include "ground_2/notation/relations/rafter_3.ma".
include "ground_2/relocation/rtmap_istot.ma".
-include "ground_2/relocation/rtmap_uni.ma".
(* RELOCATION MAP ***********************************************************)
coinductive after: relation3 rtmap rtmap rtmap ≝
| after_refl: ∀f1,f2,f,g1,g2,g.
- after f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → after g1 g2 g
+ after f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → after g1 g2 g
| after_push: ∀f1,f2,f,g1,g2,g.
- after f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → after g1 g2 g
+ after f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → after g1 g2 g
| after_next: ∀f1,f2,f,g1,g.
- after f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f = g → after g1 f2 g
+ after f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f = g → after g1 f2 g
.
interpretation "relational composition (rtmap)"
'RAfter f1 f2 f = (after f1 f2 f).
definition H_after_inj: predicate rtmap ≝
- λf1. ð\9d\90\93â¦\83f1â¦\84 →
- â\88\80f,f21,f22. f1 â\8a\9a f21 â\89¡ f â\86\92 f1 â\8a\9a f22 â\89¡ f â\86\92 f21 â\89\97 f22.
+ λf1. ð\9d\90\93â\9dªf1â\9d« →
+ â\88\80f,f21,f22. f1 â\8a\9a f21 â\89\98 f â\86\92 f1 â\8a\9a f22 â\89\98 f â\86\92 f21 â\89¡ f22.
(* Basic inversion lemmas ***************************************************)
-lemma after_inv_ppx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
- â\88\83â\88\83f. f1 â\8a\9a f2 â\89¡ f & â\86\91f = g.
+lemma after_inv_ppx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
+ â\88\83â\88\83f. f1 â\8a\9a f2 â\89\98 f & ⫯f = g.
#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
[ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
>(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
]
qed-.
-lemma after_inv_pnx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
- â\88\83â\88\83f. f1 â\8a\9a f2 â\89¡ f & ⫯f = g.
+lemma after_inv_pnx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
+ â\88\83â\88\83f. f1 â\8a\9a f2 â\89\98 f & â\86\91f = g.
#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
[ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
elim (discr_next_push … Hx2)
]
qed-.
-lemma after_inv_nxx: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89¡ g â\86\92 â\88\80f1. ⫯f1 = g1 →
- â\88\83â\88\83f. f1 â\8a\9a f2 â\89¡ f & ⫯f = g.
+lemma after_inv_nxx: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89\98 g â\86\92 â\88\80f1. â\86\91f1 = g1 →
+ â\88\83â\88\83f. f1 â\8a\9a f2 â\89\98 f & â\86\91f = g.
#g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
[ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
elim (discr_next_push … Hx1)
(* Advanced inversion lemmas ************************************************)
-lemma after_inv_ppp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g â\86\92 f1 â\8a\9a f2 â\89¡ f.
+lemma after_inv_ppp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g â\86\92 f1 â\8a\9a f2 â\89\98 f.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
#x #Hf #Hx destruct <(injective_push … Hx) -f //
qed-.
-lemma after_inv_ppn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
+lemma after_inv_ppn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → ⊥.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
#x #Hf #Hx destruct elim (discr_push_next … Hx)
qed-.
-lemma after_inv_pnn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g â\86\92 f1 â\8a\9a f2 â\89¡ f.
+lemma after_inv_pnn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g â\86\92 f1 â\8a\9a f2 â\89\98 f.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
#x #Hf #Hx destruct <(injective_next … Hx) -f //
qed-.
-lemma after_inv_pnp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → ⊥.
+lemma after_inv_pnp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
#x #Hf #Hx destruct elim (discr_next_push … Hx)
qed-.
-lemma after_inv_nxn: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89¡ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g â\86\92 f1 â\8a\9a f2 â\89¡ f.
+lemma after_inv_nxn: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89\98 g →
+ â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g â\86\92 f1 â\8a\9a f2 â\89\98 f.
#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
#x #Hf #Hx destruct <(injective_next … Hx) -f //
qed-.
-lemma after_inv_nxp: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89¡ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g → ⊥.
+lemma after_inv_nxp: â\88\80g1,f2,g. g1 â\8a\9a f2 â\89\98 g →
+ â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g → ⊥.
#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
#x #Hf #Hx destruct elim (discr_next_push … Hx)
qed-.
-lemma after_inv_pxp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g →
- â\88\83â\88\83f2. f1 â\8a\9a f2 â\89¡ f & â\86\91f2 = g2.
+lemma after_inv_pxp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g →
+ â\88\83â\88\83f2. f1 â\8a\9a f2 â\89\98 f & ⫯f2 = g2.
#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
[ elim (after_inv_pnp … Hg … H1 … H) -g1 -g -f1 -f //
| lapply (after_inv_ppp … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
]
qed-.
-lemma after_inv_pxn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g →
- â\88\83â\88\83f2. f1 â\8a\9a f2 â\89¡ f & ⫯f2 = g2.
+lemma after_inv_pxn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g →
+ â\88\83â\88\83f2. f1 â\8a\9a f2 â\89\98 f & â\86\91f2 = g2.
#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
[ lapply (after_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
| elim (after_inv_ppn … Hg … H1 … H) -g1 -g -f1 -f //
]
qed-.
-lemma after_inv_xxp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g â\86\92 â\88\80f. â\86\91f = g →
- â\88\83â\88\83f1,f2. f1 â\8a\9a f2 â\89¡ f & â\86\91f1 = g1 & â\86\91f2 = g2.
+lemma after_inv_xxp: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g â\86\92 â\88\80f. ⫯f = g →
+ â\88\83â\88\83f1,f2. f1 â\8a\9a f2 â\89\98 f & ⫯f1 = g1 & ⫯f2 = g2.
* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
[ elim (after_inv_nxp … Hg … H) -g2 -g -f //
| elim (after_inv_pxp … Hg … H) -g /2 width=5 by ex3_2_intro/
]
qed-.
-lemma after_inv_xxn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g â\86\92 â\88\80f. ⫯f = g →
- (â\88\83â\88\83f1,f2. f1 â\8a\9a f2 â\89¡ f & â\86\91f1 = g1 & ⫯f2 = g2) ∨
- â\88\83â\88\83f1. f1 â\8a\9a g2 â\89¡ f & ⫯f1 = g1.
+lemma after_inv_xxn: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g â\86\92 â\88\80f. â\86\91f = g →
+ (â\88\83â\88\83f1,f2. f1 â\8a\9a f2 â\89\98 f & ⫯f1 = g1 & â\86\91f2 = g2) ∨
+ â\88\83â\88\83f1. f1 â\8a\9a g2 â\89\98 f & â\86\91f1 = g1.
* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
[ /4 width=5 by after_inv_nxn, or_intror, ex2_intro/
| elim (after_inv_pxn … Hg … H) -g
]
qed-.
-lemma after_inv_pxx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1. â\86\91f1 = g1 →
- (â\88\83â\88\83f2,f. f1 â\8a\9a f2 â\89¡ f & â\86\91f2 = g2 & â\86\91f = g) ∨
- (â\88\83â\88\83f2,f. f1 â\8a\9a f2 â\89¡ f & ⫯f2 = g2 & ⫯f = g).
+lemma after_inv_pxx: â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1. ⫯f1 = g1 →
+ (â\88\83â\88\83f2,f. f1 â\8a\9a f2 â\89\98 f & ⫯f2 = g2 & ⫯f = g) ∨
+ (â\88\83â\88\83f2,f. f1 â\8a\9a f2 â\89\98 f & â\86\91f2 = g2 & â\86\91f = g).
#g1 * * [2: #m2 ] #g2 #g #Hg #f1 #H
[ elim (after_inv_pnx … Hg … H) -g1
/3 width=5 by or_intror, ex3_2_intro/
(* Basic properties *********************************************************)
-corec lemma after_eq_repl_back_2: ∀f1,f. eq_repl_back (λf2. f2 ⊚ f1 ≡ f).
+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/
+| cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by after_next/
]
qed-.
-lemma after_eq_repl_fwd_2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≡ f).
-#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_2/
+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_back_1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≡ f).
+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/
]
qed-.
-lemma after_eq_repl_fwd_1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≡ f).
-#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_1/
+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_back_0: ∀f1,f2. eq_repl_back (λf. f2 ⊚ f1 ≡ f).
+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/
+| cases (eq_inv_nx … H0 … H01) -g01 /3 width=5 by after_next/
]
qed-.
-lemma after_eq_repl_fwd_0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≡ f).
-#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back_0/
+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 **********************************************************)
-corec theorem after_trans1: â\88\80f0,f3,f4. f0 â\8a\9a f3 â\89¡ f4 →
- â\88\80f1,f2. f1 â\8a\9a f2 â\89¡ f0 →
- â\88\80f. f2 â\8a\9a f3 â\89¡ f â\86\92 f1 â\8a\9a f â\89¡ f4.
+corec theorem after_trans1: â\88\80f0,f3,f4. f0 â\8a\9a f3 â\89\98 f4 →
+ â\88\80f1,f2. f1 â\8a\9a f2 â\89\98 f0 →
+ â\88\80f. f2 â\8a\9a f3 â\89\98 f â\86\92 f1 â\8a\9a f â\89\98 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-.
-corec theorem after_trans2: â\88\80f1,f0,f4. f1 â\8a\9a f0 â\89¡ f4 →
- â\88\80f2, f3. f2 â\8a\9a f3 â\89¡ f0 →
- â\88\80f. f1 â\8a\9a f2 â\89¡ f â\86\92 f â\8a\9a f3 â\89¡ f4.
+corec theorem after_trans2: â\88\80f1,f0,f4. f1 â\8a\9a f0 â\89\98 f4 →
+ â\88\80f2, f3. f2 â\8a\9a f3 â\89\98 f0 →
+ â\88\80f. f1 â\8a\9a f2 â\89\98 f â\86\92 f â\8a\9a f3 â\89\98 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
(* Main inversion lemmas ****************************************************)
-corec theorem after_mono: â\88\80f1,f2,x,y. f1 â\8a\9a f2 â\89¡ x â\86\92 f1 â\8a\9a f2 â\89¡ y â\86\92 x â\89\97 y.
+corec theorem after_mono: â\88\80f1,f2,x,y. f1 â\8a\9a f2 â\89\98 x â\86\92 f1 â\8a\9a f2 â\89\98 y â\86\92 x â\89¡ 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-.
-lemma after_mono_eq: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89¡ f â\86\92 â\88\80g1,g2,g. g1 â\8a\9a g2 â\89¡ g →
- f1 â\89\97 g1 â\86\92 f2 â\89\97 g2 â\86\92 f â\89\97 g.
-/4 width=4 by after_mono, after_eq_repl_back_1, after_eq_repl_back_2/ qed-.
+lemma after_mono_eq: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89\98 f â\86\92 â\88\80g1,g2,g. g1 â\8a\9a g2 â\89\98 g →
+ f1 â\89¡ g1 â\86\92 f2 â\89¡ g2 â\86\92 f â\89¡ g.
+/4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
(* Properties on tls ********************************************************)
-lemma after_tls: â\88\80n,f1,f2,f. @â¦\830, f1â¦\84 â\89¡ n â\86\92
- f1 â\8a\9a f2 â\89¡ f â\86\92 ⫱*[n]f1 â\8a\9a f2 â\89¡ ⫱*[n]f.
+lemma after_tls: â\88\80n,f1,f2,f. @â\9dª0, f1â\9d« â\89\98 n â\86\92
+ f1 â\8a\9a f2 â\89\98 f â\86\92 ⫱*[n]f1 â\8a\9a f2 â\89\98 ⫱*[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 *******************************************************)
-corec lemma 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.
-corec lemma 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-.
+
+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: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+corec lemma after_fwd_isid1: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d«.
#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-.
-corec lemma after_fwd_isid2: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+corec lemma after_fwd_isid2: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d«.
#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-.
-lemma after_inv_isid3: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84 â\88§ ð\9d\90\88â¦\83f2â¦\84.
+lemma after_inv_isid3: â\88\80f1,f2,f. f1 â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d« â\88§ ð\9d\90\88â\9dªf2â\9d«.
/3 width=4 by after_fwd_isid2, after_fwd_isid1, conj/ qed-.
(* Properties on isuni ******************************************************)
-lemma after_isid_isuni: â\88\80f1,f2. ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\94â¦\83f1â¦\84 â\86\92 f1 â\8a\9a ⫯f2 â\89¡ ⫯f1.
+lemma after_isid_isuni: â\88\80f1,f2. ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\94â\9dªf1â\9d« â\86\92 f1 â\8a\9a â\86\91f2 â\89\98 â\86\91f1.
#f1 #f2 #Hf2 #H elim H -H
-/5 width=7 by isid_after_dx, after_eq_repl_back_2, after_next, after_push, eq_push_inv_isid/
+/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 [3: #n #m <plus_n_Sm ] (**) (* full auto fails *)
+/4 width=5 by after_uni_next2, after_isid_dx, after_isid_sn, after_next/
qed.
(* Forward lemmas on at *****************************************************)
-lemma after_at_fwd: â\88\80i,i1,f. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 â\88\80f2,f1. f2 â\8a\9a f1 â\89¡ f →
- â\88\83â\88\83i2. @â¦\83i1, f1â¦\84 â\89¡ i2 & @â¦\83i2, f2â¦\84 â\89¡ i.
+lemma after_at_fwd: â\88\80i,i1,f. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 â\88\80f2,f1. f2 â\8a\9a f1 â\89\98 f →
+ â\88\83â\88\83i2. @â\9dªi1, f1â\9d« â\89\98 i2 & @â\9dªi2, f2â\9d« â\89\98 i.
#i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
[ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
[1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
/3 width=9 by at_refl, at_push, at_next, ex2_intro/
qed-.
-lemma after_fwd_at: â\88\80i,i2,i1,f1,f2. @â¦\83i1, f1â¦\84 â\89¡ i2 â\86\92 @â¦\83i2, f2â¦\84 â\89¡ i →
- â\88\80f. f2 â\8a\9a f1 â\89¡ f â\86\92 @â¦\83i1, fâ¦\84 â\89¡ i.
+lemma after_fwd_at: â\88\80i,i2,i1,f1,f2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 @â\9dªi2, f2â\9d« â\89\98 i →
+ â\88\80f. f2 â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, fâ\9d« â\89\98 i.
#i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
[ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
#g2 [ #j2 ] #Hg2 [ #H22 ] #H20
]
qed-.
-lemma after_fwd_at2: â\88\80f,i1,i. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 â\88\80f1,i2. @â¦\83i1, f1â¦\84 â\89¡ i2 →
- â\88\80f2. f2 â\8a\9a f1 â\89¡ f â\86\92 @â¦\83i2, f2â¦\84 â\89¡ i.
+lemma after_fwd_at2: â\88\80f,i1,i. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 â\88\80f1,i2. @â\9dªi1, f1â\9d« â\89\98 i2 →
+ â\88\80f2. f2 â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi2, f2â\9d« â\89\98 i.
#f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (after_at_fwd … Hf … H) -f
#j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
qed-.
-lemma after_fwd_at1: â\88\80i,i2,i1,f,f2. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 @â¦\83i2, f2â¦\84 â\89¡ i →
- â\88\80f1. f2 â\8a\9a f1 â\89¡ f â\86\92 @â¦\83i1, f1â¦\84 â\89¡ i2.
+lemma after_fwd_at1: â\88\80i,i2,i1,f,f2. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 @â\9dªi2, f2â\9d« â\89\98 i →
+ â\88\80f1. f2 â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, f1â\9d« â\89\98 i2.
#i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
#g [ #j1 ] #Hg [ #H01 ] #H00
(* Properties with at *******************************************************)
-lemma after_isuni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. f2 ⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f.
+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 isid_after_sn, after_eq_repl_back_0/
+ /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
- /3 width=5 by after_next/
+ <tls_xn /3 width=5 by after_next/
| #g2 #Hg2 #H2 destruct
elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
- /3 width=5 by after_next/
+ <tls_xn /3 width=5 by after_next/
]
]
qed.
-lemma after_isuni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f → f2 ⊚ 𝐔❴i1❵ ≡ f.
+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 isid_after_dx, after_eq_repl_back_0/
+ /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: * |*: // ]
]
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: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\93â¦\83fâ¦\84.
+lemma after_istot_fwd: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªf2â\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\93â\9dªfâ\9d«.
#f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
#i2 #Hf1 elim (Hf2 i2) -Hf2
/3 width=7 by after_fwd_at, ex_intro/
qed-.
-lemma after_fwd_istot_dx: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84.
+lemma after_fwd_istot_dx: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d«.
#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
#i2 #Hf elim (after_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
qed-.
-lemma after_fwd_istot_sn: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84.
+lemma after_fwd_istot_sn: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d«.
#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
#i #Hf elim (after_at_fwd … Hf … H) -f
#i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
#Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
qed-.
-lemma after_inv_istot: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\88§ ð\9d\90\93â¦\83f1â¦\84.
+lemma after_inv_istot: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d« â\88§ ð\9d\90\93â\9dªf1â\9d«.
/3 width=4 by after_fwd_istot_sn, after_fwd_istot_dx, conj/ qed-.
-lemma after_at1_fwd: â\88\80f1,i1,i2. @â¦\83i1, f1â¦\84 â\89¡ i2 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 â\86\92 â\88\80f. f2 â\8a\9a f1 â\89¡ f →
- â\88\83â\88\83i. @â¦\83i2, f2â¦\84 â\89¡ i & @â¦\83i1, fâ¦\84 â\89¡ i.
+lemma after_at1_fwd: â\88\80f1,i1,i2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« â\86\92 â\88\80f. f2 â\8a\9a f1 â\89\98 f →
+ â\88\83â\88\83i. @â\9dªi2, f2â\9d« â\89\98 i & @â\9dªi1, fâ\9d« â\89\98 i.
#f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
/3 width=8 by after_fwd_at, ex2_intro/
qed-.
-lemma after_fwd_isid_sn: â\88\80f2,f1,f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 â\8a\9a f1 â\89¡ f â\86\92 f1 â\89\97 f â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+lemma after_fwd_isid_sn: â\88\80f2,f1,f. ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 â\8a\9a f1 â\89\98 f â\86\92 f1 â\89¡ f â\86\92 ð\9d\90\88â\9dªf2â\9d«.
#f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
#Hf2 #Hf1 #H @isid_at_total // -Hf2
#i2 #i #Hf2 elim (Hf1 i2) -Hf1
/3 width=7 by at_eq_repl_back, at_mono, at_id_le/
qed-.
-lemma after_fwd_isid_dx: â\88\80f2,f1,f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 â\8a\9a f1 â\89¡ f â\86\92 f2 â\89\97 f â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+lemma after_fwd_isid_dx: â\88\80f2,f1,f. ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 â\8a\9a f1 â\89\98 f â\86\92 f2 â\89¡ f â\86\92 ð\9d\90\88â\9dªf1â\9d«.
#f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
#Hf2 #Hf1 #H2 @isid_at_total // -Hf1
#i1 #i2 #Hi12 elim (after_at1_fwd … Hi12 … Hf) -f1
/3 width=8 by at_inj, at_eq_repl_back/
qed-.
-corec fact after_inj_O_aux: â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_after_inj f1.
+corec fact after_inj_O_aux: â\88\80f1. @â\9dª0, f1â\9d« â\89\98 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
/2 width=1 by after_tls, istot_tls, at_pxx_tls/
qed-.
-fact after_inj_aux: (â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_after_inj f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 â\89¡ i2 → H_after_inj f1.
+fact after_inj_aux: (â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 → H_after_inj f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« â\89\98 i2 → H_after_inj f1.
#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
#i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
projects / helm.git / blobdiff