(* SUPCLOSURE ***************************************************************)
(* activate genv *)
-(* Note: frees_total requires fqu_drop for all atoms *)
+(* Note: frees_total requires fqu_drop for all atoms
+ fqu_cpx_trans requires fqu_drop for all terms
+ frees_fqus_drops requires fqu_drop restricted on atoms
+*)
inductive fqu: tri_relation genv lenv term ≝
| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
| fqu_bind_dx: ∀p,I,G,L,V,T. fqu G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
-| fqu_drop : ∀I,I1,I2,G,L,V. ⬆*[1] ⓪{I2} ≡ ⓪{I1} →
- fqu G (L.ⓑ{I}V) (⓪{I1}) G L (⓪{I2})
+| fqu_drop : ∀I,G,L,V,T,U. ⬆*[1] T ≡ U → fqu G (L.ⓑ{I}V) U G L T
.
interpretation
(* Basic properties *********************************************************)
-lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G, L, #(i)⦄.
+lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #⫯i⦄ ⊐ ⦃G, L, #i⦄.
/2 width=1 by fqu_drop/ qed.
(* Basic inversion lemmas ***************************************************)
-fact fqu_inv_atom1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀I. L1 = ⋆ → T1 = ⓪{I} → ⊥.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #H destruct
-| #I #G #L #V #T #J #_ #H destruct
-| #p #I #G #L #V #T #J #_ #H destruct
-| #I #G #L #V #T #J #_ #H destruct
-| #I #I1 #I2 #G #L #V #_ #J #H destruct
-]
-qed-.
-
-lemma fqu_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐ ⦃G2, L2, T2⦄ → ⊥.
-/2 width=10 by fqu_inv_atom1_aux/ qed-.
-
fact fqu_inv_sort1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀I,K,V,s. L1 = K.ⓑ{I}V → T1 = ⋆s →
- ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #s #_ #H destruct
-| #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #s #H1 #H2 destruct
- lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=1 by and3_intro/
-]
-qed-.
-
-lemma fqu_inv_sort1: ∀I,G1,G2,K,L2,V,T2,s. ⦃G1, K.ⓑ{I}V, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
-/2 width=7 by fqu_inv_sort1_aux/ qed-.
-
-fact fqu_inv_zero1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀I,K,V. L1 = K.ⓑ{I}V → T1 = #0 →
- ∧∧ G1 = G2 & L2 = K & T2 = V.
+ ∀s. T1 = ⋆s →
+ ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = ⋆s.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #H1 #H2 destruct /2 width=1 by and3_intro/
-| #I #G #L #V #T #J #K #W #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #_ #H destruct
-| #I #G #L #V #T #J #K #W #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #H1 #H2 destruct
- elim (lifts_inv_lref2_uni_lt … HI12) -HI12 //
+[ #I #G #L #T #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #p #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #U #HI12 #s #H destruct
+ lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=3 by ex3_2_intro/
]
qed-.
-lemma fqu_inv_zero1: ∀I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐ ⦃G2, L2, T2⦄ →
- â\88§â\88§ G1 = G2 & L2 = K & T2 = V.
-/2 width=9 by fqu_inv_zero1_aux/ qed-.
+lemma fqu_inv_sort1: ∀G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
+ â\88\83â\88\83J,V. G1 = G2 & L1 = L2.â\93\91{J}V & T2 = â\8b\86s.
+/2 width=3 by fqu_inv_sort1_aux/ qed-.
fact fqu_inv_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀I,K,V,i. L1 = K.ⓑ{I}V → T1 = #(⫯i) →
- ∧∧ G1 = G2 & L2 = K & T2 = #i.
+ ∀i. T1 = #i →
+ (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+ ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #i #_ #H destruct
-| #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #i #H1 #H2 destruct
- lapply (lifts_inv_lref2_uni_ge … HI12) -HI12 /2 width=1 by and3_intro/
+[ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/
+| #I #G #L #V #T #i #H destruct
+| #p #I #G #L #V #T #i #H destruct
+| #I #G #L #V #T #i #H destruct
+| #I #G #L #V #T #U #HI12 #i #H destruct
+ elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_3_intro, or_intror/
]
qed-.
-lemma fqu_inv_lref1: ∀I,G1,G2,K,L2,V,T2,i. ⦃G1, K.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = #i.
-/2 width=9 by fqu_inv_lref1_aux/ qed-.
+lemma fqu_inv_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ →
+ (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+ ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j.
+/2 width=3 by fqu_inv_lref1_aux/ qed-.
fact fqu_inv_gref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀I,K,V,l. L1 = K.ⓑ{I}V → T1 = §l →
- â\88§â\88§ G1 = G2 & L2 = K & T2 = §l.
+ ∀l. T1 = §l →
+ â\88\83â\88\83J,V. G1 = G2 & L1 = L2.â\93\91{J}V & T2 = §l.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #l #_ #H destruct
-| #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #l #H1 #H2 destruct
- lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=1 by and3_intro/
+[ #I #G #L #T #l #H destruct
+| #I #G #L #V #T #l #H destruct
+| #p #I #G #L #V #T #l #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #U #HI12 #l #H destruct
+ lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_2_intro/
]
qed-.
-lemma fqu_inv_gref1: ∀I,G1,G2,K,L2,V,T2,l. ⦃G1, K.ⓑ{I}V, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
- â\88§â\88§ G1 = G2 & L2 = K & T2 = §l.
-/2 width=7 by fqu_inv_gref1_aux/ qed-.
+lemma fqu_inv_gref1: ∀G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
+ â\88\83â\88\83J,V. G1 = G2 & L1 = L2.â\93\91{J}V & T2 = §l.
+/2 width=3 by fqu_inv_gref1_aux/ qed-.
fact fqu_inv_bind1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃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).
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #q #J #W #U #H destruct
-| #I #G #L #V #T #q #J #W #U #H destruct /3 width=1 by and3_intro, or_introl/
-| #p #I #G #L #V #T #q #J #W #U #H destruct /3 width=1 by and3_intro, or_intror/
-| #I #G #L #V #T #q #J #W #U #H destruct
-| #I #I1 #I2 #G #L #V #_ #q #J #W #U #H destruct
+[ #I #G #L #T #q #J #V0 #U0 #H destruct
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
+| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct
+| #I #G #L #V #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/
]
qed-.
lemma fqu_inv_bind1: ∀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).
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1.
/2 width=4 by fqu_inv_bind1_aux/ qed-.
fact fqu_inv_flat1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
- (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
- (∧∧ G1 = G2 & L1 = L2 & U1 = T2).
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+ | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #W #U #H destruct
-| #I #G #L #V #T #J #W #U #H destruct /3 width=1 by and3_intro, or_introl/
-| #p #I #G #L #V #T #J #W #U #H destruct
-| #I #G #L #V #T #J #W #U #H destruct /3 width=1 by and3_intro, or_intror/
-| #I #I1 #I2 #G #L #V #_ #J #W #U #H destruct
+[ #I #G #L #T #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
+| #p #I #G #L #V #T #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
+| #I #G #L #V #T #U #HTU #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/
]
qed-.
lemma fqu_inv_flat1: ∀I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
- (∧∧ G1 = G2 & L1 = L2 & U1 = T2).
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+ | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1.
/2 width=4 by fqu_inv_flat1_aux/ qed-.
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqu_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐ ⦃G2, L2, T2⦄ → ⊥.
+* #x #G1 #G2 #L2 #T2 #H
+[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
+#I #V [3: #i ] #_ #H destruct
+qed-.
+
+lemma fqu_inv_sort1_pair: ∀I,G1,G2,K,L2,V,T2,s. ⦃G1, K.ⓑ{I}V, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
+#I #G1 #G2 #K #L2 #V #T2 #s #H elim (fqu_inv_sort1 … H) -H
+#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_zero1_pair: ∀I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = V.
+#I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
+#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_lref1_pair: ∀I,G1,G2,K,L2,V,T2,i. ⦃G1, K.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = #i.
+#I #G1 #G2 #K #L2 #V #T2 #i #H elim (fqu_inv_lref1 … H) -H *
+#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_gref1_pair: ∀I,G1,G2,K,L2,V,T2,l. ⦃G1, K.ⓑ{I}V, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = §l.
+#I #G1 #G2 #K #L2 #V #T2 #l #H elim (fqu_inv_gref1 … H) -H
+#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
+qed-.
+
(* Basic_2A1: removed theorems 3:
fqu_drop fqu_drop_lt fqu_lref_S_lt
*)
projects / helm.git / blobdiff