(* Advanced inversion lemmas ************************************************)
-lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+ L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
qed-.
lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
qed-.
lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
(* Main properties **********************************************************)
-theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀T2. ⦃L2,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L2 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
- ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1,T1⦄ ⊆ ⦃L1,T⦄ →
+ ∀L2,T2. ⦃L1,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
- ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1,T1⦄ ⊆ ⦃L,T⦄ →
+ ∀L2,T2. |L| = |L2| → ⦃L,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L #T1 #T #HL1
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2 #HL2
qed-.
theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
- ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+ ∀V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀p,I. ⦃L1,ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀I. ⦃L1,ⓕ{I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I2,T1,T2. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p,I1. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #HL #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
+theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀I1,I2. ⦃L1,ⓕ{I1}V1.T1⦄ ⊆ ⦃L2,ⓕ{I2}V2.T2⦄.
/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.