(* 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-.
(* Advanced properties ******************************************************)
(* 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).
+lemma rex_dec (R):
+ (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀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)
(* Main properties **********************************************************)
(* 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.
+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.
#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/
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (pr_sor_isf_bi f1 (⫰f2))
+/3 width=7 by frees_fwd_isfin, frees_bind, sex_join, pr_isf_tl, ex2_intro/
qed.
(* 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.
-#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
+theorem rex_flat (R) (I):
+ ∀L1,L2,V,T. 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 (pr_sor_isf_bi 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.
+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.
#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/
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (pr_sor_isf_bi f1 (⫰f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, pr_isf_tl, ex2_intro/
qed.
(* Negated inversion lemmas *************************************************)
(* 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 → ⊥).
+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 → ⊥).
#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 → ⊥).
+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 → ⊥).
#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.ⓧ → ⊥).
+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.ⓧ → ⊥).
#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-.