ≘ j1 & j1 ≤ j2.
#j2 elim j2 -j2 [2: #j2 #IH ] #i2 #f #Hf
[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
#g [ #x2 ] #Hg [ #H2 ] #H0
]
qed-.
-lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≘ i2 → @⦃i1, f⦄ ≘ i1.
+lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2,f⦄ ≘ i2 → @⦃i1,f⦄ ≘ i1.
#i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ]
#f #Hf elim (at_fwd_id_ex … Hf) /4 width=7 by at_inv_npn, at_push, at_refl/
qed-.
(* Main properties **********************************************************)
-theorem at_monotonic: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀j1,i1. @⦃i1, f⦄ ≘ j1 →
+theorem at_monotonic: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀j1,i1. @⦃i1,f⦄ ≘ j1 →
i1 < i2 → j1 < j2.
#j2 elim j2 -j2
[ #i2 #f #H2f elim (at_inv_xxp … H2f) -H2f //
]
qed-.
-theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1, f⦄ ≘ j1 → ∀j2,i2. @⦃i2, f⦄ ≘ j2 →
+theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1,f⦄ ≘ j1 → ∀j2,i2. @⦃i2,f⦄ ≘ j2 →
j1 < j2 → i1 < i2.
#j1 elim j1 -j1
[ #i1 #f #H1f elim (at_inv_xxp … H1f) -H1f //
]
qed-.
-theorem at_mono: ∀f,i,i1. @⦃i, f⦄ ≘ i1 → ∀i2. @⦃i, f⦄ ≘ i2 → i2 = i1.
+theorem at_mono: ∀f,i,i1. @⦃i,f⦄ ≘ i1 → ∀i2. @⦃i,f⦄ ≘ i2 → i2 = i1.
#f #i #i1 #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_inv_monotonic, eq_sym/
qed-.
-theorem at_inj: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀i2. @⦃i2, f⦄ ≘ i → i1 = i2.
+theorem at_inj: ∀f,i1,i. @⦃i1,f⦄ ≘ i → ∀i2. @⦃i2,f⦄ ≘ i → i1 = i2.
#f #i1 #i #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/
qed-.
(* Properties on tls ********************************************************)
-lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≘ n → @⦃0, ⫱*[n]f⦄ ≘ 0.
+lemma at_pxx_tls: ∀n,f. @⦃0,f⦄ ≘ n → @⦃0,⫱*[n]f⦄ ≘ 0.
#n elim n -n //
#n #IH #f #Hf
cases (at_inv_pxn … Hf) -Hf [ |*: // ] #g #Hg #H0 destruct
<tls_xn /2 width=1 by/
qed.
-lemma at_tls: ∀i2,f. ⫯⫱*[↑i2]f ≡ ⫱*[i2]f → ∃i1. @⦃i1, f⦄ ≘ i2.
+lemma at_tls: ∀i2,f. ⫯⫱*[↑i2]f ≡ ⫱*[i2]f → ∃i1. @⦃i1,f⦄ ≘ i2.
#i2 elim i2 -i2
[ /4 width=4 by at_eq_repl_back, at_refl, ex_intro/
| #i2 #IH #f <tls_xn <tls_xn in ⊢ (??%→?); #H
(* Inversion lemmas with tls ************************************************)
-lemma at_inv_nxx: ∀n,g,i1,j2. @⦃↑i1, g⦄ ≘ j2 → @⦃0, g⦄ ≘ n →
- ∃∃i2. @⦃i1, ⫱*[↑n]g⦄ ≘ i2 & ↑(n+i2) = j2.
+lemma at_inv_nxx: ∀n,g,i1,j2. @⦃↑i1,g⦄ ≘ j2 → @⦃0,g⦄ ≘ n →
+ ∃∃i2. @⦃i1,⫱*[↑n]g⦄ ≘ i2 & ↑(n+i2) = j2.
#n elim n -n
[ #g #i1 #j2 #Hg #H
elim (at_inv_pxp … H) -H [ |*: // ] #f #H0
]
qed-.
-lemma at_inv_tls: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → ⫯⫱*[↑i2]f ≡ ⫱*[i2]f.
+lemma at_inv_tls: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → ⫯⫱*[↑i2]f ≡ ⫱*[i2]f.
#i2 elim i2 -i2
[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf // #g #H1 #H destruct
/2 width=1 by eq_refl/
(* Advanced inversion lemmas on isid ****************************************)
-lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≘ i.
+lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i,f⦄ ≘ i.
#i elim i -i
[ #f #H elim (isid_inv_gen … H) -H /2 width=2 by at_refl/
| #i #IH #f #H elim (isid_inv_gen … H) -H /3 width=7 by at_push/
]
qed.
-lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1, f⦄ ≘ i2 → i1 = i2.
+lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1,f⦄ ≘ i2 → i1 = i2.
/3 width=6 by isid_inv_at, at_mono/ qed-.
(* Advanced properties on isid **********************************************)
-corec lemma isid_at: ∀f. (∀i. @⦃i, f⦄ ≘ i) → 𝐈⦃f⦄.
+corec lemma isid_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈⦃f⦄.
#f #Hf lapply (Hf 0)
#H cases (at_fwd_id_ex … H) -H
#g #H @(isid_push … H) /3 width=7 by at_inv_npn/
(* Advanced properties on id ************************************************)
-lemma id_inv_at: ∀f. (∀i. @⦃i, f⦄ ≘ i) → 𝐈𝐝 ≡ f.
+lemma id_inv_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈𝐝 ≡ f.
/3 width=1 by isid_at, eq_id_inv_isid/ qed-.
-lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≘ i.
+lemma id_at: ∀i. @⦃i,𝐈𝐝⦄ ≘ i.
/2 width=1 by isid_inv_at/ qed.
(* Advanced forward lemmas on id ********************************************)
-lemma at_id_fwd: ∀i1,i2. @⦃i1, 𝐈𝐝⦄ ≘ i2 → i1 = i2.
+lemma at_id_fwd: ∀i1,i2. @⦃i1,𝐈𝐝⦄ ≘ i2 → i1 = i2.
/2 width=4 by at_mono/ qed.
(* Main properties on id ****************************************************)
(* RELOCATION MAP ***********************************************************)
-definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @⦃i, f⦄ ≘ j.
+definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @⦃i,f⦄ ≘ j.
interpretation "test for totality (rtmap)"
'IsTotal f = (istot f).
(* Main forward lemmas on at ************************************************)
corec theorem at_ext: ∀f1,f2. 𝐓⦃f1⦄ → 𝐓⦃f2⦄ →
- (∀i,i1,i2. @⦃i, f1⦄ ≘ i1 → @⦃i, f2⦄ ≘ i2 → i1 = i2) →
+ (∀i,i1,i2. @⦃i,f1⦄ ≘ i1 → @⦃i,f2⦄ ≘ i2 → i1 = i2) →
f1 ≡ f2.
#f1 cases (pn_split f1) * #g1 #H1
#f2 cases (pn_split f2) * #g2 #H2
(* Advanced properties on at ************************************************)
-lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1, f⦄ ≘ i2).
+lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1,f⦄ ≘ i2).
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
#j2 #Hf elim (eq_nat_dec i2 j2)
[ #H destruct /2 width=1 by or_introl/
]
qed-.
-lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1, f⦄ ≘ i2 → ⊥) →
- Decidable (∃i1. @⦃i1, f⦄ ≘ i2).
+lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1,f⦄ ≘ i2 → ⊥) →
+ Decidable (∃i1. @⦃i1,f⦄ ≘ i2).
#f #i2 #i #Hf elim i -i
[ #Ht @or_intror * /3 width=3 by at_increasing/
| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/
]
qed-.
-lemma is_at_dec: ∀f,i2. 𝐓⦃f⦄ → Decidable (∃i1. @⦃i1, f⦄ ≘ i2).
+lemma is_at_dec: ∀f,i2. 𝐓⦃f⦄ → Decidable (∃i1. @⦃i1,f⦄ ≘ i2).
#f #i2 #Hf @(is_at_dec_le ?? (↑i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on isid **********************************************)
-lemma isid_at_total: ∀f. 𝐓⦃f⦄ → (∀i1,i2. @⦃i1, f⦄ ≘ i2 → i1 = i2) → 𝐈⦃f⦄.
+lemma isid_at_total: ∀f. 𝐓⦃f⦄ → (∀i1,i2. @⦃i1,f⦄ ≘ i2 → i1 = i2) → 𝐈⦃f⦄.
#f #H1f #H2f @isid_at
#i lapply (H1f i) -H1f *
#j #Hf >(H2f … Hf) in ⊢ (???%); -H2f //
inductive eq_t: relation rtc ≝
| eq_t_intro: ∀ri1,ri2,rs1,rs2,ti,ts.
- eq_t (〈ri1, rs1, ti, ts〉) (〈ri2, rs2, ti, ts〉)
+ eq_t (〈ri1,rs1,ti,ts〉) (〈ri2,rs2,ti,ts〉)
.
(* Basic properties *********************************************************)
(* RT-TRANSITION COUNTER ****************************************************)
definition isrt: relation2 nat rtc ≝ λts,c.
- ∃∃ri,rs. 〈ri, rs, 0, ts〉 = c.
+ ∃∃ri,rs. 〈ri,rs,0,ts〉 = c.
interpretation "test for costrained rt-transition counter (rtc)"
'IsRedType ts c = (isrt ts c).
(* Basic properties *********************************************************)
-lemma isr_00: 𝐑𝐓⦃0, 𝟘𝟘⦄.
+lemma isr_00: 𝐑𝐓⦃0,𝟘𝟘⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isr_10: 𝐑𝐓⦃0, 𝟙𝟘⦄.
+lemma isr_10: 𝐑𝐓⦃0,𝟙𝟘⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isrt_01: 𝐑𝐓⦃1, 𝟘𝟙⦄.
+lemma isrt_01: 𝐑𝐓⦃1,𝟘𝟙⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n, c2⦄.
+lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 * #ri1 #rs1 #H destruct
#H elim (eq_t_inv_dx … H) -H /2 width=3 by ex1_2_intro/
qed-.
(* Basic inversion properties ***********************************************)
-lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n, 𝟘𝟘⦄ → 0 = n.
+lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n,𝟘𝟘⦄ → 0 = n.
#n * #ri #rs #H destruct //
qed-.
-lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n, 𝟙𝟘⦄ → 0 = n.
+lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n,𝟙𝟘⦄ → 0 = n.
#n * #ri #rs #H destruct //
qed-.
-lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n, 𝟘𝟙⦄ → 1 = n.
+lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n,𝟘𝟙⦄ → 1 = n.
#n * #ri #rs #H destruct //
qed-.
(* Main inversion properties ************************************************)
-theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1, c⦄ → 𝐑𝐓⦃n2, c⦄ → n1 = n2.
+theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1,c⦄ → 𝐑𝐓⦃n2,c⦄ → n1 = n2.
#n1 #n2 #c * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
qed-.
-theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → eq_t c1 c2.
+theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → eq_t c1 c2.
#n #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
qed-.
definition max (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
- mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉
+ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉
]
].
(* Basic properties *********************************************************)
lemma max_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
- 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 =
+ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 =
(〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉).
// qed.
(* Properties with test for constrained rt-transition counter ***************)
-lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1∨n2, c1∨c2⦄.
+lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1∨n2,c1∨c2⦄.
#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
/2 width=3 by ex1_2_intro/
qed.
-lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
/2 width=1 by isrt_max/ qed.
-lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
qed.
-lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
#n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/
qed.
(* Inversion properties with test for constrained rt-transition counter *****)
-lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ →
- ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & (n1 ∨ n2) = n.
+lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ →
+ ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & (n1 ∨ n2) = n.
#n #c1 #c2 * #ri #rs #H
elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
qed-.
-lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0, c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃0, c2⦄.
+lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0,c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃0,c2⦄.
#c1 #c2 #H
elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H
elim (max_inv_O3 … H) -H #H1 #H2 destruct
/2 width=1 by conj/
qed-.
-lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄.
#n #c1 #c2 #H #H2
elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
qed-.
-lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → eq_t c1 c2 →
- ∧∧ 𝐑𝐓⦃n, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → eq_t c1 c2 →
+ ∧∧ 𝐑𝐓⦃n,c1⦄ & 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 #H #Hc12
elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
(* Inversion lemmaswith shift ***********************************************)
-lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n, ↕*c1 ∨ c2⦄ →
- ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n,↕*c1 ∨ c2⦄ →
+ ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 #H
elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1
definition plus (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
- mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉
+ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉
]
].
(**) (* plus is not disambiguated parentheses *)
lemma plus_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
- 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 =
+ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉 =
(〈ri1,rs1,ti1,ts1〉) + (〈ri2,rs2,ti2,ts2〉).
// qed.
(* Properties with test for constrained rt-transition counter ***************)
-lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1+n2, c1+c2⦄.
+lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1+n2,c1+c2⦄.
#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
/2 width=3 by ex1_2_intro/
qed.
-lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄.
+lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
/2 width=1 by isrt_plus/ qed.
-lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄.
+lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by isrt_plus/
qed.
-lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n, c⦄ → 𝐑𝐓⦃↑n, c+𝟘𝟙⦄.
+lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n,c⦄ → 𝐑𝐓⦃↑n,c+𝟘𝟙⦄.
/2 width=1 by isrt_plus/ qed.
(* Inversion properties with test for constrained rt-transition counter *****)
-lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ →
- ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & n1 + n2 = n.
+lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ →
+ ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & n1 + n2 = n.
#n #c1 #c2 * #ri #rs #H
elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
elim (plus_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
qed-.
-lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄.
#n #c1 #c2 #H #H2
elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
qed-.
-lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃1, c2⦄ →
- ∃∃m. 𝐑𝐓⦃m, c1⦄ & n = ↑m.
+lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃1,c2⦄ →
+ ∃∃m. 𝐑𝐓⦃m,c1⦄ & n = ↑m.
#n #c1 #c2 #H #H2
elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct
(* RT-TRANSITION COUNTER ****************************************************)
definition shift (c:rtc): rtc ≝ match c with
-[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs, 0, ti∨ts, 0〉 ].
+[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs,0,ti∨ts,0〉 ].
interpretation "shift (rtc)"
'UpDownArrowStar c = (shift c).
(* Basic properties *********************************************************)
-lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs, 0, ti∨ts, 0〉 = ↕*〈ri, rs, ti, ts〉.
+lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs,0,ti∨ts,0〉 = ↕*〈ri,rs,ti,ts〉.
normalize //
qed.
(* Basic inversion properties ***********************************************)
-lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri, rs, ti, ts〉 = ↕*c →
+lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri,rs,ti,ts〉 = ↕*c →
∃∃ri0,rs0,ti0,ts0. (ri0∨rs0) = ri & 0 = rs & (ti0∨ts0) = ti & 0 = ts &
- 〈ri0, rs0, ti0, ts0〉 = c.
+ 〈ri0,rs0,ti0,ts0〉 = c.
#ri #rs #ti #ts * #ri0 #rs0 #ti0 #ts0 <shift_rew #H destruct
/2 width=7 by ex5_4_intro/
qed-.
(* Properties with test for costrained rt-transition counter ****************)
-lemma isr_shift: ∀c. 𝐑𝐓⦃0, c⦄ → 𝐑𝐓⦃0, ↕*c⦄.
+lemma isr_shift: ∀c. 𝐑𝐓⦃0,c⦄ → 𝐑𝐓⦃0,↕*c⦄.
#c * #ri #rs #H destruct /2 width=3 by ex1_2_intro/
qed.
(* Inversion properties with test for costrained rt-counter *****************)
-lemma isrt_inv_shift: ∀n,c. 𝐑𝐓⦃n, ↕*c⦄ → 𝐑𝐓⦃0, c⦄ ∧ 0 = n.
+lemma isrt_inv_shift: ∀n,c. 𝐑𝐓⦃n,↕*c⦄ → 𝐑𝐓⦃0,c⦄ ∧ 0 = n.
#n #c * #ri #rs #H
elim (shift_inv_dx … H) -H #rt0 #rs0 #ti0 #ts0 #_ #_ #H1 #H2 #H3
elim (max_inv_O3 … H1) -H1 /3 width=3 by ex1_2_intro, conj/
qed-.
-lemma isr_inv_shift: ∀c. 𝐑𝐓⦃0, ↕*c⦄ → 𝐑𝐓⦃0, c⦄.
+lemma isr_inv_shift: ∀c. 𝐑𝐓⦃0,↕*c⦄ → 𝐑𝐓⦃0,c⦄.
#c #H elim (isrt_inv_shift … H) -H //
qed-.
#!/bin/sh
-for SRC in `find grond_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
- sed "s!$1!$2!g" ${SRC} > ${SRC}.new
+for SRC in `find ground_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
+ sed "/$1/s!$2!$3!g" ${SRC} > ${SRC}.new
if [ ! -s ${SRC}.new ] || diff ${SRC} ${SRC}.new > /dev/null;
then rm -f ${SRC}.new;
else echo ${SRC}; mv -f ${SRC} ${SRC}.old; mv -f ${SRC}.new ${SRC};
--- /dev/null
+definition sh_N: sh ≝ mk_sh S ….
+// defined.
+
+axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
+
+axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
(* Basic properties *********************************************************)
-lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L →
- ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R,T] L →
+ ∀L2. L ⪤[R,T] L2 → L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
qed-.
-lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L →
- ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R,T] L →
+ ∀L2. L ⪤*[R,T] L2 → L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
qed-.
-lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
+lemma rexs_atom: ∀R,I. ⋆ ⪤*[R,⓪{I}] ⋆.
/2 width=1 by inj/ qed.
lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
+ L1 ⪤*[R,⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R,⋆s] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
/3 width=4 by rex_sort, rexs_step_dx, inj/
qed.
lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2,V. L1 ⪤*[R, V] L2 →
- L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V.
+ ∀I,L1,L2,V. L1 ⪤*[R,V] L2 →
+ L1.ⓑ{I}V ⪤*[R,#0] L2.ⓑ{I}V.
#R #HR #I #L1 #L2 #V #H elim H -L2
/3 width=5 by rex_pair, rexs_step_dx, inj/
qed.
-lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
+ L1.ⓤ{I} ⪤*[R,#0] L2.ⓤ{I}.
/3 width=3 by rex_unit, inj/ qed.
lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2.
+ L1 ⪤*[R,#i] L2 → L1.ⓑ{I}V1 ⪤*[R,#↑i] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
/3 width=4 by rex_lref, rexs_step_dx, inj/
qed.
lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
+ L1 ⪤*[R,§l] L2 → L1.ⓑ{I}V1 ⪤*[R,§l] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
/3 width=4 by rex_gref, rexs_step_dx, inj/
qed.
lemma rexs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+ ∀L1,L2,T. L1 ⪤*[R1,T] L2 → L1 ⪤*[R2,T] L2.
#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
/4 width=5 by rex_co, rexs_step_dx, inj/
qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆.
+lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R,⓪{I}] Y2 → Y2 = ⋆.
#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/
qed-.
(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R,⓪{I}] ⋆ → Y1 = ⋆.
#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
/3 width=3 by inj, rex_inv_atom_dx/
qed-.
-lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
+lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R,⋆s] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R,⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #s #H elim H -Y2
[ #Y2 #H elim (rex_inv_sort … H) -H *
]
qed-.
-lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
+lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R,§l] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R,§l] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #l #H elim H -Y2
[ #Y2 #H elim (rex_inv_gref … H) -H *
qed-.
lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/
]
qed-.
-lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
+lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1 ⪤*[R,T] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/
(* Advanced inversion lemmas ************************************************)
-lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R,⋆s] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R,⋆s] L2 & Y2 = L2.ⓘ{I2}.
#R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H *
[ #H destruct
| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R,⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R,⋆s] L2 & Y1 = L1.ⓘ{I1}.
#R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H *
[ #_ #H destruct
| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R,§l] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R,§l] L2 & Y2 = L2.ⓘ{I2}.
#R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H *
[ #H destruct
| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R,§l] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R,§l] L2 & Y1 = L1.ⓘ{I1}.
#R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H *
[ #_ #H destruct
| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
+lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R,②{I}V.T] L2 → L1 ⪤*[R,V] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
/3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/
qed-.
lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H //
qed-.
-lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 → L1 ⪤*[R,T] L2.
#R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H //
qed-.
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤*[R,T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤*[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤*[R,U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤*[R,T] K2 & ⬇*[b,f] L2 ≘ K2.
definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+ λR. ∀L1,L2,U. L1 ⪤*[R,U] L2 →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤*[R,T] K2.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
lemma rexs_pair_refl: ∀R. c_reflexive … R →
- ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
+ ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R,T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
/3 width=3 by rexs_step_dx, rex_pair_refl, inj/
qed.
lemma rexs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (sex cfull (cext2 R) f) L1 L2 →
- L1 ⪤*[R, T] L2.
+ L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #f #Hf #H elim H -L2
[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
/5 width=7 by sex_sdj, rexs_step_dx, sdj_isid_sn, inj, ex2_intro/
lemma rexs_ind_sn: ∀R. c_reflexive … R →
∀L1,T. ∀Q:predicate …. Q L1 →
- (∀L,L2. L1 ⪤*[R, T] L → L ⪤[R, T] L2 → Q L → Q L2) →
- ∀L2. L1 ⪤*[R, T] L2 → Q L2.
+ (∀L,L2. L1 ⪤*[R,T] L → L ⪤[R,T] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤*[R,T] L2 → Q L2.
#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by rex_refl/
qed-.
lemma rexs_ind_dx: ∀R. c_reflexive … R →
∀L2,T. ∀Q:predicate …. Q L2 →
- (∀L1,L. L1 ⪤[R, T] L → L ⪤*[R, T] L2 → Q L → Q L1) →
- ∀L1. L1 ⪤*[R, T] L2 → Q L1.
+ (∀L1,L. L1 ⪤[R,T] L → L ⪤*[R,T] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤*[R,T] L2 → Q L1.
#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by rex_refl/
qed-.
(* Advanced inversion lemmas ************************************************)
lemma rexs_inv_bind_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H @(rexs_ind_sn … HR … H) -L2
[ /3 width=1 by rexs_refl, conj/
| #L #L2 #_ #H * elim (rex_inv_bind_void … H) -H /3 width=3 by rexs_step_dx, conj/
(* Advanced forward lemmas **************************************************)
lemma rexs_fwd_bind_dx_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind_void … H) -H //
qed-.
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
-lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
+lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R,T] L2 → |L1| = |L2|.
#R #L1 #L2 #T #H elim H -L2
[ #L2 #HL12 >(rex_fwd_length … HL12) -HL12 //
| #L #L2 #_ #HL2 #IHL1
(* Properties with generic extension of a context sensitive relation ********)
lemma rexs_lex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R, T] L2.
+ ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R,T] L2.
#R #HR #L1 #L2 #T *
/5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/
qed.
lemma rexs_lex_req: ∀R. c_reflexive … R →
∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
- L1 ⪤*[R, T] L2.
+ L1 ⪤*[R,T] L2.
/3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed.
(* Inversion lemmas with generic extension of a context sensitive relation **)
rex_fsge_compatible R →
s_rs_transitive … R (λ_.lex R) →
req_transitive R →
- ∀L1,L2,T. L1 ⪤*[R, T] L2 →
+ ∀L1,L2,T. L1 ⪤*[R,T] L2 →
∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓓ T1 )"
+notation "hvbox( L. ⓓ break T1 )"
left associative with precedence 50
for @{ 'DxAbbr $L $T1 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓛ T1 )"
+notation "hvbox( L. ⓛ break T1 )"
left associative with precedence 51
for @{ 'DxAbst $L $T1 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓤ { term 46 I } )"
+notation "hvbox( L. ⓤ { break term 46 I } )"
non associative with precedence 47
for @{ 'DxBind1 $L $I }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓑ { term 46 I } break term 49 T1 )"
+notation "hvbox( L. ⓑ { break term 46 I } break term 49 T1 )"
non associative with precedence 48
for @{ 'DxBind2 $L $I $T1 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⫯[ term 46 h ] break term 46 s )"
+ non associative with precedence 46
+ for @{ 'UpSpoon $h $s }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermPlus $b $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermStar $b $G1 $L1 $T1 $G2 $L2 $T2 }.
'RDropStar b f L1 L2 = (drops b f L1 L2).
definition d_liftable1: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U. ⬆*[f] T ≘ U → R L U.
definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ →
∀U. ⬆*[f] T ≘ U → R L U.
definition d_deliftable1: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T. ⬆*[f] T ≘ U → R K T.
definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ →
∀T. ⬆*[f] T ≘ U → R K T.
definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U1. S f T1 U1 →
∃∃U2. S f T2 U2 & R L U1 U2.
definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T1. S f T1 U1 →
∃∃T2. S f T2 U2 & R K T1 T2.
definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U1. S f T1 U1 →
∀U2. S f T2 U2 → R L U1 U2.
definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T1. S f T1 U1 →
∀T2. S f T2 U2 → R K T1 T2.
definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≘ K2.
+ ∃∃K2. R f1 K1 K2 & ⬇*[b,f] L2 ≘ K2.
definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
λR. ∀f2,L1,L2. R f2 L1 L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & R f1 K1 K2.
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & R f1 K1 K2.
definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
∀f2. f ~⊚ f1 ≘ f2 →
- ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ ∃∃L2. R f2 L1 L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
(* Basic properties *********************************************************)
-lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆.
+lemma drops_atom_F: ∀f. ⬇*[Ⓕ,f] ⋆ ≘ ⋆.
#f @drops_atom #H destruct
qed.
-lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L2).
+lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b,f] L1 ≘ L2).
#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
[ /4 width=3 by drops_atom, isid_eq_repl_back/
| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
]
qed-.
-lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≘ L2).
+lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b,f] L1 ≘ L2).
#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_2A1: includes: drop_FT *)
-lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2.
+lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[b,f] L1 ≘ L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic_2A1: includes: drop_T *)
-lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_F: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ →
+fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → X = ⋆ →
Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
#b #f #X #Y * -f -X -Y
[ /3 width=1 by conj/
(* Basic_1: includes: drop_gen_sort *)
(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b,f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
/2 width=3 by drops_inv_atom1_aux/ qed-.
-fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
- ⬇*[b, g] K ≘ Y.
+fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
+ ⬇*[b,g] K ≘ Y.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J #K #H destruct
| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
(* Basic_1: includes: drop_gen_drop *)
(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b, ↑f] K.ⓘ{I} ≘ Y → ⬇*[b, f] K ≘ Y.
+lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b,↑f] K.ⓘ{I} ≘ Y → ⬇*[b,f] K ≘ Y.
/2 width=6 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
- ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
+ ∃∃I2,K2. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J1 #K1 #H destruct
| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_l *)
(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b, ⫯f] K1.ⓘ{I1} ≘ Y →
- ∃∃I2,K2. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b,⫯f] K1.ⓘ{I1} ≘ Y →
+ ∃∃I2,K2. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
/2 width=5 by drops_inv_skip1_aux/ qed-.
-fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
- ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
+fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
+ ∃∃I1,K1. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J2 #K2 #H destruct
| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_r *)
(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b, ⫯f] X ≘ K2.ⓘ{I2} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
+lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b,⫯f] X ≘ K2.ⓘ{I2} →
+ ∃∃I1,K1. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
/2 width=5 by drops_inv_skip2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K.
#b #f2 #X #Y #H elim H -f2 -X -Y
[ #f2 #Hf2 #J #K #H destruct
| #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
]
qed-.
-lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K.
/2 width=4 by drops_fwd_drop2_aux/ qed-.
(* Properties with test for identity ****************************************)
(* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≘ L.
+lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b,f] L ≘ L.
#b #L elim L -L /2 width=1 by drops_atom/
#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
/3 width=1 by drops_skip, liftsb_refl/
(* Basic_1: includes: drop_gen_refl *)
(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
+lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2 //
[ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
| /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
]
qed-.
-lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b, f] X ≘ K.
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} →
+ ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b,f] X ≘ K.
#b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
#g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
/3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
(* Forward lemmas with test for finite colength *****************************)
-lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐅⦃f⦄.
+lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐅⦃f⦄.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by isfin_next, isfin_push, isfin_isid/
qed-.
(* Properties with test for uniformity **************************************)
-lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≘ K.
+lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ,f] L ≘ K.
#f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
#f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
#L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
(* Inversion lemmas with test for uniformity ********************************)
-lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
(𝐈⦃f⦄ ∧ L1 = L2) ∨
- ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
+ ∃∃g,I,K. ⬇*[Ⓣ,g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
#f #L1 #L2 * -f -L1 -L2
[ /4 width=1 by or_introl, conj/
| /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 →
+lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b,f] K.ⓘ{I} ≘ L2 →
(𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
- ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≘ L2 & f = ↑g.
+ ∃∃g. 𝐔⦃g⦄ & ⬇*[b,g] K ≘ L2 & f = ↑g.
#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
[ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
<(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
qed-.
(* Basic_2A1: was: drop_inv_O1_pair2 *)
-lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
+lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K.ⓘ{I} →
(𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
- ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
+ ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b,g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
#b #f #I #K *
[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
]
qed-.
-lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{I} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,↑f] L1 ≘ K.ⓘ{I} →
+ ∃∃I1,K1. ⬇*[b,f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
#b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
[ #H elim (isid_inv_next … H) -H //
| /2 width=4 by ex2_2_intro/
]
qed-.
-fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
- ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}.
+fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ,f] L1 ≘ K.ⓘ{I}.
#f #L1 #L2 #H elim H -f -L1 -L2
[ #f #_ #_ #J #K #H destruct
| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
qed-.
(* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}.
/2 width=3 by drops_inv_TF_aux/ qed-.
(* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b,f] L ≘ K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_1: was: drop_S *)
(* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{I} → ⬇*[b, ↑f] X ≘ K.
+lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b,f] X ≘ K.ⓘ{I} → ⬇*[b,↑f] X ≘ K.
/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
(* Inversion lemmas with uniform relocations ********************************)
-lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ →
- ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
+lemma drops_inv_atom2: ∀b,L,f. ⬇*[b,f] L ≘ ⋆ →
+ ∃∃n,f1. ⬇*[b,𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
#b #L elim L -L
[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
(* Properties with uniform relocations **************************************)
-lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
#L elim L -L /2 width=1 by or_introl/
#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
qed-.
(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
- ∃∃L. ⬇*[b, f1] L1 ≘ L & ⬇*[b, f2] L ≘ L2.
+lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[b,f1] L1 ≘ L & ⬇*[b,f2] L ≘ L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2
[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
#H lapply (H0f H) -b
]
qed-.
-lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
- ∃∃L2. ⬇*[Ⓕ, f2] L ≘ L2 & ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b,f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
+ ∃∃L2. ⬇*[Ⓕ,f2] L ≘ L2 & ⬇*[Ⓕ,f] L1 ≘ L2.
#b #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
⬇*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
-lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≘ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≘ i →
- ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b, ⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b,f] L ≘ K0.ⓘ{I} → ∀i. @⦃O,f⦄ ≘ i →
+ ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b,⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
#b #f #I #L #K0 #H #i #Hf
elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
lapply (drops_tls_at … Hf … H) -H #H
(* Main properties **********************************************************)
(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 →
- ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2.
+theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f,L2. ⬇*[b2,f] L1 ≘ L2 →
+ ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2,f2] L ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
#H #Hf destruct @drops_atom
(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
drops_drop_trans
*)
-theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 →
- ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2.
+theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f2,L2. ⬇*[b2,f2] L ≘ L2 →
+ ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2,f] L1 ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
#H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
(* Advanced properties ******************************************************)
(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 →
- ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2.
+lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1,f] L ≘ L1 →
+ ∀b2,L2. ⬇*[b2,f] L ≘ L2 → L1 = L2.
#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
/3 width=8 by drops_conf, drops_fwd_isid/
qed-.
-lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
+lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
#L #i #H1 #I #K #H2
lapply (drops_F … H2) -H2 #H2
lapply (drops_mono … H2 … H1) -L -i #H destruct
qed-.
(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 →
- ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} →
+lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2,f] L ≘ L2 →
+ ∀b1,f1,I1,K1. ⬇*[b1,f1] L ≘ K1.ⓘ{I1} →
∀f2. f1 ⊚ ⫯f2 ≘ f →
∃∃I2,K2. L2 = K2.ⓘ{I2} &
- ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
+ ⬇*[b2,f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} →
+lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f2,I2,K2. ⬇*[b2,f2] L ≘ K2.ⓘ{I2} →
∀f. f1 ⊚ f2 ≘ ⫯f →
∃∃I1,K1. L1 = K1.ⓘ{I1} &
- ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
+ ⬇*[b1∧b2,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
lapply (drops_trans … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
(* Basic_2A1: includes: drops_conf_div *)
lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
- ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
+ ⬇*[Ⓣ,f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ,f2] L ≘ K.ⓘ{I2} →
𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: includes: drop_fwd_length_le4 *)
-lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → |L2| ≤ |L1|.
+lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → |L2| ≤ |L1|.
#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq1 *)
-theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1, f] L1 ≘ K1 →
- ∀L2,K2. ⬇*[b2, f] L2 ≘ K2 →
+theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1,f] L1 ≘ K1 →
+ ∀L2,K2. ⬇*[b2,f] L2 ≘ K2 →
|L1| = |L2| → |K1| = |K2|.
#b1 #b2 #f #L1 #K1 #HLK1 elim HLK1 -f -L1 -K1
[ #f #_ #L2 #K2 #HLK2 #H lapply (length_inv_zero_sn … H) -H
(* forward lemmas with finite colength assignment ***************************)
-lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
∃∃n. 𝐂⦃f⦄ ≘ n & |L1| = |L2| + n.
#f #L1 #L2 #H elim H -f -L1 -L2
[ /4 width=3 by fcla_isid, ex2_intro/
qed-.
(* Basic_2A1: includes: drop_fwd_length *)
-lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
|L1| = |L2| + n.
#f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
-lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
∃∃n. 𝐂⦃f⦄ ≘ n & n ≤ |L1|.
#f #L1 #L2 #H elim (drops_fwd_fcla … H) -H /2 width=3 by ex2_intro/
qed-.
(* Basic_2A1: includes: drop_fwd_length_le2 *)
-lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
n ≤ |L1|.
#f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
-lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} →
+lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} →
∃∃n. 𝐂⦃f⦄ ≘ n & n < |L1|.
#f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H
#n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/
(* Basic_2A1: includes: drop_fwd_length_lt2 *)
lemma drops_fcla_fwd_lt2: ∀f,L1,I2,K2,n.
- ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
+ ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
n < |L1|.
#f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
(* Basic_2A1: includes: drop_fwd_length_lt4 *)
-lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
+lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
|L2| < |L1|.
#f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f
/2 width=1 by lt_minus_to_plus_r/ qed-.
(* Basic_2A1: includes: drop_inv_length_eq *)
-lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
+lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
#f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H
#n #Hn <HL12 -L2 #H lapply (discr_plus_x_xy … H) -H
/2 width=3 by fcla_inv_xp/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq2 *)
-theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ, f] L1 ≘ K1 → ⬇*[Ⓣ, f] L2 ≘ K2 →
+theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ,f] L1 ≘ K1 → ⬇*[Ⓣ,f] L2 ≘ K2 →
|K1| = |K2| → |L1| = |L2|.
#f #L1 #L2 #K1 #K2 #HLK1 #HLK2 #HL12
elim (drops_fwd_fcla … HLK1) -HLK1 #n1 #Hn1 #H1 >H1 -L1
<(fcla_mono … Hn2 … Hn1) -f //
qed-.
-theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 →
+theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 →
∃∃n. 𝐂⦃f1⦄ ≘ n & 𝐂⦃f2⦄ ≘ n.
#f1 #f2 #L1 #L2 #H1 #H2
elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1
qed-.
theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2.
- ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
+ ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
n1 = n2.
#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2
lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
definition dedropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
- ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
+ ∃∃L2. L1 ⪤[R] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition dropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
- ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & ⬇*[b,f] L2 ≘ K2.
definition dropable_dx: predicate … ≝
- λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
+ λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R] K2.
(* Properties with generic extension ****************************************)
(* Basic_2A1: includes: lpx_sn_drop_conf *)
lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
- ∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
+ ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
+ ∃∃K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf
elim (sex_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
[ #Z2 #K2 #HLK2 #HK12 #H
(* Basic_2A1: includes: lpx_sn_drop_trans *)
lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
- ∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
+ ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
+ ∃∃K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf
elim (sex_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
[ #Z1 #K1 #HLK1 #HK12 #H
(* Basic_2A1: includes: lreq_drop_trans_be *)
lemma seq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀b,f,I,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
#f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf
#I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2
(* Basic_2A1: includes: lreq_drop_conf_be *)
lemma seq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀b,f,I,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
+ ∃∃K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
#f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (seq_drops_trans_next … (seq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
/3 width=3 by seq_sym, ex2_intro/
qed-.
lemma drops_seq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
- ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 →
+ ∀b,f,I,L1. ⬇*[b,f] L1.ⓘ{I} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+ ∃∃L2. ⬇*[b,f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
#f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (drops_sex_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
/2 width=6 by cfull_lift_sn, ceq_lift_sn/
qed-.
lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f2,L1,L2. L1 ⪤[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- f ~⊚ f1 ≘ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
]
qed-.
-fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∀f2,L1. L1 ⪤[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[RN, RP, f1] K2.
+fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H
#H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/
/2 width=5 by sex_dropable_dx_aux/ qed-.
lemma sex_drops_conf_next: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
qed-.
lemma sex_drops_conf_push: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ⫯f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
-lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
- ∀f,L2. L1 ⪤[RN, RP, f] L2 →
- ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆.
+lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b,f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
+ ∀f,L2. L1 ⪤[RN,RP,f] L2 →
+ ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b,f1] L2 ≘ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
#L #H #HL2 lapply (sex_inv_atom1 … H) -H //
definition d_liftable1_all: predicate (relation2 lenv term) ≝
λR. ∀K,Ts. all … (R K) Ts →
- ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀b,f,L. ⬇*[b,f] L ≘ K →
∀Us. ⬆*[f] Ts ≘ Us → all … (R L) Us.
(* Properties with generic relocation for term vectors **********************)
(* Forward lemmas with weight for local environments ************************)
(* Basic_2A1: includes: drop_fwd_lw *)
-lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
+lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
#b #f #L1 #L2 #H elim H -f -L1 -L2 //
[ /2 width=3 by transitive_le/
| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 normalize
qed-.
(* Basic_2A1: includes: drop_fwd_lw_lt *)
-lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
(𝐈⦃f⦄ → ⊥) → ♯{L2} < ♯{L1}.
#f #L1 #L2 #H elim H -f -L1 -L2
[ #f #Hf #Hnf elim Hnf -Hnf /2 width=1 by/
(* Forward lemmas with restricted weight for closures ***********************)
(* Basic_2A1: includes: drop_fwd_rfw *)
-lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b, f] L ≘ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b,f] L ≘ K.ⓑ{I}V → ∀T. ♯{K,V} < ♯{L,T}.
#b #f #I #L #K #V #HLK lapply (drops_fwd_lw … HLK) -HLK
normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt, monotonic_lt_plus_r/
qed-.
(* Advanced inversion lemma *************************************************)
-lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b, f] L ≘ L.ⓘ{I} → ⊥.
+lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b,f] L ≘ L.ⓘ{I} → ⊥.
#b #f #I #L #H lapply (drops_fwd_lw … H) -b -f
/2 width=4 by lt_le_false/ (**) (* full auto is a bit slow: 19s *)
qed-.
(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
definition lex (R): relation lenv ≝
- λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull, cext2 R, f] L2.
+ λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull,cext2 R,f] L2.
interpretation "generic extension (local environment)"
'Relation R L1 L2 = (lex R L1 L2).
*)
inductive lifts: rtmap → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → lifts f (#i1) (#i2)
+| lifts_lref: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → lifts f (#i1) (#i2)
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts (⫯f) T1 T2 →
/2 width=4 by lifts_inv_sort1_aux/ qed-.
fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
∨∨ ∃∃s. I = Sort s & Y = ⋆s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j
+ | ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j
| ∃∃l. I = GRef l & Y = §l.
#f * #n #Y #H
[ lapply (lifts_inv_sort1 … H)
lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
∨∨ ∃∃s. X = ⋆s & I = Sort s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j
+ | ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j
| ∃∃l. X = §l & I = GRef l.
#f * #n #X #H
[ lapply (lifts_inv_sort2 … H)
relation3 rtmap lenv bind ≝
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
- ∀L1. L0 ⪤[RN1, RP1, f] L1 → ∀L2. L0 ⪤[RN2, RP2, f] L2 →
+ ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
∃∃I. R2 L1 I1 I & R1 L2 I2 I.
definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 rtmap lenv bind ≝
λR1,R2,R3,RN,RP,f,L1,I1.
- ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
∀I2. R2 L2 I I2 → R3 L1 I1 I2.
(* Basic inversion lemmas ***************************************************)
-fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆.
+lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆.
/2 width=6 by sex_inv_atom1_aux/ qed-.
-fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ↑g] Y →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,↑g] Y →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by sex_inv_next1_aux/ qed-.
-fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ⫯g] Y →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,⫯g] Y →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by sex_inv_push1_aux/ qed-.
-fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆.
+lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆.
/2 width=6 by sex_inv_atom2_aux/ qed-.
-fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J2 #K2 #H destruct
| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ↑g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by sex_inv_next2_aux/ qed-.
-fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #J2 #K2 #g #H destruct
| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ⫯g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by sex_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2.
+ L1.ⓘ{I1} ⪤[RN,RP,↑f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2.
+ L1.ⓘ{I1} ⪤[RN,RP,⫯f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
-lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 →
+lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫱f] L2 →
RN L1 I1 I2 → RP L1 I1 I2 →
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}.
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2}.
#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
/2 width=1 by sex_next, sex_push/
qed-.
(* Basic forward lemmas *****************************************************)
lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, ⫱f] L2.
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,⫱f] L2.
#RN #RP #f #I1 #I2 #L1 #L2 #Hf
elim (pn_split f) * #g #H destruct
[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
(* Basic properties *********************************************************)
-lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
]
qed-.
-lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
qed-.
qed-.
lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} →
∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
- L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}.
+ L1.ⓘ{J1} ⪤[RN,RP,f] L2.ⓘ{J2}.
/3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-.
lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2.
+ ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → L1 ⪤[RN2,RP2,f] L2.
#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_atom, sex_next, sex_push/
qed-.
lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
- L1 ⪤[RN2, RP2, f] L2.
+ ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → 𝐈⦃f⦄ →
+ L1 ⪤[RN2,RP2,f] L2.
#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
[ elim (isid_inv_next … H) -H //
qed-.
lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2.
+ ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
qed-.
lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2.
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
[ elim (pn_split f1) * ]
qed-.
lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
[2: elim (pn_split f2) * ]
qed-.
lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ⊆ g →
- ∃∃L. L1 ⪤[R1, RP, g] L & L ⪤[R2, cfull, f] L2.
+ ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ⊆ g →
+ ∃∃L. L1 ⪤[R1,RP,g] L & L ⪤[R2,cfull,f] L2.
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
qed-.
lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ∥ g →
- ∃∃L. L1 ⪤[RP, R1, g] L & L ⪤[R2, cfull, f] L2.
+ ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g →
+ ∃∃L. L1 ⪤[RP,R1,g] L & L ⪤[R2,cfull,f] L2.
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
lemma sex_dec: ∀RN,RP.
(∀L,I1,I2. Decidable (RN L I1 I2)) →
(∀L,I1,I2. Decidable (RP L I1 I2)) →
- ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2).
+ ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2).
#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
[ /2 width=1 by sex_atom, or_introl/
| #L2 #I2 #f @or_intror #H
(* Forward lemmas with length for local environments ************************)
(* Note: "#f #I1 #I2 #L1 #L2 >length_bind >length_bind //" was needed to conclude *)
-lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → |L1| = |L2|.
+lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → |L1| = |L2|.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 //
qed-.
(* Properties with length for local environments ****************************)
-lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull, cfull, f] L2.
+lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull,cfull,f] L2.
#L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
| #L1 #I1 #IH #Y2 #H #f
qed.
lemma sex_length_isid: ∀R,L1,L2. |L1| = |L2| →
- ∀f. 𝐈⦃f⦄ → L1 ⪤[R, cfull, f] L2.
+ ∀f. 𝐈⦃f⦄ → L1 ⪤[R,cfull,f] L2.
#R #L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
| #L1 #I1 #IH #Y2 #H #f #Hf
∀L1,f.
(∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
(∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) →
- ∀L0. L1 ⪤[RN1, RP1, f] L0 →
- ∀L2. L0 ⪤[RN2, RP2, f] L2 →
- L1 ⪤[RN, RP, f] L2.
+ ∀L0. L1 ⪤[RN1,RP1,f] L0 →
+ ∀L2. L0 ⪤[RN2,RP2,f] L2 →
+ L1 ⪤[RN,RP,f] L2.
#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
[ #f #_ #_ #L0 #H1 #L2 #H2
lapply (sex_inv_atom1 … H1) -H1 #H destruct
Transitive … (sex RN RP f).
/2 width=9 by sex_trans_gen/ qed-.
-theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1, cfull, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤[R2, cfull, f] L2 → L1 ⪤[R3, cfull, f] L2.
+theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2.
#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
[ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ]
#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
/3 width=3 by/ qed-.
lemma sex_meet: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. L1 ⪤[RN, RP, f2] L2 →
- ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+ ∀f1. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
qed-.
lemma sex_join: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. L1 ⪤[RN, RP, f2] L2 →
- ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+ ∀f1. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
qed.
lemma sex_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 →
TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by sex_refl, sex_next, step, inj/
qed.
lemma sex_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 →
TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by sex_refl, sex_push, step, inj/
qed.
-lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[CTC … RN, RP, f] L2.
+lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[CTC … RN,RP,f] L2.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_push, sex_next, inj/
qed.
-lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[RN, CTC … RP, f] L2.
+lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[RN,CTC … RP,f] L2.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_push, sex_next, inj/
qed.
(* Basic_2A1: uses: TC_lpx_sn_ind *)
theorem sex_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f,L1,L. L1 ⪤[RN, RP, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤[RN, CTC … RP, f] L2 → L1⪤ [RN, CTC … RP, f] L2.
+ ∀f,L1,L. L1 ⪤[RN,RP,f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[RN,CTC … RP,f] L2 → L1⪤ [RN,CTC … RP,f] L2.
#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
[ #f #_ #Y #H -HRP >(sex_inv_atom1 … H) -Y // ]
#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
(* Advanced properties ******************************************************)
lemma sex_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN, CTC … RP, f] L2.
+ ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN,CTC … RP,f] L2.
#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
/3 width=3 by sex_tc_step_dx, sex_tc_inj_dx/
qed.
(* Advanced inversion lemmas ************************************************)
lemma sex_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤[CTC … RN, RP, f] L2 → TC … (sex RN RP f) L1 L2.
+ ∀f,L1,L2. L1 ⪤[CTC … RN,RP,f] L2 → TC … (sex RN RP f) L1 L2.
#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/2 width=1 by sex_tc_next, sex_tc_push_sn, sex_atom, inj/
qed-.
lemma sex_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤[RN, CTC … RP, f] L2 → TC … (sex RN RP f) L1 L2.
+ ∀f,L1,L2. L1 ⪤[RN,CTC … RP,f] L2 → TC … (sex RN RP f) L1 L2.
#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/2 width=1 by sex_tc_push, sex_tc_next_sn, sex_atom, inj/
qed-.
(* Basic properties *********************************************************)
-lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed.
lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed.
-lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
+lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G,L,②{I}V.T⦄ ⊐+[b] ⦃G,L,V⦄.
/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
+lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G,L.ⓑ{I}V,T⦄.
/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
+lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G,L.ⓧ,T⦄.
/3 width=1 by fqu_clear, fqu_fqup/ qed.
-lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
+lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G,L,ⓕ{I}V.T⦄ ⊐+[b] ⦃G,L,T⦄.
/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
+lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G,L,V2⦄.
/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
+lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G,L,ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I1}V1,T⦄.
/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
+lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I2}V2,T⦄.
/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
(* Basic eliminators ********************************************************)
lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
qed-.
lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
qed-.
(* Properties with generic slicing for local environments *******************)
lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ U →
- ⦃G, L, U⦄ ⊐+[b] ⦃G, K, T⦄.
+ ⦃G,L,U⦄ ⊐+[b] ⦃G,K,T⦄.
#b #G #K #T #i elim i -i
[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
#I #Y #HY #H destruct <(drops_fwd_isid … HY) -K //
qed.
lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ⬆*[i] T1 ≘ U1 →
- ⦃G1, K1, T1⦄ ⊐[b] ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, K2, T2⦄.
+ ⦃G1,K1,T1⦄ ⊐[b] ⦃G2,K2,T2⦄ → ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,K2,T2⦄.
#b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
[ #HLK1 #HTU1 #HT12
>(drops_fwd_isid … HLK1) -L1 //
]
qed-.
-lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+[b] ⦃G, K, V⦄.
+lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L,#i⦄ ⊐+[b] ⦃G,K,V⦄.
/2 width=6 by fqup_drops_strap1/ qed.
(* Forward lemmas with weight for closures **********************************)
-lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} < ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=3 by fqu_fwd_fw, transitive_lt/
qed-.
(* Advanced eliminators *****************************************************)
lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
qed-.
lemma fqup_wf_ind_eq: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
(* Basic eliminators ********************************************************)
lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
qed-.
lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
qed-.
lemma fqus_refl: ∀b. tri_reflexive … (fqus b).
/2 width=1 by tri_inj/ qed.
-lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
-lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed-.
-lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
#G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
* #HG #HL #HT #_ destruct //
qed-.
-lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
- ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+ ∃∃J,L. ⦃G1,L,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}.
#b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
qed-.
-lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
- | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
- | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
+ | ∃∃J,L,V. ⦃G1,L,V⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓑ{J}V & i = 0
+ | ∃∃J,L,j. ⦃G1,L,#j⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
#b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
qed-.
-lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
- ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+ ∃∃J,L. ⦃G1,L,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}.
#b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
qed-.
-lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+ | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄
+ | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄
+ | ⦃G1,L1.ⓧ,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ ∧ b = Ⓕ
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
#b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/
#G #L #T #H elim (fqu_inv_bind1 … H) -H *
[4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct
qed-.
-lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+ | ⦃G1,L1,V1⦄ ⊐* ⦃G2,L2,T2⦄
+ | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐* ⦃G2,L2,T2⦄
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐* ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ]
/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
#_ #H destruct
qed-.
-lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
+ | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
#b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
#G #L #T #H elim (fqu_inv_flat1 … H) -H *
[3: #J ] #H1 #H2 #H3 #H destruct
(* Advanced inversion lemmas ************************************************)
-lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
#b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
#G #L #T #H elim (fqu_inv_atom1 … H)
qed-.
-lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1,L1.ⓘ{I},⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_sort1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1,L1.ⓑ{I}V1,#0⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_zero1_pair … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1,L1.ⓘ{I},#↑i⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_lref1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1,L1.ⓘ{I},§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_gref1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
(* Properties with generic slicing for local environments *******************)
lemma fqus_drops: ∀b,G,L,K,T,U,i. ⬇*[i] L ≘ K → ⬆*[i] T ≘ U →
- ⦃G, L, U⦄ ⊐*[b] ⦃G, K, T⦄.
+ ⦃G,L,U⦄ ⊐*[b] ⦃G,K,T⦄.
#b #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
#HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
qed.
(* Alternative definition with plus-iterated supclosure *********************)
-lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
qed.
(* Basic_2A1: was: fqus_inv_gen *)
-lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
#G #G2 #L #L2 #T #T2 #_ *
[ #H2 * /3 width=5 by fqup_strap1, or_introl/
(* Advanced properties ******************************************************)
-lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
[ /2 width=5 by fqup_strap1/
| * /2 width=1 by fqu_fqup/
]
qed-.
-lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
[ /2 width=5 by fqup_strap2/
| * /2 width=1 by fqu_fqup/
]
qed-.
-lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
/2 width=5 by fqus_strap1_fqu, fqup_strap1/
qed-.
-lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
/3 width=5 by fqus_strap2_fqu, fqup_strap2/
qed-.
(* Advanced inversion lemmas for plus-iterated supclosure *******************)
-lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
qed-.
(* Forward lemmas with weight for closures **********************************)
-lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
/3 width=3 by fquq_fwd_fw, transitive_le/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G, L, ⓪{I}⦄ ⊐*[b] ⦃G, L, X⦄ → ⓪{I} = X.
+lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G,L,⓪{I}⦄ ⊐*[b] ⦃G,L,X⦄ → ⓪{I} = X.
#b #I #G #L #X #H elim (fqus_inv_fqu_sn … H) -H * //
#G0 #L0 #T0 #H1 #H2 lapply (fqu_fwd_fw … H1) lapply (fqus_fwd_fw … H2) -H2 -H1
#H2 #H1 lapply (le_to_lt_to_lt … H2 H1) -G0 -L0 -T0
(* Basic properties *********************************************************)
-lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄.
+lemma fqu_sort: ∀b,I,G,L,s. ⦃G,L.ⓘ{I},⋆s⦄ ⊐[b] ⦃G,L,⋆s⦄.
/2 width=1 by fqu_drop/ qed.
-lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄.
+lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G,L.ⓘ{I},#↑i⦄ ⊐[b] ⦃G,L,#i⦄.
/2 width=1 by fqu_drop/ qed.
-lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄.
+lemma fqu_gref: ∀b,I,G,L,l. ⦃G,L.ⓘ{I},§l⦄ ⊐[b] ⦃G,L,§l⦄.
/2 width=1 by fqu_drop/ qed.
(* Basic inversion lemmas ***************************************************)
-fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀s. T1 = ⋆s →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
]
qed-.
-lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
/2 width=4 by fqu_inv_sort1_aux/ qed-.
-fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀i. T1 = #i →
(∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
]
qed-.
-lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐[b] ⦃G2,L2,T2⦄ →
(∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
/2 width=4 by fqu_inv_lref1_aux/ qed-.
-fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀l. T1 = §l →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
]
qed-.
-lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
/2 width=4 by fqu_inv_gref1_aux/ qed-.
-fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
]
qed-.
-lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
| ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
/2 width=4 by fqu_inv_bind1_aux/ qed-.
-lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
+lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
* #_ #_ #_ #H destruct
qed-.
-fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1 = L2 & U1 = T2
]
qed-.
-lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1 = L2 & U1 = T2
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
(* Advanced inversion lemmas ************************************************)
-lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥.
+lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⊐[b] ⦃G2,L2,T2⦄ → ⊥.
#b * #x #G1 #G2 #L2 #T2 #H
[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
#I [2: #V |3: #i ] #_ #H destruct
qed-.
-lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1,K.ⓘ{I},⋆s⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
#b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H
#Z #X #H1 #H2 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1,K.ⓑ{I}V,#0⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = V.
#b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1,K.ⓘ{I},#(↑i)⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = #i.
#b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H *
#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1,K.ⓘ{I},§l⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = §l.
#b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H
#Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
(* Forward lemmas with length for local environments ************************)
-fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
∀i. T1 = #i → |L2| < |L1|.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2,3: #p]
#I #G #L #V #T [2: #_ ] #j #H destruct
qed-.
-lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐[b] ⦃G2,L2,T2⦄ →
|L2| < |L1|.
/2 width=8 by fqu_fwd_length_lref1_aux/
qed-.
(* Inversion lemmas with context-free sort-irrelevant equivalence for terms *)
-fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
G1 = G2 → |L1| = |L2| → T1 ≛ T2 → ⊥.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H)
qed-.
(* Basic_2A1: uses: fqu_inv_eq *)
-lemma fqu_inv_tdeq: ∀b,G,L1,L2,T1,T2. ⦃G, L1, T1⦄ ⊐[b] ⦃G, L2, T2⦄ →
+lemma fqu_inv_tdeq: ∀b,G,L1,L2,T1,T2. ⦃G,L1,T1⦄ ⊐[b] ⦃G,L2,T2⦄ →
|L1| = |L2| → T1 ≛ T2 → ⊥.
#b #G #L1 #L2 #T1 #T2 #H
@(fqu_inv_tdeq_aux … H) // (**) (* full auto fails *)
(* Forward lemmas with weight for closures **********************************)
-lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} < ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
#I #I1 #I2 #G #L #HI12 normalize in ⊢ (?%%); -I1
<(lifts_fwd_tw … HI12) /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/
(* Advanced eliminators *****************************************************)
lemma fqu_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=2 by fqu_fwd_fw/
lemma fquq_refl: ∀b. tri_reflexive … (fquq b).
// qed.
-lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄.
+lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄.
/2 width=1 by or_introl/ qed.
(* Basic_2A1: removed theorems 8:
(* Forward lemmas with length for local environments ************************)
-lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
|L2| ≤ |L1|.
#b #G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ]
/3 width=6 by fqu_fwd_length_lref1, lt_to_le/
(* Forward lemmas with weight for closures **********************************)
-lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ]
/3 width=2 by fqu_fwd_fw, lt_to_le/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
#G #L #T #A * -G -L -T -A //
[ #I #G #L #V #B #_ #s #H destruct
| #I #G #L #A #i #_ #s #H destruct
]
qed-.
-lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
+lemma aaa_inv_sort: ∀G,L,A,s. ⦃G,L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
/2 width=6 by aaa_inv_sort_aux/ qed-.
-fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → T = #0 →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
[ #G #L #s #H destruct
| #I #G #L #A #i #_ #H destruct
]
qed-.
-lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_zero: ∀G,L,A. ⦃G,L⦄ ⊢ #0 ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
/2 width=3 by aaa_inv_zero_aux/ qed-.
-fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #j #H destruct
| #I #G #L #V #B #_ #j #H destruct
]
qed-.
-lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G,L⦄ ⊢ #↑i ⁝ A →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A.
/2 width=3 by aaa_inv_lref_aux/ qed-.
-fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
#G #L #T #A * -G -L -T -A
[ #G #L #s #k #H destruct
| #I #G #L #V #B #_ #k #H destruct
]
qed-.
-lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
+lemma aaa_inv_gref: ∀G,L,A,l. ⦃G,L⦄ ⊢ §l ⁝ A → ⊥.
/2 width=7 by aaa_inv_gref_aux/ qed-.
-fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
+ ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L.ⓓW⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #q #W #U #H destruct
| #I #G #L #V #B #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
+lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G,L⦄ ⊢ ⓓ{p}V.T ⁝ A →
+ ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L.ⓓV⦄ ⊢ T ⁝ A.
/2 width=4 by aaa_inv_abbr_aux/ qed-.
-fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
+ ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
#G #L #T #A * -G -L -T -A
[ #G #L #s #q #W #U #H destruct
| #I #G #L #V #B #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
+lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G,L⦄ ⊢ ⓛ{p}W.T ⁝ A →
+ ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
/2 width=4 by aaa_inv_abst_aux/ qed-.
-fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
+fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
+ ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L⦄ ⊢ U ⁝ ②B.A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #W #U #H destruct
| #I #G #L #V #B #_ #W #U #H destruct
]
qed-.
-lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
+lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G,L⦄ ⊢ ⓐV.T ⁝ A →
+ ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L⦄ ⊢ T ⁝ ②B.A.
/2 width=3 by aaa_inv_appl_aux/ qed-.
-fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
+fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
+ ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #W #U #H destruct
| #I #G #L #V #B #_ #W #U #H destruct
]
qed-.
-lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
+lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G,L⦄ ⊢ ⓝW.T ⁝ A →
+ ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ T ⁝ A.
/2 width=3 by aaa_inv_cast_aux/ qed-.
(* Main inversion lemmas ****************************************************)
-theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
+theorem aaa_mono: ∀G,L,T,A1. ⦃G,L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T ⁝ A2 → A1 = A2.
#G #L #T #A1 #H elim H -G -L -T -A1
[ #G #L #s #A2 #H >(aaa_inv_sort … H) -H //
| #I1 #G #L #V1 #B #_ #IH #A2 #H
(* Advanced properties ******************************************************)
(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ B → ⦃G,L⦄ ⊢ #i ⁝ B.
#I #G #K #V #B #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
#H destruct /2 width=1 by aaa_zero/
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: aaa_inv_lref *)
-lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G,L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
#G #A #i elim i -i
[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
| #i #IH #L #H elim (aaa_inv_lref … H) -H
(* Basic_2A1: includes: aaa_lift *)
(* Note: it should use drops_split_trans_pair2 *)
-lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
- ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b,f] L2 ≘ L1 →
+ ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
[ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: includes: aaa_inv_lift *)
-lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 →
- ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G,L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b,f] L2 ≘ L1 →
+ ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G,L1⦄ ⊢ T1 ⁝ A.
@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
[ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
(* Properties with sort-irrelevant equivalence on referred entries **********)
-lemma aaa_fdeq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ →
- ∀A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → ⦃G2, L2⦄ ⊢ T2 ⁝ A.
+lemma aaa_fdeq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∀A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → ⦃G2,L2⦄ ⊢ T2 ⁝ A.
#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/2 width=5 by aaa_tdeq_conf_rdeq/ qed-.
(* Properties on supclosure *************************************************)
-lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H
#J #K #V #H #HA destruct /2 width=2 by ex_intro/
]
qed-.
-lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/
* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
qed-.
-lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
/2 width=6 by aaa_fqu_conf/
qed-.
-lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/
* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
qed-.
(* Properties with sort-irrelevant equivalence on referred entries **********)
-lemma aaa_tdeq_conf_rdeq: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛ T2 →
- ∀L2. L1 ≛[T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+lemma aaa_tdeq_conf_rdeq: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛ T2 →
+ ∀L2. L1 ≛[T1] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
[ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 //
| #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
(* Basic_properties *********************************************************)
lemma fdeq_intro_dx (G): ∀L1,L2,T2. L1 ≛[T2] L2 →
- ∀T1. T1 ≛ T2 → ⦃G, L1, T1⦄ ≛ ⦃G, L2, T2⦄.
+ ∀T1. T1 ≛ T2 → ⦃G,L1,T1⦄ ≛ ⦃G,L2,T2⦄.
/3 width=3 by fdeq_intro_sn, tdeq_rdeq_div/ qed.
(* Basic inversion lemmas ***************************************************)
-lemma fdeq_inv_gen_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ →
+lemma fdeq_inv_gen_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L1 ≛[T1] L2 & T1 ≛ T2.
#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
qed-.
-lemma fdeq_inv_gen_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ →
+lemma fdeq_inv_gen_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L1 ≛[T2] L2 & T1 ≛ T2.
#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=3 by tdeq_rdeq_conf, and3_intro/
/4 width=5 by fdeq_intro_sn, rdeq_trans, tdeq_rdeq_div, tdeq_trans/
qed-.
-theorem fdeq_canc_sn: ∀G,G1,L,L1,T,T1. ⦃G, L, T⦄ ≛ ⦃G1, L1, T1⦄→
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄.
+theorem fdeq_canc_sn: ∀G,G1,L,L1,T,T1. ⦃G,L,T⦄ ≛ ⦃G1,L1,T1⦄→
+ ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄.
/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
-theorem fdeq_canc_dx: ∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G2, L2, T2⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄.
+theorem fdeq_canc_dx: ∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G2,L2,T2⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄.
/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
(* Main inversion lemmas with degree-based equivalence on terms *************)
-theorem fdeq_tdneq_repl_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ →
- ∀U1,U2. ⦃G1, L1, U1⦄ ≛ ⦃G2, L2, U2⦄ →
+theorem fdeq_tdneq_repl_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∀U1,U2. ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄ →
(T2 ≛ U2 → ⊥) → (T1 ≛ U1 → ⊥).
#G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
elim (fdeq_inv_gen_sn … HT) -HT #_ #_ #HT
(* Properties with sort-irrelevant equivalence for terms ********************)
lemma tdeq_fdeq: ∀T1,T2. T1 ≛ T2 →
- ∀G,L. ⦃G, L, T1⦄ ≛ ⦃G, L, T2⦄.
+ ∀G,L. ⦃G,L,T1⦄ ≛ ⦃G,L,T2⦄.
/2 width=1 by fdeq_intro_sn/ qed.
(* Advanced properties ******************************************************)
(* Properties with star-iterated structural successor for closures **********)
-lemma fdeq_fqus_trans: ∀b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄.
+lemma fdeq_fqus_trans: ∀b,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ →
+ ∃∃G,L0,T0. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L0,T0⦄ & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄.
#b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
elim(fdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct
elim (rdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2
(* Properties with syntactic equivalence on referred entries ****************)
lemma req_rdeq_trans: ∀L1,L,T1. L1 ≡[T1] L →
- ∀G1,G2,L2,T2. ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄.
+ ∀G1,G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄.
#L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H
elim (fdeq_inv_gen_sn … H) -H #H #HL2 #T12 destruct
/3 width=3 by fdeq_intro_sn, req_rdeq_trans/
(* Advanced properties ******************************************************)
-lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
+lemma frees_atom_drops: ∀b,L,i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ →
∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
#b #L elim L -L /2 width=1 by frees_atom/
#L #I #IH *
(* Advanced inversion lemmas ************************************************)
lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
- ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+ ∨∨ ∃∃g. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
| ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
| ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
(* Properties with generic slicing for local environments *******************)
lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+ ∀f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
]
qed-.
-lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+lemma frees_lifts_SO: ∀b,L,K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
#b #L #K #HLK #T #U #HTU #f #Hf
@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
(* Forward lemmas with generic slicing for local environments ***************)
lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
/3 width=9 by frees_fwd_coafter, ex2_intro/
qed-.
lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
- ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
+ ∀K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
qed-.
lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
/3 width=7 by frees_eq_repl_back, coafter_inj/
definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2.
∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+ L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
interpretation "free variables inclusion (restricted closure)"
'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2).
(* Basic properties *********************************************************)
-lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
+lemma fsle_sort: ∀L,s1,s2. ⦃L,⋆s1⦄ ⊆ ⦃L,⋆s2⦄.
/3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed.
-lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
+lemma fsle_gref: ∀L,l1,l2. ⦃L,§l1⦄ ⊆ ⦃L,§l2⦄.
/3 width=8 by frees_gref, sle_refl, ex4_4_intro/ qed.
(* Advanced properties ******************************************************)
lemma fsle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
- ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄.
+ ∀T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ⦃L1.ⓧ,U1⦄ ⊆ ⦃L2,T2⦄.
#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
* #n #m #f #g #Hf #Hg #H2L #Hfg
lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_dx/ (**) (* explict constructor *)
qed-.
-lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1}, W1⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ →
+ ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1},W1⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#W1 #HVW1 #I1 #I2
/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/
qed.
-lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
+lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1,T1⦄ ⊆ ⦃K2,T2⦄ →
∀U1,U2. ⬆*[1] T1 ≘ U1 → ⬆*[1] T2 ≘ U2 →
- ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
+ ∀I1,I2. ⦃K1.ⓘ{I1},U1⦄ ⊆ ⦃K2.ⓘ{I2},U2⦄.
#K1 #K2 #HK #T1 #T2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#U1 #U2 #HTU1 #HTU2 #I1 #I2
(* Advanced inversion lemmas ************************************************)
lemma fsle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 →
- ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2, U2⦄ →
- ∀p. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.U2⦄.
+ ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2,U2⦄ →
+ ∀p. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.U2⦄.
#T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2
* #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p
elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct
qed.
lemma fsle_shift: ∀L1,L2. |L1| = |L2| →
- ∀I,T1,T2,V. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ →
- ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄.
+ ∀I,T1,T2,V. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I}V,T2⦄ →
+ ∀p. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V.T2⦄.
#L1 #L2 #H1L #I #T1 #T2 #V
* #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct
/4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/
qed.
-lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀p,I,T2. ⦃L1,V1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄.
#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
-lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
- ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2.ⓧ,T2⦄ → |L1| ≤ |L2| →
+ ∀p,I,V2. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄.
#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct
<tls_xn in Hfg2; #Hfg2
/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
qed.
-lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I,T2. ⦃L1,V1⦄ ⊆ ⦃L2,ⓕ{I}V2.T2⦄.
#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
elim (frees_total L2 T2) #g2 #Hg2
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
-lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀I,V2. ⦃L1,T1⦄ ⊆ ⦃L2,ⓕ{I}V2.T2⦄.
#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
elim (frees_total L2 V2) #g1 #Hg1
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
(* Advanced forward lemmas ***************************************************)
-lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
+lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄.
#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
(* Advanced inversion lemmas ************************************************)
-lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+ L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
qed-.
lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
qed-.
lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
(* Main properties **********************************************************)
-theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀T2. ⦃L2,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L2 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
- ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1,T1⦄ ⊆ ⦃L1,T⦄ →
+ ∀L2,T2. ⦃L1,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
- ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1,T1⦄ ⊆ ⦃L,T⦄ →
+ ∀L2,T2. |L| = |L2| → ⦃L,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L #T1 #T #HL1
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2 #HL2
qed-.
theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
- ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+ ∀V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀p,I. ⦃L1,ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀I. ⦃L1,ⓕ{I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I2,T1,T2. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p,I1. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #HL #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
+theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀I1,I2. ⦃L1,ⓕ{I1}V1.T1⦄ ⊆ ⦃L2,ⓕ{I2}V2.T2⦄.
/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.
(* Properties with length for local environments ****************************)
-lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
+lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1,⋆s1⦄ ⊆ ⦃L2,⋆s2⦄.
/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
-lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
+lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1,§l1⦄ ⊆ ⦃L2,§l2⦄.
/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
-lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ →
+ ∀I1,I2. ⦃K1.ⓑ{I1}V1,#O⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#I1 #I2
qed.
lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
- ∀I1,I2. ⦃K1.ⓤ{I1}, #O⦄ ⊆ ⦃K2.ⓤ{I2}, #O⦄.
+ ∀I1,I2. ⦃K1.ⓤ{I1},#O⦄ ⊆ ⦃K2.ⓤ{I2},#O⦄.
/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_intro/
qed.
(* Basic_1: was: sc3_arity_csubc *)
theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≘ L1 →
+ ∀G,L1,T,A. ⦃G,L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b,f] L0 ≘ L1 →
∀T0. ⬆*[f] T ≘ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
- ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
+ ⦃G,L2,T0⦄ ϵ[RP] 〚A〛.
#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq (Ⓣ) … G L1 T) -G -L1 -T
#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
(* Basic_1: was: sc3_arity *)
lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L,T⦄ ϵ[RP] 〚A〛.
/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → RP G L T.
#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
lapply (acr_gcr … H1RP H2RP A) #HA
@(s1 … HA) /2 width=4 by acr_aaa/
(* the functional construction for candidates *)
definition cfun: candidate → candidate → candidate ≝
λC1,C2,G,K,T. ∀f,L,W,U.
- ⬇*[Ⓕ, f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
+ ⬇*[Ⓕ,f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
(* the reducibility candidate associated to an atomic arity *)
rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
qed.
lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀p,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀b,f,L0,V0,W0,T0. ⬇*[b, f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
- ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
+ ∀p,G,L,W,T,A,B. ⦃G,L,W⦄ ϵ[RP] 〚B〛 → (
+ ∀b,f,L0,V0,W0,T0. ⬇*[b,f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
+ ⦃G,L0,V0⦄ ϵ[RP] 〚B〛 → ⦃G,L0,W0⦄ ϵ[RP] 〚B〛 → ⦃G,L0.ⓓⓝW0.V0,T0⦄ ϵ[RP] 〚A〛
) →
- ⦃G, L, ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
+ ⦃G,L,ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #HL0 #H #HB
lapply (acr_gcr … H1RP H2RP A) #HCA
lapply (acr_gcr … H1RP H2RP B) #HCB
inductive lsuba (G:genv): relation lenv ≝
| lsuba_atom: lsuba G (⋆) (⋆)
| lsuba_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G,L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G,L2⦄ ⊢ W ⁝ A →
lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
.
fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
(∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,W,V,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃⁝ K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
#G #L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
(∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ ∃∃K2,W,V,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsuba_inv_bind1_aux/ qed-.
fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
#G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ ∃∃K1,V,W,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsuba_inv_bind2_aux/ qed-.
(* Properties with atomic arity assignment **********************************)
-lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
- ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
+lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G,L1⦄ ⊢ V ⁝ A →
+ ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G,L2⦄ ⊢ V ⁝ A.
#G #L1 #V #A #H elim H -G -L1 -V -A
[ //
| #I #G #L1 #V #A #HA #IH #L2 #H
]
qed-.
-lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
- ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
+lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G,L2⦄ ⊢ V ⁝ A →
+ ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G,L1⦄ ⊢ V ⁝ A.
#G #L2 #V #A #H elim H -G -L2 -V -A
[ //
| #I #G #L2 #V #A #HA #IH #L1 #H
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsuba_drop_O1_conf *)
lemma lsuba_drops_conf_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K1 →
- ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L2 ≘ K2.
+ ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K1 →
+ ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b,f] L2 ≘ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsuba_drop_O1_trans *)
lemma lsuba_drops_trans_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L1 ≘ K1.
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
+ ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b,f] L1 ≘ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
inductive lsubc (RP) (G): relation lenv ≝
| lsubc_atom: lsubc RP G (⋆) (⋆)
| lsubc_bind: ∀I,L1,L2. lsubc RP G L1 L2 → lsubc RP G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G,L1,V⦄ ϵ[RP] 〚A〛 → ⦃G,L1,W⦄ ϵ[RP] 〚A〛 → ⦃G,L2⦄ ⊢ W ⁝ A →
lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
.
fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was: csubc_gen_head_r *)
lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
/2 width=3 by lsubc_inv_bind1_aux/ qed-.
fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was just: csubc_gen_head_l *)
lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
/2 width=3 by lsubc_inv_bind2_aux/ qed-.
(* Basic_1: includes: csubc_drop_conf_O *)
(* Basic_2A1: includes: lsubc_drop_O1_trans *)
lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
#RP #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
(* Basic_1: includes: csubc_drop_conf_rev *)
(* Basic_2A1: includes: drop_lsubc_trans *)
lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
- ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
- ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≘ K2.
+ ∀b,f,G,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+ ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b,f] L2 ≘ K2.
#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
#H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
(* Basic inversion lemmas ***************************************************)
-fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
+fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L1 = ⋆ →
f1 ≡ f2 ∧ L2 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
]
qed-.
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
+lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
/2 width=3 by lsubf_inv_atom1_aux/ qed-.
-fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+ ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
]
qed-.
-lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
+ ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
/2 width=6 by lsubf_inv_push1_aux/ qed-.
-fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
L2 = K2.ⓤ{J}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
]
qed-.
-lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X,↑g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
+ ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
L2 = K2.ⓤ{J}.
/2 width=5 by lsubf_inv_pair1_aux/ qed-.
-fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
]
qed-.
-lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I}, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I},↑g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
+ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
/2 width=5 by lsubf_inv_unit1_aux/ qed-.
-fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
+fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L2 = ⋆ →
f1 ≡ f2 ∧ L1 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
]
qed-.
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
+lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1,f1⦄ ⫃𝐅* ⦃⋆,f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
/2 width=3 by lsubf_inv_atom2_aux/ qed-.
-fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+ ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
]
qed-.
-lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, ⫯g2⦄ →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2},⫯g2⦄ →
+ ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
/2 width=6 by lsubf_inv_push2_aux/ qed-.
-fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
I = Abst & L1 = K1.ⓓⓝW.V.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
]
qed-.
-lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W,↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
I = Abst & L1 = K1.ⓓⓝW.V.
/2 width=5 by lsubf_inv_pair2_aux/ qed-.
-fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
L1 = K1.ⓑ{J}V.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
]
qed-.
-lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓤ{I}, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓤ{I},↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
L1 = K1.ⓑ{J}V.
/2 width=5 by lsubf_inv_unit2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2.
+lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆,f1⦄ ⫃𝐅* ⦃⋆,f2⦄ → f1 ≡ f2.
#f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
qed-.
-lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2.
+lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2},f2⦄ →
+ ∃∃g2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2.
#g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
#g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
qed-.
-lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I}, ↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2.
+lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I},f2⦄ →
+ ∃∃g2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2.
#g1 #f2 * #I [2: #X ] #K1 #K2 #H
[ elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓛW, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅* ⦃K2.ⓛW,f2⦄ →
+ ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
#g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #_ #_ #H destruct
| #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
]
qed-.
-lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J}, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J},f2⦄ →
+ ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
#g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #_ #_ #H destruct
| #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
(* Basic forward lemmas *****************************************************)
lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
- ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ → ⦃L1, ⫱f1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄.
+ ⦃L1.ⓘ{I},f1⦄ ⫃𝐅* ⦃L2.ⓘ{I},f2⦄ → ⦃L1,⫱f1⦄ ⫃𝐅* ⦃L2,⫱f2⦄.
#f1 #f2 #I #L1 #L2 #H
elim (pn_split f1) * #g1 #H0 destruct
[ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
#g2 #H12 #H destruct //
qed-.
-lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
+lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /2 width=3 by isid_eq_repl_fwd/
| /4 width=3 by isid_inv_push, isid_push/
]
qed-.
-lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
+lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /2 width=3 by isid_eq_repl_back/
| /4 width=3 by isid_inv_push, isid_push/
]
qed-.
-lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
+lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → f2 ⊆ f1.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
/3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
qed-.
(* Basic properties *********************************************************)
-axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
-lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
#f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
qed-.
-axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
-lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
#f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
qed-.
/2 width=1 by lsubf_push, lsubf_bind/
qed.
-lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
+lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄.
/2 width=3 by lsubf_eq_repl_back2/ qed.
-lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1, g1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ & g1 = ⫱f1.
+lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1,g1⦄ ⫃𝐅* ⦃L2,⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅* ⦃L2.ⓘ{I},f2⦄ & g1 = ⫱f1.
#g1 #f2 #I #L1 #L2 #H
elim (pn_split f2) * #g2 #H2 destruct
@ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
qed-.
lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
- ∀f2,L2,W. ⦃L1, f0⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓓⓝW.V, f1⦄ ⫃𝐅* ⦃L2.ⓛW, f2⦄ & ⫱f1 ⊆ g1.
+ ∀f2,L2,W. ⦃L1,f0⦄ ⫃𝐅* ⦃L2,⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓓⓝW.V,f1⦄ ⫃𝐅* ⦃L2.ⓛW,f2⦄ & ⫱f1 ⊆ g1.
#f #f0 #g1 #L1 #V #Hf #Hg1 #f2
elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
[ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
qed-.
(* Note: this might be moved *)
-lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
∀f2l,f2r. f2l⋓f2r ≘ f2 →
- ∃∃f1l,f1r. ⦃L1, f1l⦄ ⫃𝐅* ⦃L2, f2l⦄ & ⦃L1, f1r⦄ ⫃𝐅* ⦃L2, f2r⦄ & f1l⋓f1r ≘ f1.
+ ∃∃f1l,f1r. ⦃L1,f1l⦄ ⫃𝐅* ⦃L2,f2l⦄ & ⦃L1,f1r⦄ ⫃𝐅* ⦃L2,f2r⦄ & f1l⋓f1r ≘ f1.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
| #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
(* Properties with context-sensitive free variables *************************)
lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f1,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≘ f1.
+ ∀f1,L1. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≘ f1.
#f2 #L2 #T #H elim H -f2 -L2 -T
[ /3 width=5 by lsubf_fwd_isid_dx, frees_sort/
| #f2 #i #Hf2 #g1 #Y1 #H
(* Main properties **********************************************************)
-theorem lsubf_sor: ∀K,L,g1,f1. ⦃K, g1⦄ ⫃𝐅* ⦃L, f1⦄ →
- ∀g2,f2. ⦃K, g2⦄ ⫃𝐅* ⦃L, f2⦄ →
- ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K, g⦄ ⫃𝐅* ⦃L, f⦄.
+theorem lsubf_sor: ∀K,L,g1,f1. ⦃K,g1⦄ ⫃𝐅* ⦃L,f1⦄ →
+ ∀g2,f2. ⦃K,g2⦄ ⫃𝐅* ⦃L,f2⦄ →
+ ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K,g⦄ ⫃𝐅* ⦃L,f⦄.
#K elim K -K
[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
(* Forward lemmas with restricted refinement for local environments *********)
-lemma lsubf_fwd_lsubr_isdiv: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+lemma lsubf_fwd_lsubr_isdiv: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
𝛀⦃f1⦄ → 𝛀⦃f2⦄ → L1 ⫃ L2.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
/4 width=3 by lsubr_bind, isdiv_inv_next/
(* Properties with restricted refinement for local environments *************)
lemma lsubr_lsubf_isid: ∀L1,L2. L1 ⫃ L2 →
- ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+ ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄.
#L1 #L2 #H elim H -L1 -L2
[ /3 width=1 by lsubf_atom, isid_inv_eq_repl/
| #I #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V
qed.
lemma lsubr_lsubf: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 → ∀L1. L1 ⫃ L2 →
- ∀f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+ ∀f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄.
#f2 #L2 #T #H elim H -f2 -L2 -T
[ #f2 #L2 #s #Hf2 #L1 #HL12 #f1 #Hf1
lapply (frees_inv_sort … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
lemma lsubr_fwd_drops2_bind: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
+ ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
#L1 #L2 #H elim H -L1 -L2
[ #b #f #I #K2 #_ #H
elim (drops_inv_atom1 … H) -H #H destruct
(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓV.
+ ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓓV.
#L1 #L2 #HL12 #b #f #K2 #V #Hf #HLK2
elim (lsubr_fwd_drops2_bind … HL12 … Hf HLK2) -L2 -Hf // *
[ #K1 #W #V #_ #_ #H destruct
qed-.
lemma tdeq_rex_div (R): ∀T1,T2. T1 ≛ T2 →
- ∀L1,L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+ ∀L1,L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2.
/3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-.
lemma tdeq_rdeq_conf: s_r_confluent1 … cdeq rdeq.
L1 ≛[V1] L2 → V1 ≛ V2 → L1.ⓑ{I}V1 ≛[#0] L2.ⓑ{I}V2.
/2 width=1 by rex_pair/ qed.
(*
-lemma rdeq_unit: ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext, cfull, f] L2 →
+lemma rdeq_unit: ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext,cfull,f] L2 →
L1.ⓤ{I} ≛[#0] L2.ⓤ{I}.
/2 width=3 by rex_unit/ qed.
*)
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
| ∃∃I,L1,L2,V1,V2. L1 ≛[V1] L2 & V1 ≛ V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o, cfull, f] L2 &
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o,cfull,f] L2 &
Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
#Y1 #Y2 #H elim (rex_inv_zero … H) -H *
/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
/2 width=5 by rex_dropable_dx/ qed-.
lemma rdeq_inv_lifts_bi: ∀L1,L2,U. L1 ≛[U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
∀T. ⬆*[f] T ≘ U → K1 ≛[T] K2.
/2 width=10 by rex_inv_lifts_bi/ qed-.
(* Properties with extended structural successor for closures ***************)
-lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,T1⦄ →
∀U2. U1 ≛ U2 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[T1] L & T1 ≛ T2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐[b] ⦃G2,L,T2⦄ & L2 ≛[T1] L & T1 ≛ T2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
/2 width=5 by fqu_lref_O, ex3_2_intro/
]
qed-.
-lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
elim (fqu_tdeq_conf … H12 U2) -H12
/3 width=5 by rdeq_sym, tdeq_sym, ex3_2_intro/
qed-.
(* Basic_2A1: uses: lleq_fqu_trans *)
-lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
[ #I #G #L2 #V2 #L1 #H elim (rdeq_inv_zero_pair_dx … H) -H
#K1 #V1 #HV1 #HV12 #H destruct
(* Properties with optional structural successor for closures ***************)
-lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐⸮[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fquq, ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fquq_trans *)
-lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐⸮[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐⸮[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H elim H -H
[ #H #L1 #HL12 elim (rdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
(* Properties with plus-iterated structural successor for closures **********)
(* Basic_2A1: was just: lleq_fqup_trans *)
-lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐+[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐+[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
[ #G2 #K2 #U #HTU #L1 #HL12 elim (rdeq_fqu_trans … HTU … HL12) -L2
/3 width=5 by fqu_fqup, ex3_2_intro/
]
qed-.
-lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐+[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fqup, ex3_2_intro/
(* Properties with star-iterated structural successor for closures **********)
-lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐*[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fqus_trans *)
-lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐*[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐*[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
[ #H elim (rdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
(* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *)
lemma rdeq_lifts_bi: ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
∀U. ⬆*[f] T ≘ U → L1 ≛[U] L2.
/3 width=9 by rex_lifts_bi, tdeq_lifts_sn/ qed-.
(* Basic_2A1: was: llpx_sn_lrefl *)
(* Basic_2A1: this should have been lleq_fwd_llpx_sn *)
lemma req_fwd_rex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R, T] L2.
+ ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R,T] L2.
#R #HR #L1 #L2 #T * #f #Hf #HL12
/4 width=7 by sex_co, cext2_co, ex2_intro/
qed-.
(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
lemma req_inv_lifts_bi: ∀L1,L2,U. L1 ≡[U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
∀T. ⬆*[f] T ≘ U → K1 ≡[T] K2.
/2 width=10 by rex_inv_lifts_bi/ qed-.
(* Forward lemmas with free variables inclusion for restricted closures *****)
lemma req_rex_trans: ∀R. req_transitive R →
- ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R, T] L2 → L1 ⪤[R, T] L2.
+ ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R,T] L2 → L1 ⪤[R,T] L2.
/4 width=16 by req_fsle_comp, rex_trans_fsle, rex_trans_next/ qed-.
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition rex (R) (T): relation lenv ≝
- λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R, cfull, f] L2.
+ λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R,cfull,f] L2.
interpretation "generic extension on referred entries (local environment)"
'Relation R T L1 L2 = (rex R T L1 L2).
(relation3 lenv term term) … ≝
λR1,R2,RP1,RP2.
∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⪤[RP1, T0] L1 → ∀L2. L0 ⪤[RP2, T0] L2 →
+ ∀L1. L0 ⪤[RP1,T0] L1 → ∀L2. L0 ⪤[RP2,T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
definition rex_confluent: relation … ≝
λR1,R2.
- ∀K1,K,V1. K1 ⪤[R1, V1] K → ∀V. R1 K1 V1 V →
- ∀K2. K ⪤[R2, V] K2 → K ⪤[R2, V1] K2.
+ ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V →
+ ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2.
definition rex_transitive: relation3 ? (relation3 ?? term) … ≝
λR1,R2,R3.
- ∀K1,K,V1. K1 ⪤[R1, V1] K →
+ ∀K1,K,V1. K1 ⪤[R1,V1] K →
∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
(* Basic inversion lemmas ***************************************************)
-lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R, T] Y2 → Y2 = ⋆.
+lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R,T] Y2 → Y2 = ⋆.
#R #Y2 #T * /2 width=4 by sex_inv_atom1/
qed-.
-lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R, T] ⋆ → Y1 = ⋆.
+lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R,T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by sex_inv_atom2/
qed-.
-lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 →
+lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R,⋆s] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, ⋆s] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
+lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R,#0] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R, cfull, f] L2 &
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R,cfull,f] L2 &
Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
#R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 →
+lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R,#↑i] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, #i] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,#i] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 →
+lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R,§l] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, §l] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,§l] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
-lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- ∧∧ L1 ⪤[R, V1] L2 & L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ ∧∧ L1 ⪤[R,V1] L2 & L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
/6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_flat *)
-lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤[R, V] L2 & L1 ⪤[R, T] L2.
+lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤[R,V] L2 & L1 ⪤[R,T] L2.
#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
/5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R, ⋆s] L2 →
- ∃∃I2,K2. K1 ⪤[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R,⋆s] L2 →
+ ∃∃I2,K2. K1 ⪤[R,⋆s] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R, ⋆s] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R,⋆s] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,⋆s] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 →
- ∃∃K2,V2. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
+lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R,#0] L2 →
+ ∃∃K2,V2. K1 ⪤[R,V1] K2 & R K1 V1 V2 &
L2 = K2.ⓑ{I}V2.
#R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H *
[ #H destruct
]
qed-.
-lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 →
- ∃∃K1,V1. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
+lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R,#0] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R,V1] K2 & R K1 V1 V2 &
L1 = K1.ⓑ{I}V1.
#R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H *
[ #_ #H destruct
]
qed-.
-lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 →
- ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
+lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R,#0] L2 →
+ ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 &
L2 = K2.ⓤ{I}.
#R #I #K1 #L2 #H elim (rex_inv_zero … H) -H *
[ #H destruct
]
qed-.
-lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} →
- ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
+lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R,#0] K2.ⓤ{I} →
+ ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 &
L1 = K1.ⓤ{I}.
#R #I #L1 #K2 #H elim (rex_inv_zero … H) -H *
[ #_ #H destruct
]
qed-.
-lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R, #↑i] L2 →
- ∃∃I2,K2. K1 ⪤[R, #i] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R,#↑i] L2 →
+ ∃∃I2,K2. K1 ⪤[R,#i] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R, #↑i] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, #i] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R,#↑i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,#i] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R, §l] L2 →
- ∃∃I2,K2. K1 ⪤[R, §l] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R,§l] L2 →
+ ∃∃I2,K2. K1 ⪤[R,§l] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R, §l] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, §l] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R,§l] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,§l] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
(* Basic forward lemmas *****************************************************)
lemma rex_fwd_zero_pair (R): ∀I,K1,K2,V1,V2.
- K1.ⓑ{I}V1 ⪤[R, #0] K2.ⓑ{I}V2 → K1 ⪤[R, V1] K2.
+ K1.ⓑ{I}V1 ⪤[R,#0] K2.ⓑ{I}V2 → K1 ⪤[R,V1] K2.
#R #I #K1 #K2 #V1 #V2 #H
elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
-lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R, ②{I}V.T] L2 → L1 ⪤[R, V] L2.
+lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R,②{I}V.T] L2 → L1 ⪤[R,V] L2.
#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
/4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
-lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
-lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → L1 ⪤[R, T] L2.
+lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → L1 ⪤[R,T] L2.
#R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H //
qed-.
-lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R, T] K2.ⓘ{I2} →
+lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ{I2} →
∃∃I1,K1. L1 = K1.ⓘ{I1}.
#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
[ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
(* Basic properties *********************************************************)
-lemma rex_atom (R): ∀I. ⋆ ⪤[R, ⓪{I}] ⋆.
+lemma rex_atom (R): ∀I. ⋆ ⪤[R,⓪{I}] ⋆.
#R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/
qed.
lemma rex_sort (R): ∀I1,I2,L1,L2,s.
- L1 ⪤[R, ⋆s] L2 → L1.ⓘ{I1} ⪤[R, ⋆s] L2.ⓘ{I2}.
+ L1 ⪤[R,⋆s] L2 → L1.ⓘ{I1} ⪤[R,⋆s] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
lapply (frees_inv_sort … Hf) -Hf
/4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/
qed.
-lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, #0] L2.ⓑ{I}V2.
+lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,#0] L2.ⓑ{I}V2.
#R #I1 #I2 #L1 #L2 #V1 *
/4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/
qed.
-lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
+ L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}.
/4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed.
lemma rex_lref (R): ∀I1,I2,L1,L2,i.
- L1 ⪤[R, #i] L2 → L1.ⓘ{I1} ⪤[R, #↑i] L2.ⓘ{I2}.
+ L1 ⪤[R,#i] L2 → L1.ⓘ{I1} ⪤[R,#↑i] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/
qed.
lemma rex_gref (R): ∀I1,I2,L1,L2,l.
- L1 ⪤[R, §l] L2 → L1.ⓘ{I1} ⪤[R, §l] L2.ⓘ{I2}.
+ L1 ⪤[R,§l] L2 → L1.ⓘ{I1} ⪤[R,§l] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
lapply (frees_inv_gref … Hf) -Hf
/4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/
qed.
lemma rex_bind_repl_dx (R): ∀I,I1,L1,L2,T.
- L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I1} →
+ L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I1} →
∀I2. cext2 R L1 I I2 →
- L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I2}.
+ L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I2}.
#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
/3 width=5 by sex_pair_repl, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_co *)
lemma rex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤[R1, T] L2 → L1 ⪤[R2, T] L2.
+ ∀L1,L2,T. L1 ⪤[R1,T] L2 → L1 ⪤[R2,T] L2.
#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/
qed-.
lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2.
(∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
(∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) →
- L1 ⪤[R1, T1] L2 → L1 ⪤[R2, T2] L2.
+ L1 ⪤[R1,T1] L2 → L1 ⪤[R2,T2] L2.
#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
/4 width=7 by sex_co_isid, ex2_intro/
qed-.
lemma rex_unit_sn (R1) (R2):
- ∀I,K1,L2. K1.ⓤ{I} ⪤[R1, #0] L2 → K1.ⓤ{I} ⪤[R2, #0] L2.
+ ∀I,K1,L2. K1.ⓤ{I} ⪤[R1,#0] L2 → K1.ⓤ{I} ⪤[R2,#0] L2.
#R1 #R2 #I #K1 #L2 #H
elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
/3 width=7 by rex_unit, sex_co_isid/
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤[R,T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤[R,U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤[R,T] K2 & ⬇*[b,f] L2 ≘ K2.
definition f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R, T] K2.
+ λR. ∀L1,L2,U. L1 ⪤[R,U] L2 →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R,T] K2.
definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
qed-.
(* Basic_2A1: uses: llpx_sn_inv_lift_O *)
-lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀T. ⬆*[f] T ≘ U → K1 ⪤[R, T] K2.
+lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R,U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀T. ⬆*[f] T ≘ U → K1 ⪤[R,T] K2.
#R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
elim (rex_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct //
qed-.
-lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R,V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (rex_inv_zero_pair_sn … HY) -HY
#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R,V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (rex_inv_zero_pair_dx … HY) -HY
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
lemma rex_inv_lref_pair_bi (R) (L1) (L2) (i):
- L1 ⪤[R, #i] L2 →
+ L1 ⪤[R,#i] L2 →
∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 →
∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 →
- ∧∧ K1 ⪤[R, V1] K2 & R K1 V1 V2 & I1 = I2.
+ ∧∧ K1 ⪤[R,V1] K2 & R K1 V1 V2 & I1 = I2.
#R #L1 #L2 #i #H12 #I1 #K1 #V1 #H1 #I2 #K2 #V2 #H2
elim (rex_inv_lref_pair_sn … H12 … H1) -L1 #Y2 #X2 #HLY2 #HK12 #HV12
lapply (drops_mono … HLY2 … H2) -HLY2 -H2 #H destruct
/2 width=1 by and3_intro/
qed-.
-lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
- ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
+ ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄.
#R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (rex_inv_zero_unit_sn … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
- ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
+ ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄.
#R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (rex_inv_zero_unit_dx … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: llpx_sn_refl *)
-lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R, T] L.
+lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R,T] L.
#R #HR #L #T elim (frees_total L T)
/4 width=3 by sex_refl, ext2_refl, ex2_intro/
qed.
lemma rex_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R, T] L.ⓑ{I}V2.
+ ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R,T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #HV12 #I #T
elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
elim (pn_split f) * #g #H destruct
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
- L1 ⪤[R, V] L2 ∧ L1.ⓧ ⪤[R, T] L2.ⓧ.
+lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 →
+ L1 ⪤[R,V] L2 ∧ L1.ⓧ ⪤[R,T] L2.ⓧ.
#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind_void … Hf) -Hf
/6 width=6 by sle_sex_trans, sex_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced forward lemmas **************************************************)
-lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤[R, T] L2.ⓧ.
+lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤[R,T] L2.ⓧ.
#R #p #I #L1 #L2 #V #T #H elim (rex_inv_bind_void … H) -H //
qed-.
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
+ ∀L,T1,T2. RN L T1 T2 → ⦃L,T2⦄ ⊆ ⦃L,T1⦄.
definition rex_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
+ ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L2,T⦄ ⊆ ⦃L1,T⦄.
definition rex_fsle_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+ ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L1,T⦄ ⊆ ⦃L2,T⦄.
(* Basic inversions with free variables inclusion for restricted closures ***)
lemma frees_sex_conf: ∀R. rex_fsge_compatible R →
∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀L2. L1 ⪤[cext2 R, cfull, f1] L2 →
+ ∀L2. L1 ⪤[cext2 R,cfull,f1] L2 →
∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 & f2 ⊆ f1.
#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
(* Properties with free variables inclusion for restricted closures *********)
-(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
- ∀L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+(* Note: we just need lveq_inv_refl: ∀L, n1, n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L1,T2⦄ →
+ ∀L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2.
#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
/4 width=5 by rex_inv_frees, sle_sex_trans, ex2_intro/
lemma rex_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
rex_fsge_compatible R1 →
- ∀L1,L2,V. L1 ⪤[R1, V] L2 → ∀I,T.
- ∃∃L. L1 ⪤[R1, ②{I}V.T] L & L ⪤[R2, V] L2.
+ ∀L1,L2,V. L1 ⪤[R1,V] L2 → ∀I,T.
+ ∃∃L. L1 ⪤[R1,②{I}V.T] L & L ⪤[R2,V] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
lemma rex_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
rex_fsge_compatible R1 →
- ∀L1,L2,T. L1 ⪤[R1, T] L2 → ∀I,V.
- ∃∃L. L1 ⪤[R1, ⓕ{I}V.T] L & L ⪤[R2, T] L2.
+ ∀L1,L2,T. L1 ⪤[R1,T] L2 → ∀I,V.
+ ∃∃L. L1 ⪤[R1,ⓕ{I}V.T] L & L ⪤[R2,T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
lemma rex_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
rex_fsge_compatible R1 →
- ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1, T] L2 → ∀p.
- ∃∃L,V. L1 ⪤[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2, T] L2 & R1 L1 V1 V.
+ ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1,T] L2 → ∀p.
+ ∃∃L,V. L1 ⪤[R1,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2,T] L2 & R1 L1 V1 V.
#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg
elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy
lemma rex_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
rex_fsge_compatible R1 →
- ∀L1,L2,T. L1.ⓧ ⪤[R1, T] L2 → ∀p,I,V.
- ∃∃L. L1 ⪤[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2, T] L2.
+ ∀L1,L2,T. L1.ⓧ ⪤[R1,T] L2 → ∀p,I,V.
+ ∃∃L. L1 ⪤[R1,ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2,T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy
theorem rex_trans_fsle: ∀R1,R2,R3.
rex_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
- ∀L1,L,T. L1 ⪤[R1, T] L →
- ∀L2. L ⪤[R2, T] L2 → L1 ⪤[R3, T] L2.
+ ∀L1,L,T. L1 ⪤[R1,T] L →
+ ∀L2. L ⪤[R2,T] L2 → L1 ⪤[R3,T] L2.
#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
lapply (H1R … H) -H1R #H0
cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: llpx_sn_fwd_length *)
-lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R, T] L2 → |L1| = |L2|.
+lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R,T] L2 → |L1| = |L2|.
#R #L1 #L2 #T * /2 width=4 by sex_fwd_length/
qed-.
(* Properties with length for local environments ****************************)
(* Basic_2A1: uses: llpx_sn_sort *)
-lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R, ⋆s] L2.
+lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R,⋆s] L2.
#R #L1 elim L1 -L1
[ #Y #H #s >(length_inv_zero_sn … H) -H //
| #K1 #I1 #IH #Y #H #s
qed.
(* Basic_2A1: uses: llpx_sn_gref *)
-lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R, §l] L2.
+lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R,§l] L2.
#R #L1 elim L1 -L1
[ #Y #H #s >(length_inv_zero_sn … H) -H //
| #K1 #I1 #IH #Y #H #s
]
qed.
-lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}.
/3 width=3 by rex_unit, sex_length_isid/ qed.
(* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *)
lemma rex_lifts_bi (R): d_liftable2_sn … lifts R →
- ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R, T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀U. ⬆*[f] T ≘ U → L1 ⪤[R, U] L2.
+ ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R,T] K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ⪤[R,U] L2.
#R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU
elim (frees_total L1 U) #f2 #Hf2
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
(* Inversion lemmas with length for local environment ***********************)
-lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
+lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R,#0] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
| ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
#R #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
(* Properties with generic extension of a context-sensitive relation ********)
-lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R, T] L2.
+lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R,T] L2.
#R #L1 #L2 * #f #Hf #HL12 #T
elim (frees_total L1 T) #g #Hg
/4 width=5 by sex_sdj, sdj_isid_sn, ex2_intro/
lemma rex_inv_lex_req: ∀R. c_reflexive … R →
rex_fsge_compatible R →
- ∀L1,L2,T. L1 ⪤[R, T] L2 →
+ ∀L1,L2,T. L1 ⪤[R,T] L2 →
∃∃L. L1 ⪤[R] L & L ≡[T] L2.
#R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
elim (sex_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 →
- ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2.
+lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R,T] L2 →
+ ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R,cfull,f] L2.
#R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/
qed-.
(* Basic_2A1: uses: llpx_sn_dec *)
lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2).
+ ∀L1,L2,T. Decidable (L1 ⪤[R,T] L2).
#R #HR #L1 #L2 #T
elim (frees_total L1 T) #f #Hf
elim (sex_dec (cext2 R) cfull … L1 L2 f)
(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T.
- L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 →
- L1 ⪤[R, ⓑ{p,I}V1.T] L2.
+ L1 ⪤[R,V1] L2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2 →
+ L1 ⪤[R,ⓑ{p,I}V1.T] L2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
/3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/
(* Basic_2A1: llpx_sn_flat *)
theorem rex_flat: ∀R,I,L1,L2,V,T.
- L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 →
- L1 ⪤[R, ⓕ{I}V.T] L2.
+ L1 ⪤[R,V] L2 → L1 ⪤[R,T] L2 →
+ L1 ⪤[R,ⓕ{I}V.T] L2.
#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
/3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/
qed.
theorem rex_bind_void: ∀R,p,I,L1,L2,V,T.
- L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ →
- L1 ⪤[R, ⓑ{p,I}V.T] L2.
+ L1 ⪤[R,V] L2 → L1.ⓧ ⪤[R,T] L2.ⓧ →
+ L1 ⪤[R,ⓑ{p,I}V.T] L2.
#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
/3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/
(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥).
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤[R,V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R,T] L2.ⓑ{I}V → ⊥).
#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=2 by rex_bind, or_intror, or_introl/
qed-.
(* Basic_2A1: uses: nllpx_sn_inv_flat *)
lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥).
+ ∀I,L1,L2,V,T. (L1 ⪤[R,ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ⪤[R,V] L2 → ⊥) ∨ (L1 ⪤[R,T] L2 → ⊥).
#R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=1 by rex_flat, or_intror, or_introl/
qed-.
lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥).
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤[R,V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R,T] L2.ⓧ → ⊥).
#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=2 by rex_bind_void, or_intror, or_introl/
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: flt_shift *)
-lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{p,I}V.T}.
+lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V,T} < ♯{K,ⓑ{p,I}V.T}.
normalize /2 width=1 by monotonic_le_plus_r/
qed.
-lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1}, T} < ♯{K, ⓑ{p,I2}V.T}.
+lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1},T} < ♯{K,ⓑ{p,I2}V.T}.
normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
qed.
-lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
+lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?%%); //
qed.
-lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
+lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?%%); //
qed.
-lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
+lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
-lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}.
+lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
(* Basic properties *********************************************************)
(* Basic_1: was: flt_shift *)
-lemma fw_shift: ∀p,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{p,I}V.T}.
+lemma fw_shift: ∀p,I,G,K,V,T. ♯{G,K.ⓑ{I}V,T} < ♯{G,K,ⓑ{p,I}V.T}.
normalize /2 width=1 by monotonic_le_plus_r/
qed.
-lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G, K.ⓤ{I1}, T} < ♯{G, K, ⓑ{p,I2}V.T}.
+lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G,K.ⓤ{I1},T} < ♯{G,K,ⓑ{p,I2}V.T}.
normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
qed.
-lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
+lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?→?%%); //
qed.
-lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
+lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G,L,T} < ♯{G,L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?→?%%); //
qed.
-lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
+lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/item_sh.ma".
-
-(* SORT DEGREE **************************************************************)
-
-(* sort degree specification *)
-record sd (h:sh): Type[0] ≝ {
- deg : relation nat; (* degree of the sort *)
- deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
- deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
- deg_next : ∀s,d. deg s d → deg (next h s) (↓d) (* compatibility condition *)
-}.
-
-(* Notable specifications ***************************************************)
-
-definition deg_O: relation nat ≝ λs,d. d = 0.
-
-definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
-/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
-
-(* Basic_2A1: includes: deg_SO_pos *)
-inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
-| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
-| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
-.
-
-fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n →
- (next h)^n s0 = s.
-#h #s #s0 #n0 * -n0
-[ #n #Hn #x #H destruct //
-| #_ #x #H destruct
-]
-qed-.
-
-(* Basic_2A1: was: deg_SO_inv_pos *)
-lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
-/2 width=3 by deg_SO_inv_succ_aux/ qed-.
-
-lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
-#h #s @(deg_SO_succ … 0 ?) //
-qed.
-
-lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
-#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
-[ #H destruct
- elim (lt_refl_false … HK12)
-| #n #_ #H
- lapply (next_lt h ((next h)^n s2)) >H -H #H
- lapply (transitive_lt … H HK12) -s1 #H1
- lapply (nexts_le h s2 n) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -n #H
- elim (lt_refl_false … H)
-]
-qed.
-
-definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
-[ #s0
- lapply (nexts_dec h s0 s) *
- [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
- [ < H2 in H1; -H2 #H
- lapply (nexts_inj … H) -H #H destruct //
- | elim H1 /2 width=2 by ex_intro/
- | elim H2 /2 width=2 by ex_intro/
- ]
-| #s0 #n *
- [ #d #H destruct elim d -d normalize
- /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
- | #H1 @deg_SO_zero * #d #H2 destruct
- @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
- ]
-]
-defined.
-
-rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
- match d with
- [ O ⇒ sd_O h
- | S d ⇒ match d with
- [ O ⇒ sd_SO h s
- | _ ⇒ sd_d h (next h s) d
- ]
- ].
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (↑d) → deg h o s (↑↑d).
-#h #o #s #d #H1
-elim (deg_total h o s) #n #H0
-lapply (deg_next … H0) #H2
-lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
-qed-.
-
-lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (↑d) → deg h o s (↑(d+n)).
-#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
-#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
-qed.
-
-lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
-/2 width=1 by deg_next/ qed-.
-
-lemma sd_d_SS: ∀h,s,d. sd_d h s (↑↑d) = sd_d h (next h s) (↑d).
-// qed.
-
-lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
-#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/arith.ma".
-
-(* SORT HIERARCHY ***********************************************************)
-
-(* sort hierarchy specification *)
-record sh: Type[0] ≝ {
- next : nat → nat; (* next sort in the hierarchy *)
- next_lt: ∀s. s < next s (* strict monotonicity condition *)
-}.
-
-definition sh_N: sh ≝ mk_sh S ….
-// defined.
-
-(* Basic properties *********************************************************)
-
-lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
-#h #s #n elim n -n // normalize #n #IH
-lapply (next_lt h ((next h)^n s)) #H
-lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
-qed.
-
-lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s.
-#h #s #n normalize
-lapply (nexts_le h s n) #H
-@(le_to_lt_to_lt … H) //
-qed.
-
-axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
-
-axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
(* Basic properties *********************************************************)
-lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
+lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L.
#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
0 = n1 → 0 = n2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: /3 width=1 by or_introl, conj/
|2: /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
/2 width=5 by lveq_inv_zero_aux/ qed-.
-fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1. ↑m1 = n1 →
- ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
+ ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m #H destruct
|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
]
qed-.
-lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 →
- ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
+lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 →
+ ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2.
/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
-lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 →
- ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
+lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 →
+ ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1.
#K1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
qed-.
-fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m1 #m2 #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
+lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1,↑n2] L2 → ⊥.
/2 width=9 by lveq_inv_succ_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
+lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0,0] K2.ⓘ{I2} → K1 ≋ⓧ*[0,0] K2.
#I1 #I2 #K1 #K2 #H
elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
qed-.
-lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
* [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
- ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
+lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1,n2] ⋆ →
+ ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
- ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ{I2} →
+ ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
#I2 #K2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_bind_atom … H) -H
/3 width=3 by lveq_sym, ex4_intro/
qed-.
-lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 →
- ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2.
+lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 →
+ ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2.
#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 →
- ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
+lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 →
+ ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2.
#L1 #L2 #n1 #n2 #H
elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
qed-.
-lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ →
- ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
+lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ →
+ ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1.
#L1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_void_succ_sn … H) -H
(* Advanced forward lemmas **************************************************)
-lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∨∨ 0 = n1 | 0 = n2.
#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H) ]
/2 width=1 by or_introl, or_intror/
qed-.
-lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
+lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2 → 0 = n1.
#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2.
+lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 → 0 = n2.
/3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.
(* Properties with length for local environments ****************************)
-lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0, 0] L2.
+lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0,0] L2.
#L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 /2 width=3 by lveq_atom, ex_intro/
| #K1 #I1 #IH #Y2 #H
(* Forward lemmas with length for local environments ************************)
-lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n1 ≤ |L1|.
+lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n1 ≤ |L1|.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=1 by le_S_S/
qed-.
-lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n2 ≤ |L2|.
+lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n2 ≤ |L2|.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=1 by le_S_S/
qed-.
-lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∧∧ |L1|-|L2| = n1 & |L2|-|L1| = n2.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 /2 width=1 by conj/
#K1 #K2 #n #_ * #H1 #H2 >length_bind /3 width=1 by minus_Sn_m, conj/
qed-.
-lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L1| ≤ |L2| → 0 = n1.
+lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| ≤ |L2| → 0 = n1.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>(eq_minus_O … HL) //
qed-.
-lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → 0 = n2.
+lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| → 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>(eq_minus_O … HL) //
qed-.
-lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| = |L2| → ∧∧ 0 = n1 & 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>HL -HL /2 width=1 by conj/
qed-.
-lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| + n2 = |L2| + n1.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=2 by injective_plus_r/
qed-.
-lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → |L1| = |L2|.
+lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 → |L1| = |L2|.
/3 width=2 by lveq_fwd_length_plus, injective_plus_l/ qed-.
-lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| - n1 = |L2| - n2.
/3 width=3 by lveq_fwd_length_plus, lveq_fwd_length_le_dx, lveq_fwd_length_le_sn, plus_to_minus_2/ qed-.
lemma lveq_fwd_abst_bind_length_le: ∀I1,I2,L1,L2,V1,n1,n2.
- L1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
+ L1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
#I1 #I2 #L1 #L2 #V1 #n1 #n2 #HL
lapply (lveq_fwd_pair_sn … HL) #H destruct
elim (lveq_fwd_length … HL) -HL >length_bind >length_bind //
qed-.
lemma lveq_fwd_bind_abst_length_le: ∀I1,I2,L1,L2,V2,n1,n2.
- L1.ⓘ{I1} ≋ⓧ*[n1, n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
+ L1.ⓘ{I1} ≋ⓧ*[n1,n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
/3 width=6 by lveq_fwd_abst_bind_length_le, lveq_sym/ qed-.
(* Inversion lemmas with length for local environments **********************)
-lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2.ⓧ → |L1| ≤ |L2| →
- ∃∃m2. L1 ≋ ⓧ*[n1, m2] L2 & 0 = n1 & ↑m2 = n2.
+lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2.ⓧ → |L1| ≤ |L2| →
+ ∃∃m2. L1 ≋ ⓧ*[n1,m2] L2 & 0 = n1 & ↑m2 = n2.
#L1 #L2 #n1 #n2 #H #HL12
lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0
lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0
elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/
qed-.
-lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| →
- ∃∃m1. L1 ≋ ⓧ*[m1, n2] L2 & ↑m1 = n1 & 0 = n2.
+lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| →
+ ∃∃m1. L1 ≋ ⓧ*[m1,n2] L2 & ↑m1 = n1 & 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
lapply (lveq_sym … H) -H #H
elim (lveq_inv_void_dx_length … H HL) -H -HL
(* Main inversion lemmas ****************************************************)
-theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 →
- ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
+theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0,0] K2 →
+ ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1,m2] K2.ⓘ{I2} →
∧∧ 0 = m1 & 0 = m2.
#K1 #K2 #HK #I1 #I2 #m1 #m2 #H
lapply (lveq_fwd_length_eq … HK) -HK #HK
elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
qed-.
-theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
+theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
+ ∀m1,m2. L1 ≋ⓧ*[m1,m2] L2 →
∧∧ n1 = m1 & n2 = m2.
#L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm
elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
qed-.
theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
+ ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 →
+ ∀m1,m2. K1.ⓧ ≋ⓧ*[m1,m2] K2 →
∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2.
#L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm
elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
qed-.
theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
+ ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 →
+ ∀m1,m2. K1 ≋ⓧ*[m1,m2] K2.ⓧ →
∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1.
/3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/item_sh.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* sort degree specification *)
+record sd (h:sh): Type[0] ≝ {
+ deg : relation nat; (* degree of the sort *)
+ deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
+ deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
+ deg_next : ∀s,d. deg s d → deg (next h s) (↓d) (* compatibility condition *)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition deg_O: relation nat ≝ λs,d. d = 0.
+
+definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
+
+(* Basic_2A1: includes: deg_SO_pos *)
+inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
+| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
+| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
+.
+
+fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n →
+ (next h)^n s0 = s.
+#h #s #s0 #n0 * -n0
+[ #n #Hn #x #H destruct //
+| #_ #x #H destruct
+]
+qed-.
+
+(* Basic_2A1: was: deg_SO_inv_pos *)
+lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
+/2 width=3 by deg_SO_inv_succ_aux/ qed-.
+
+lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
+#h #s @(deg_SO_succ … 0 ?) //
+qed.
+
+lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
+#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
+[ #H destruct
+ elim (lt_refl_false … HK12)
+| #n #_ #H
+ lapply (next_lt h ((next h)^n s2)) >H -H #H
+ lapply (transitive_lt … H HK12) -s1 #H1
+ lapply (nexts_le h s2 n) #H2
+ lapply (le_to_lt_to_lt … H2 H1) -h -n #H
+ elim (lt_refl_false … H)
+]
+qed.
+
+definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
+[ #s0
+ lapply (nexts_dec h s0 s) *
+ [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
+| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
+ [ < H2 in H1; -H2 #H
+ lapply (nexts_inj … H) -H #H destruct //
+ | elim H1 /2 width=2 by ex_intro/
+ | elim H2 /2 width=2 by ex_intro/
+ ]
+| #s0 #n *
+ [ #d #H destruct elim d -d normalize
+ /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
+ | #H1 @deg_SO_zero * #d #H2 destruct
+ @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+ ]
+]
+defined.
+
+rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
+ match d with
+ [ O ⇒ sd_O h
+ | S d ⇒ match d with
+ [ O ⇒ sd_SO h s
+ | _ ⇒ sd_d h (next h s) d
+ ]
+ ].
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (↑d) → deg h o s (↑↑d).
+#h #o #s #d #H1
+elim (deg_total h o s) #n #H0
+lapply (deg_next … H0) #H2
+lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
+qed-.
+
+lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (↑d) → deg h o s (↑(d+n)).
+#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
+#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
+qed.
+
+lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
+/2 width=1 by deg_next/ qed-.
+
+lemma sd_d_SS: ∀h,s,d. sd_d h s (↑↑d) = sd_d h (next h s) (↑d).
+// qed.
+
+lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
+#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/arith.ma".
+include "static_2/notation/functions/upspoon_2.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specification *)
+record sh: Type[0] ≝ {
+ next: nat → nat (* next sort in the hierarchy *)
+}.
+
+interpretation "next sort (sort hierarchy)"
+ 'UpSpoon h s = (next h s).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sort.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+record is_lt (h): Prop ≝
+{
+ next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *)
+}.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s.
+#h #Hh #s #n elim n -n [ // ] normalize #n #IH
+lapply (next_lt … Hh ((next h)^n s)) #H
+lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
+qed.
+
+lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s.
+#h #Hh #s #n normalize
+lapply (nexts_le … Hh s n) #H
+@(le_to_lt_to_lt … H) /2 width=1 by next_lt/
+qed.
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