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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:
40 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L1 = ⋆ →
42 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
44 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
45 | #f1 #f2 #I #L1 #L2 #_ #H destruct
46 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
47 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
51 lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → ∧∧ f1 ≡ f2 & L2 = ⋆.
52 /2 width=3 by lsubf_inv_atom1_aux/ qed-.
54 fact lsubf_inv_push1_aux:
55 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
56 ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
57 ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
58 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
59 [ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
60 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
61 <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
62 | #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
63 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
64 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
68 lemma lsubf_inv_push1:
69 ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
70 ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
71 /2 width=6 by lsubf_inv_push1_aux/ qed-.
73 fact lsubf_inv_pair1_aux:
74 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
75 ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
76 ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
77 | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
78 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
79 I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
80 | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
81 K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
82 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
83 [ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
84 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
85 | #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
86 <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
87 | #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
88 <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
89 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
90 <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
94 lemma lsubf_inv_pair1:
95 ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X,↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
96 ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
97 | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
98 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
99 I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
100 | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
101 K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
102 /2 width=5 by lsubf_inv_pair1_aux/ qed-.
104 fact lsubf_inv_unit1_aux:
105 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
106 ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
107 ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
108 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
109 [ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
110 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
111 | #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
112 <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
113 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
114 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
118 lemma lsubf_inv_unit1:
119 ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I},↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
120 ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
121 /2 width=5 by lsubf_inv_unit1_aux/ qed-.
123 fact lsubf_inv_atom2_aux:
124 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L2 = ⋆ →
126 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
127 [ /2 width=1 by conj/
128 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
129 | #f1 #f2 #I #L1 #L2 #_ #H destruct
130 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
131 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
135 lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → ∧∧f1 ≡ f2 & L1 = ⋆.
136 /2 width=3 by lsubf_inv_atom2_aux/ qed-.
138 fact lsubf_inv_push2_aux:
139 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
140 ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
141 ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
142 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
143 [ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
144 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
145 <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
146 | #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
147 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
148 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
152 lemma lsubf_inv_push2:
153 ∀f1,g2,I2,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},⫯g2⦄ →
154 ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
155 /2 width=6 by lsubf_inv_push2_aux/ qed-.
157 fact lsubf_inv_pair2_aux:
158 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
159 ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
160 ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
161 | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
162 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
163 I = Abst & L1 = K1.ⓓⓝW.V.
164 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
165 [ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
166 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
167 | #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
168 <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
169 | #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
170 <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
171 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
175 lemma lsubf_inv_pair2:
176 ∀f1,g2,I,L1,K2,W. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓑ{I}W,↑g2⦄ →
177 ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
178 | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
179 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
180 I = Abst & L1 = K1.ⓓⓝW.V.
181 /2 width=5 by lsubf_inv_pair2_aux/ qed-.
183 fact lsubf_inv_unit2_aux:
184 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
185 ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
186 ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
187 | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
188 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
189 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
190 [ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
191 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
192 | #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
193 <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
194 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
195 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
196 <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
200 lemma lsubf_inv_unit2:
201 ∀f1,g2,I,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓤ{I},↑g2⦄ →
202 ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
203 | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
204 K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
205 /2 width=5 by lsubf_inv_unit2_aux/ qed-.
207 (* Advanced inversion lemmas ************************************************)
209 lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → f1 ≡ f2.
210 #f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
213 lemma lsubf_inv_push_sn:
214 ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},f2⦄ →
215 ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2.
216 #g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
217 #g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
220 lemma lsubf_inv_bind_sn:
221 ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I},f2⦄ →
222 ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2.
223 #g1 #f2 * #I [2: #X ] #K1 #K2 #H
224 [ elim (lsubf_inv_pair1 … H) -H *
225 [ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
226 | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
227 | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
229 | elim (lsubf_inv_unit1 … H) -H
230 #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
234 lemma lsubf_inv_beta_sn:
235 ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓛW,f2⦄ →
236 ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
237 #g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
238 [ #z2 #Y2 #_ #_ #H destruct
239 | #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
240 /2 width=7 by ex4_3_intro/
241 | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
245 lemma lsubf_inv_unit_sn:
246 ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓤ{J},f2⦄ →
247 ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
248 #g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
249 [ #z2 #Y2 #_ #_ #H destruct
250 | #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
251 | #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
252 /2 width=7 by ex4_3_intro/
256 lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄ → f1 ≡ f2.
257 #L elim L -L /2 width=1 by lsubf_inv_atom/
258 #L #I #IH #f1 #f2 #H12
259 elim (pn_split f1) * #g1 #H destruct
260 [ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
261 #g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
264 (* Basic forward lemmas *****************************************************)
266 lemma lsubf_fwd_bind_tl:
267 ∀f1,f2,I,L1,L2. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ → ⦃L1,⫱f1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄.
268 #f1 #f2 #I #L1 #L2 #H
269 elim (pn_split f1) * #g1 #H0 destruct
270 [ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
271 #g2 #H12 #H destruct //
274 lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
275 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
276 [ /2 width=3 by isid_eq_repl_fwd/
277 | /4 width=3 by isid_inv_push, isid_push/
278 | #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
279 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
280 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
284 lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
285 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
286 [ /2 width=3 by isid_eq_repl_back/
287 | /4 width=3 by isid_inv_push, isid_push/
288 | #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
289 | #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
290 | #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
294 lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → f2 ⊆ f1.
295 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
296 /3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
299 (* Basic properties *********************************************************)
301 axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
303 lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
304 #f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
307 axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
309 lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
310 #f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
313 lemma lsubf_refl: bi_reflexive … lsubf.
314 #L elim L -L /2 width=1 by lsubf_atom, eq_refl/
315 #L #I #IH #f elim (pn_split f) * #g #H destruct
316 /2 width=1 by lsubf_push, lsubf_bind/
319 lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄.
320 /2 width=3 by lsubf_eq_repl_back2/ qed.
322 lemma lsubf_bind_tl_dx:
323 ∀g1,f2,I,L1,L2. ⦃L1,g1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
324 ∃∃f1. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ & g1 = ⫱f1.
325 #g1 #f2 #I #L1 #L2 #H
326 elim (pn_split f2) * #g2 #H2 destruct
327 @ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
330 lemma lsubf_beta_tl_dx:
331 ∀f,f0,g1,L1,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
332 ∀f2,L2,W. ⦃L1,f0⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
333 ∃∃f1. ⦃L1.ⓓⓝW.V,f1⦄ ⫃𝐅+ ⦃L2.ⓛW,f2⦄ & ⫱f1 ⊆ g1.
334 #f #f0 #g1 #L1 #V #Hf #Hg1 #f2
335 elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
336 [ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
337 | @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
341 (* Note: this might be moved *)
342 lemma lsubf_inv_sor_dx:
343 ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
344 ∀f2l,f2r. f2l⋓f2r ≘ f2 →
345 ∃∃f1l,f1r. ⦃L1,f1l⦄ ⫃𝐅+ ⦃L2,f2l⦄ & ⦃L1,f1r⦄ ⫃𝐅+ ⦃L2,f2r⦄ & f1l⋓f1r ≘ f1.
346 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
347 [ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
348 | #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
349 elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
350 elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
351 | #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
352 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
353 elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
354 | #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
355 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
356 elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
357 [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
358 /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
359 | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
360 /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
361 | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
362 /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
364 | #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
365 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
366 elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
367 [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
368 /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
369 | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
370 /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
371 | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
372 /3 width=11 by lsubf_unit, sor_nn, ex3_2_intro/