(**************************************************************************)
include "basic_2/notation/relations/lrsubeqf_4.ma".
+include "ground_2/relocation/nstream_sor.ma".
include "basic_2/static/frees.ma".
(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
-| lsubf_atom: ∀f1,f2. f2 ⊆ f1 → lsubf (⋆) f1 (⋆) f2
-| lsubf_pair: ∀f1,f2,I,L1,L2,V. f2 ⊆ f1 → lsubf L1 (⫱f1) L2 (⫱f2) →
- lsubf (L1.ⓑ{I}V) f1 (L2.ⓑ{I}V) f2
-| lsubf_beta: ∀f,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≡ f → f ⊆ ⫱f1 → f2 ⊆ f1 →
- lsubf L1 (⫱f1) L2 (⫱f2) → lsubf (L1.ⓓⓝW.V) f1 (L2.ⓛW) f2
+| lsubf_atom: ∀f1,f2. f1 ≡ f2 → lsubf (⋆) f1 (⋆) f2
+| lsubf_push: ∀f1,f2,I1,I2,L1,L2. lsubf L1 (f1) L2 (f2) →
+ lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
+| lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
+ lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
+| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+ lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
+| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+ lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
.
interpretation
(* Basic inversion lemmas ***************************************************)
fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
- L2 = ⋆ ∧ f2 ⊆ f1.
+ f1 ≡ f2 ∧ L2 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
-| #f1 #f2 #I #L1 #L2 #V #_ #_ #H destruct
-| #f #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
]
qed-.
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ ∧ f2 ⊆ f1.
+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⦄ →
+ ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
+ ∃∃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
+ <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+]
+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}.
+/2 width=6 by lsubf_inv_push1_aux/ qed-.
+
fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L2 = K2.ⓑ{I}X) ∨
- ∃∃f,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
- f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+ ∀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⦄ &
+ 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⦄ &
+ K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ L2 = K2.ⓤ{J}.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
+]
+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⦄ &
+ 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⦄ &
+ 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⦄ →
+ ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
+ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #J #K1 #X #H destruct
-| #f1 #f2 #I #L1 #L2 #V #H21 #HL12 #J #K1 #X #H destruct
- /3 width=3 by ex3_intro, or_introl/
-| #f #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H21 #HL12 #J #K1 #X #H destruct
- /3 width=11 by ex7_4_intro, or_intror/
+[ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
+ <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
]
qed-.
-lemma lsubf_inv_pair1: ∀f1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- (∃∃K2. f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L2 = K2.ⓑ{I}X) ∨
- ∃∃f,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
- f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=3 by lsubf_inv_pair1_aux/ 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}.
+/2 width=5 by lsubf_inv_unit1_aux/ qed-.
fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
- L1 = ⋆ ∧ f2 ⊆ f1.
+ f1 ≡ f2 ∧ L1 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
-| #f1 #f2 #I #L1 #L2 #V #_ #_ #H destruct
-| #f #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
]
qed-.
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → L1 = ⋆ ∧ f2 ⊆ f1.
+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⦄ →
+ ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
+ ∃∃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
+ <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+]
+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}.
+/2 width=6 by lsubf_inv_push2_aux/ qed-.
+
fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀I,K2,W. L2 = K2.ⓑ{I}W →
- (∃∃K1.f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L1 = K1.ⓑ{I}W) ∨
- ∃∃f,K1,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
- f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abst & L1 = K1.ⓓⓝW.V.
+ ∀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⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #J #K2 #X #H destruct
-| #f1 #f2 #I #L1 #L2 #V #H21 #HL12 #J #K2 #X #H destruct
- /3 width=3 by ex3_intro, or_introl/
-| #f #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H21 #HL12 #J #K2 #X #H destruct
- /3 width=7 by ex6_3_intro, or_intror/
+[ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
]
qed-.
-lemma lsubf_inv_pair2: ∀f1,f2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, f2⦄ →
- (∃∃K1.f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L1 = K1.ⓑ{I}W) ∨
- ∃∃f,K1,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
- f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abst & L1 = K1.ⓓⓝW.V.
+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⦄ →
+ ∀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⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ L1 = K1.ⓑ{J}V.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
+]
+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⦄ &
+ 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.
+#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.
+#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.
+#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/
+ | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
+ | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
+ ]
+| elim (lsubf_inv_unit1 … 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.
+#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
+ /2 width=7 by ex4_3_intro/
+| #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H 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.
+#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
+| #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
+ /2 width=7 by ex4_3_intro/
+]
+qed-.
+
+lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
+#L elim L -L /2 width=1 by lsubf_inv_atom/
+#L #I #IH #f1 #f2 #H12
+elim (pn_split f1) * #g1 #H destruct
+[ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
+#g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
+qed-.
+
(* 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⦄.
+#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⦄.
+#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/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
+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/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
-#f1 #f2 #L1 #L2 * //
+#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 *********************************************************)
-lemma lsubf_pair_nn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀I,V. ⦃L1.ⓑ{I}V, ⫯f1⦄ ⫃𝐅* ⦃L2.ⓑ{I}V, ⫯f2⦄.
-/4 width=5 by lsubf_fwd_sle, lsubf_pair, sle_next/ qed.
+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⦄).
+#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⦄).
+
+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-.
-lemma lsubf_refl: ∀L,f1,f2. f2 ⊆ f1 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
-#L elim L -L /4 width=1 by lsubf_atom, lsubf_pair, sle_tl/
+lemma lsubf_refl: bi_reflexive … lsubf.
+#L elim L -L /2 width=1 by lsubf_atom, eq_refl/
+#L #I #IH #f elim (pn_split f) * #g #H destruct
+/2 width=1 by lsubf_push, lsubf_bind/
qed.
-lemma lsubf_sle_div: ∀f,f2,L1,L2. ⦃L1, f⦄ ⫃ 𝐅* ⦃L2, f2⦄ →
- ∀f1. f1 ⊆ f2 → ⦃L1, f⦄ ⫃ 𝐅* ⦃L2, f1⦄.
-#f #f2 #L1 #L2 #H elim H -f -f2 -L1 -L2
-/4 width=3 by lsubf_beta, lsubf_pair, lsubf_atom, sle_tl, sle_trans/
+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.
+#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.
+#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/
+| @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
+]
+qed-.
+
+(* Note: this might be moved *)
+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.
+#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
+ elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
+| #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
+| #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
+ [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
+ | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
+ | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+ /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
+ ]
+| #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
+ [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
+ | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
+ | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+ /3 width=11 by lsubf_unit, sor_nn, ex3_2_intro/
+ ]
+]
qed-.
projects / helm.git / blobdiff