(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_2_3.ma".
+include "ground_2/xoa/ex_3_2.ma".
+include "ground_2/xoa/ex_3_3.ma".
+include "ground_2/xoa/ex_3_4.ma".
include "static_2/notation/relations/lrsubeqc_2.ma".
include "static_2/syntax/lenv.ma".
#L1 #L2 * -L1 -L2 //
[ #I #L1 #L2 #_ #H destruct
| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
]
qed-.
lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
+fact lsubr_inv_bind1_aux:
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
#L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
qed-.
(* Basic_2A1: uses: lsubr_inv_pair1 *)
-lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
+lemma lsubr_inv_bind1:
+ ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
/2 width=3 by lsubr_inv_bind1_aux/ qed-.
fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
/2 width=3 by lsubr_inv_atom2_aux/ qed-.
-fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+fact lsubr_inv_bind2_aux:
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
#L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
]
qed-.
-lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+lemma lsubr_inv_bind2:
+ ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
/2 width=3 by lsubr_inv_bind2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
- | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+lemma lsubr_inv_abst1:
+ ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
#K2 #V2 #W2 #_ #_ #H destruct
qed-.
-lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
- ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+lemma lsubr_inv_unit1:
+ ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
| #K2 #V #W #_ #_ #H destruct
-| #I1 #I2 #K2 #V #_ #_ #H destruct
+| #J1 #J2 #K2 #V #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
+lemma lsubr_inv_pair2:
+ ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
-| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
-| #I1 #I1 #K2 #V #_ #_ #H destruct
+| #K1 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
+| #J1 #J1 #K1 #V #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
+lemma lsubr_inv_abbr2:
+ ∀L1,K2,V. L1 ⫃ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
[ /2 width=3 by ex2_intro/
| #K1 #X #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
+lemma lsubr_inv_abst2:
+ ∀L1,K2,W. L1 ⫃ K2.ⓛW →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
qed-.
-lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
- | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
+lemma lsubr_inv_unit2:
+ ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
+ | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
| #K1 #W #V #_ #_ #H destruct
-| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
+| #J1 #J2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
- ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
+lemma lsubr_fwd_bind1:
+ ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
+ ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #J1 #J2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
]
qed-.
-lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
+lemma lsubr_fwd_bind2:
+ ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #J1 #J2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
]
qed-.