]
qed-.
+(* Basic properties *********************************************************)
+
+corec lemma after_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ⊚ f1 ≡ f).
+#f1 #f #f2 * -f2 -f1 -f
+#f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
+[ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_refl/
+| cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_push/
+| cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by after_next/
+]
+qed-.
+
+lemma after_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≡ f).
+#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back2/
+qed-.
+
+corec lemma after_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≡ f).
+#f2 #f #f1 * -f2 -f1 -f
+#f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
+[ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by after_refl/
+| cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by after_push/
+| @(after_next … H2 H) /2 width=5 by/
+]
+qed-.
+
+lemma after_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≡ f).
+#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back1/
+qed-.
+
+corec lemma after_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ⊚ f1 ≡ f).
+#f2 #f1 #f * -f2 -f1 -f
+#f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
+[ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by after_refl/
+| cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by after_push/
+| cases (eq_inv_nx … H0 … H01) -g01 /3 width=5 by after_next/
+]
+qed-.
+
+lemma after_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≡ f).
+#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back0/
+qed-.
+
(* Main properties **********************************************************)
-let corec after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
- ∀f1,f2. f1 ⊚ f2 ≡ f0 →
- ∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4 ≝ ?.
+corec theorem after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
+ ∀f1,f2. f1 ⊚ f2 ≡ f0 →
+ ∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4.
#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
cases (after_inv_xxp … Hg0 … H0) -g0
]
qed-.
-let corec after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≡ f4 →
- ∀f2, f3. f2 ⊚ f3 ≡ f0 →
- ∀f. f1 ⊚ f2 ≡ f → f ⊚ f3 ≡ f4 ≝ ?.
+corec theorem after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≡ f4 →
+ ∀f2, f3. f2 ⊚ f3 ≡ f0 →
+ ∀f. f1 ⊚ f2 ≡ f → f ⊚ f3 ≡ f4.
#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
cases (after_inv_xxp … Hg0 … H0) -g0
]
qed-.
-(* Main inversion lemmas on after *******************************************)
+(* Main inversion lemmas ****************************************************)
-let corec after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≡ x → f1 ⊚ f2 ≡ y → x ≗ y ≝ ?.
+corec theorem after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≡ x → f1 ⊚ f2 ≡ y → x ≗ y.
#f1 #f2 #x #y * -f1 -f2 -x
#f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
[ cases (after_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
]
qed-.
-(* Properties on minus ******************************************************)
+lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
+/4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
+
+(* Properties on tls ********************************************************)
-lemma after_minus: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
- f1 ⊚ f2 ≡ f → f1-n ⊚ f2 ≡ f-n.
+lemma after_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
+ f1 ⊚ f2 ≡ f → ⫱*[n]f1 ⊚ f2 ≡ ⫱*[n]f.
#n elim n -n //
-#n #IH #f1 #f2 #f #Hf1 #Hf cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ]
-#g1 #Hg1 #H1 cases (after_inv_nxx … Hf … H1) -Hf /2 width=1 by/
+#n #IH #f1 #f2 #f #Hf1 #Hf
+cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
+cases (after_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+<tls_xn <tls_xn /2 width=1 by/
qed.
-(* Inversion lemmas on isid *************************************************)
+(* Properties on isid *******************************************************)
-let corec isid_after_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊚ f2 ≡ f2 ≝ ?.
+corec lemma after_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊚ f2 ≡ f2.
#f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
/3 width=7 by after_push, after_refl/
-qed-.
+qed.
-let corec isid_after_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⊚ f2 ≡ f1 ≝ ?.
+corec lemma after_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⊚ f2 ≡ f1.
#f2 * -f2 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) * #g1 #H1
[ /3 width=7 by after_refl/
| @(after_next … H1 H1) /3 width=3 by isid_push/
]
-qed-.
+qed.
-lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
-/3 width=6 by isid_after_sn, after_mono/
-qed-.
+(* Inversion lemmas on isid *************************************************)
-lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≗ f.
-/3 width=6 by isid_after_dx, after_mono/
-qed-.
+lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
+/3 width=6 by after_isid_sn, after_mono/ qed-.
-let corec after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ≝ ?.
+lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≗ f.
+/3 width=6 by after_isid_dx, after_mono/ qed-.
+
+corec lemma after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
#f1 #f2 #f * -f1 -f2 -f
#f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
[ /4 width=6 by isid_inv_push, isid_push/ ]
cases (isid_inv_next … H … H0)
qed-.
-let corec after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄ ≝ ?.
+corec lemma after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
#f1 #f2 #f * -f1 -f2 -f
#f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
[ /4 width=6 by isid_inv_push, isid_push/ ]
lemma after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
/3 width=4 by after_fwd_isid2, after_fwd_isid1, conj/ qed-.
+(* Properties on isuni ******************************************************)
+
+lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ⫯f2 ≡ ⫯f1.
+#f1 #f2 #Hf2 #H elim H -H
+/5 width=7 by after_isid_dx, after_eq_repl_back2, after_next, after_push, eq_push_inv_isid/
+qed.
+
+lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ⊚ f1 ≡ f → f2 ⊚ ⫯f1 ≡ f.
+#f2 #H elim H -f2
+[ #f2 #Hf2 #f1 #f #Hf
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+ /4 width=7 by after_isid_inv_sn, after_isid_sn, after_eq_repl_back0, eq_next/
+| #f2 #_ #g2 #H2 #IH #f1 #f #Hf
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+ /3 width=5 by after_next/
+]
+qed.
+
+(* Properties on uni ********************************************************)
+
+lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+@nat_elim2
+/4 width=5 by after_uni_next2, after_isid_sn, after_isid_dx, after_next/
+qed.
+
(* Forward lemmas on at *****************************************************)
lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
]
qed-.
+(* Properties with at *******************************************************)
+
+lemma after_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. f2 ⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ lapply (after_isid_inv_dx … Hf ?) -Hf
+ /3 width=3 by after_isid_sn, after_eq_repl_back0/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct
+ elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ <tls_xn /3 width=5 by after_next/
+ | #g2 #Hg2 #H2 destruct
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ <tls_xn /3 width=5 by after_next/
+ ]
+]
+qed.
+
+lemma after_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f → f2 ⊚ 𝐔❴i1❵ ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ lapply (after_isid_inv_sn … Hf ?) -Hf
+ /3 width=3 by after_isid_dx, after_eq_repl_back0/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by after_push/
+ | #g2 #Hg2 #H2 destruct /3 width=5 by after_next/
+ ]
+]
+qed-.
+
+lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. f2 ⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
+ lapply (after_isid_inv_dx … Hg ?) -Hg
+ /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct
+ elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ <tls_xn /3 width=5 by after_next/
+ | #g2 #Hg2 #H2 destruct
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ <tls_xn /3 width=5 by after_next/
+ ]
+]
+qed.
+
+lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f → f2 ⊚ 𝐔❴⫯i1❵ ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ lapply (after_isid_inv_sn … Hg ?) -Hg
+ /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
+ | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
+ ]
+]
+qed-.
+
+lemma after_uni_one_dx: ∀f2,f. ↑f2 ⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ⊚ f2 ≡ f.
+#f2 #f #H @(after_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
+qed.
+
+lemma after_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ⊚ f1 ≡ f → ↑f1 ⊚ 𝐔❴⫯O❵ ≡ f.
+/3 width=3 by after_uni_succ_sn, at_refl/ qed-.
+
(* Forward lemmas on istot **************************************************)
lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
/3 width=8 by at_inj, at_eq_repl_back/
qed-.
-let corec after_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_after_inj f1 ≝ ?.
+corec fact after_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_after_inj f1.
#f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
| cases (after_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
@(eq_next … H21 H22) -f21 -f22
]
-@(after_inj_O_aux (g1-n) … (g-n)) -after_inj_O_aux
-/2 width=1 by after_minus, istot_minus, at_pxx_minus/
+@(after_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -after_inj_O_aux
+/2 width=1 by after_tls, istot_tls, at_pxx_tls/
qed-.
fact after_inj_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_after_inj f1) →
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