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15 include "static_2/notation/relations/lrsubeqf_4.ma".
16 include "ground_2/relocation/nstream_sor.ma".
17 include "static_2/static/frees.ma".
19 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
21 inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
22 | lsubf_atom: ∀f1,f2. f1 ≡ f2 → lsubf (⋆) f1 (⋆) f2
23 | lsubf_push: ∀f1,f2,I1,I2,L1,L2. lsubf L1 (f1) L2 (f2) →
24 lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
25 | lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
26 lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
27 | lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
28 lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
29 | lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
30 lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
34 "local environment refinement (context-sensitive free variables)"
35 'LRSubEqF L1 f1 L2 f2 = (lsubf L1 f1 L2 f2).
37 (* Basic inversion lemmas ***************************************************)
39 fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
41 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
43 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
44 | #f1 #f2 #I #L1 #L2 #_ #H destruct
45 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
46 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
50 lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
51 /2 width=3 by lsubf_inv_atom1_aux/ qed-.
53 fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
54 ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
55 ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
56 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
57 [ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
58 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
59 <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
60 | #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
61 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
62 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
66 lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
67 ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
68 /2 width=6 by lsubf_inv_push1_aux/ qed-.
70 fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
71 ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
72 ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
73 | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
74 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
75 I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
76 | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
77 K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
79 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
80 [ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
81 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
82 | #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
83 <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
84 | #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
85 <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
86 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
87 <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
91 lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
92 ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
93 | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
94 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
95 I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
96 | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
97 K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
99 /2 width=5 by lsubf_inv_pair1_aux/ qed-.
101 fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
102 ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
103 ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
104 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
105 [ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
106 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
107 | #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
108 <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
109 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
110 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
114 lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I}, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
115 ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
116 /2 width=5 by lsubf_inv_unit1_aux/ qed-.
118 fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
120 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
121 [ /2 width=1 by conj/
122 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
123 | #f1 #f2 #I #L1 #L2 #_ #H destruct
124 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
125 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
129 lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
130 /2 width=3 by lsubf_inv_atom2_aux/ qed-.
132 fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
133 ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
134 ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
135 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
136 [ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
137 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
138 <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
139 | #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
140 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
141 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
145 lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, ⫯g2⦄ →
146 ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
147 /2 width=6 by lsubf_inv_push2_aux/ qed-.
149 fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
150 ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
151 ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
152 | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
153 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
154 I = Abst & L1 = K1.ⓓⓝW.V.
155 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
156 [ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
157 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
158 | #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
159 <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
160 | #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
161 <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
162 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
166 lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, ↑g2⦄ →
167 ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
168 | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
169 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
170 I = Abst & L1 = K1.ⓓⓝW.V.
171 /2 width=5 by lsubf_inv_pair2_aux/ qed-.
173 fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
174 ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
175 ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
176 | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
177 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
179 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
180 [ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
181 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
182 | #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
183 <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
184 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
185 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
186 <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
190 lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓤ{I}, ↑g2⦄ →
191 ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
192 | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
193 K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
195 /2 width=5 by lsubf_inv_unit2_aux/ qed-.
197 (* Advanced inversion lemmas ************************************************)
199 lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2.
200 #f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
203 lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, f2⦄ →
204 ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2.
205 #g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
206 #g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
209 lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I}, ↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I}, f2⦄ →
210 ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2.
211 #g1 #f2 * #I [2: #X ] #K1 #K2 #H
212 [ elim (lsubf_inv_pair1 … H) -H *
213 [ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
214 | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
215 | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
217 | elim (lsubf_inv_unit1 … H) -H
218 #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
222 lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓛW, f2⦄ →
223 ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
224 #g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
225 [ #z2 #Y2 #_ #_ #H destruct
226 | #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
227 /2 width=7 by ex4_3_intro/
228 | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
232 lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J}, f2⦄ →
233 ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
234 #g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
235 [ #z2 #Y2 #_ #_ #H destruct
236 | #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
237 | #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
238 /2 width=7 by ex4_3_intro/
242 lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
243 #L elim L -L /2 width=1 by lsubf_inv_atom/
244 #L #I #IH #f1 #f2 #H12
245 elim (pn_split f1) * #g1 #H destruct
246 [ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
247 #g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
250 (* Basic forward lemmas *****************************************************)
252 lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
253 ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ → ⦃L1, ⫱f1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄.
254 #f1 #f2 #I #L1 #L2 #H
255 elim (pn_split f1) * #g1 #H0 destruct
256 [ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
257 #g2 #H12 #H destruct //
260 lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
261 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
262 [ /2 width=3 by isid_eq_repl_fwd/
263 | /4 width=3 by isid_inv_push, isid_push/
264 | #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
265 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
266 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
270 lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
271 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
272 [ /2 width=3 by isid_eq_repl_back/
273 | /4 width=3 by isid_inv_push, isid_push/
274 | #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
275 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
276 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
280 lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
281 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
282 /3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
285 (* Basic properties *********************************************************)
287 axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
289 lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
290 #f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
293 axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
295 lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
296 #f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
299 lemma lsubf_refl: bi_reflexive … lsubf.
300 #L elim L -L /2 width=1 by lsubf_atom, eq_refl/
301 #L #I #IH #f elim (pn_split f) * #g #H destruct
302 /2 width=1 by lsubf_push, lsubf_bind/
305 lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
306 /2 width=3 by lsubf_eq_repl_back2/ qed.
308 lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1, g1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
309 ∃∃f1. ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ & g1 = ⫱f1.
310 #g1 #f2 #I #L1 #L2 #H
311 elim (pn_split f2) * #g2 #H2 destruct
312 @ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
315 lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
316 ∀f2,L2,W. ⦃L1, f0⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
317 ∃∃f1. ⦃L1.ⓓⓝW.V, f1⦄ ⫃𝐅* ⦃L2.ⓛW, f2⦄ & ⫱f1 ⊆ g1.
318 #f #f0 #g1 #L1 #V #Hf #Hg1 #f2
319 elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
320 [ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
321 | @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
325 (* Note: this might be moved *)
326 lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
327 ∀f2l,f2r. f2l⋓f2r ≘ f2 →
328 ∃∃f1l,f1r. ⦃L1, f1l⦄ ⫃𝐅* ⦃L2, f2l⦄ & ⦃L1, f1r⦄ ⫃𝐅* ⦃L2, f2r⦄ & f1l⋓f1r ≘ f1.
329 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
330 [ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
331 | #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
332 elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
333 elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
334 | #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
335 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
336 elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
337 | #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
338 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
339 elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
340 [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
341 /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
342 | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
343 /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
344 | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
345 /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
347 | #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
348 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
349 elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
350 [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
351 /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
352 | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
353 /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
354 | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
355 /3 width=11 by lsubf_unit, sor_nn, ex3_2_intro/