1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 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Option α) := Option.map Indexed.idx (find_some_idxd f) private def find_some_bounded_acc { α } (f: ℕ → Option α) (idx: ℕ): ℕ -> Option (Indexed α) | 0 => none | n + 1 => match f idx with | some val => some ⟨ val, idx ⟩ | none => find_some_bounded_acc f (idx + 1) n def find_some_bounded_idx { α } (f: ℕ → Option α) (k: ℕ) := find_some_bounded_acc f 0 k def find_some_bounded { α } (f: ℕ → Option α) := Option.map Indexed.val ∘ find_some_bounded_idx f lemma find_some_idxd_eq_none_iff { α } { f: ℕ → Option α }: find_some_idxd f = none ↔ ∀ i, f i = none := by unfold find_some_idxd split_ifs case pos h => simp only [reduceDIte, h, reduceCtorEq, false_iff, not_forall] rcases h with ⟨n, hn⟩ use n apply Aesop.BuiltinRules.not_intro intro a simp_all only [Option.isSome_none, Bool.false_eq_true] case neg h => simp_all [h] lemma find_some_idxd_eq_some_iff { α } { f: ℕ → Option α } (val): find_some_idxd f = some val ↔ f val.idx = some val.val ∧ ∀ i < val.idx, f i = none := by let _dec := Classical.dec; unfold find_some_idxd simp only [Option.some_get, Option.dite_none_right_eq_some] constructor · intro h let ⟨h', h⟩ := h let ⟨n, h''⟩ := h' simp only [Option.some.injEq] at h simp only [←h] simp_all [] · intro h let ⟨k, hk⟩ := h have h'': ∃ n, (f n).isSome = true := by use val.idx simp [k] use h'' simp have hk2 : (f val.idx).isSome = true ∧ ∀ j < val.idx, (f j).isSome ≠ true := by simp_all rw [←Nat.find_eq_iff h''] at hk2 simp_all lemma unroll_all2 (b) (p: ℕ → Prop): (∀ j, b ≤ j → p j) ↔ p b ∧ ∀ j, b + 1 ≤ j → p j := by apply Iff.intro · intro a simp_all only [le_refl, true_and] intro j a_1 have y: b ≤ j := by omega simp [a j y] · intro a j a_1 obtain ⟨left, right⟩ := a have right := right j by_cases c: b + 1 ≤ j · exact right c · have h: j = b := by omega simp_all private lemma find_some_bounded_acc_eq_none_iff { α } { f: ℕ → Option α } (s len): find_some_bounded_acc f s len = none ↔ ∀ j ∈ Set.Ico s (s + len), f j = none := by induction len generalizing s case zero => unfold find_some_bounded_acc simp [imp_false, true_iff] case succ n ih => unfold find_some_bounded_acc have ih := ih (s+1) cases c: f s · simp [ih] conv => arg 2 rw [unroll_all2] simp_all [c] have h: s + (n + 1) = s + 1 + n := by omega simp [h] · simp use s simp [c] lemma find_some_bounded_eq_none_iff { α } { f: ℕ → Option α } (k): find_some_bounded_idx f k = none ↔ ∀ j < k, f j = none := by simp [find_some_bounded_idx, find_some_bounded_acc_eq_none_iff] private lemma find_some_bounded_acc_eq_some_iff { α } { f: ℕ → Option α } (s len val): find_some_bounded_acc f s len = some val ↔ f val.idx = some val.val ∧ val.idx ∈ Set.Ico s (s + len) ∧ ∀ j ∈ Set.Ico s val.idx, f j = none := by induction len generalizing s case zero => unfold find_some_bounded_acc simp case succ n ih => unfold find_some_bounded_acc have ih := ih (s+1) cases c: f s case none => simp [ih] intro h conv => arg 2 rw [unroll_all2] simp [c] intro hh have x: val.idx ≠ s := by simp_all only [mem_Ico, and_imp, true_and, ne_eq] apply Aesop.BuiltinRules.not_intro intro a subst a simp_all only [reduceCtorEq] omega case some => simp constructor · intro h simp [←h, c] intro j intro h1 intro h2 exfalso exact Nat.lt_irrefl s (Nat.lt_of_le_of_lt h1 h2) · intro h cases x: val with | mk val_val val_idx => simp_all [x] by_cases c2: s = val_idx · simp_all · have x: s < val_idx := by have y := h.2.1.1 omega have y := h.2.2 s simp at y have y := y x rename_i x_1 y_1 subst x_1 simp_all only [le_refl, reduceCtorEq] lemma find_some_bounded_idx_eq_some_iff { α } { f: ℕ → Option α } { k val }: find_some_bounded_idx f k = some val ↔ f val.idx = some val.val ∧ val.idx < k ∧ ∀ j < val.idx, f j = none := by simp [find_some_bounded_idx, find_some_bounded_acc_eq_some_iff] lemma find_some_bounded_eq_some_iff { α } { f: ℕ → Option α } { k val }: find_some_bounded f k = some val ↔ ∃ t, f t = some val ∧ t < k ∧ ∀ j < t, f j = none := by simp [find_some_bounded, find_some_bounded_idx_eq_some_iff] constructor · intro h have ⟨ a, h ⟩ := h use a.idx simp_all · intro h have ⟨ a, h ⟩ := h use ⟨ val, a ⟩ lemma find_some_eq_none_iff_find_some_bounded_eq_none { α } { f: ℕ → Option α }: (∀ k, find_some_bounded_idx f k = none) ↔ find_some_idxd f = none := by rw [find_some_idxd_eq_none_iff] conv => pattern find_some_bounded_idx _ _ = _ rw [find_some_bounded_eq_none_iff] apply Iff.intro · intro a k simp [a (k+1) k] · intro a k j a_1 simp_all only lemma find_some_of_find_some_bounded { α } { f: ℕ → Option α } {val} (h: find_some_bounded_idx f k = some val): find_some_idxd f = some val := by rw [find_some_idxd_eq_some_iff] rw [find_some_bounded_idx_eq_some_iff] at h simp_all lemma find_some_bounded_eq_none_iff_find_some_eq_none { α } { f: ℕ → Option α } (val): ∃ k, find_some_bounded_idx f k = some val ↔ find_some_idxd f = some val := by rw [find_some_idxd_eq_some_iff] simp [find_some_bounded_idx_eq_some_iff] use val.idx + 1 simp structure RepeatingFunction (f: ℕ → M) where k: ℕ inv: f '' univ = f '' { n | n < k } def iterative_function {M} (f: ℕ → M) := ∀ i j, f i = f j → f (i+1) = f (j+1) def k' (k a b: ℕ) := if k ≤ a then k else a + ((k-a) % b) lemma k'_bounded (k a b) (h: b > 0): k' k a b ≤ a + (b - 1) := by unfold k' by_cases c: k ≤ a · simp [c] omega · simp [c] have x := (Nat.mod_lt (k-a) h) omega def apply_iterated (f: α → α) (a: α) (k: ℕ) := Nat.iterate f k a @[simp] theorem apply_iterated_fixed {α: Type u} {m: α} {f: α -> α} {t: ℕ} (h: f m = m): apply_iterated f m t = m := by unfold apply_iterated apply Function.iterate_fixed h @[simp] theorem apply_iterated_zero {α: Type u} {m: α} {f: α -> α}: apply_iterated f m 0 = m := by simp [apply_iterated] theorem apply_iterated_succ_apply' {α: Type u} {m: α} {f: α -> α}: apply_iterated f m (n+1) = f (apply_iterated f m n) := by simp [apply_iterated, Function.iterate_succ_apply'] lemma f_not_inj (h: Fintype M) (f: ℕ → M): ∃ b, ∃ a < b, b ≤ h.card ∧ f a = f b := by set n := Fintype.card M + 1 with nh let f' : Fin n → M := fun i => f i.val have : ¬Function.Injective f' := by have x := Fintype.card_le_of_injective f' simp_all [n] -- So f' is not injective ⇒ ∃ i ≠ j, f' i = f' j simp only [Function.Injective] at this push_neg at this obtain ⟨i, j, hne, heq⟩ := this wlog hij : i < j generalizing i j · exact this j i (Eq.symm hne) (Ne.symm heq) (lt_of_le_of_ne (le_of_not_gt hij) (Ne.symm heq)) use j.val use i.val constructor · exact hij constructor · omega · exact hne lemma apply_iterated_mod {M} (h: Fintype M) (f: M -> M) (m:M): ∃ a b, a + (b - 1) < h.card ∧ b > 0 ∧ ∀ k, (apply_iterated f m) k = (apply_iterated f m) (k' k a b) := by have ⟨ b, a, h ⟩ := f_not_inj h (apply_iterated f m) set g := apply_iterated f m use a set bb := b - a have m1 (i k: ℕ): g (a + i) = g (a + i + bb) := by induction i case zero => simp [h, bb, Nat.add_sub_cancel' (Nat.le_of_lt h.1)] case succ n ih => simp [g, apply_iterated] at ih have x1: a + (n + 1) = 1 + (a + n) := by omega have x2: a + (n + 1) + bb = 1 + (a + n + bb) := by omega rw [x2, x1] unfold g simp [apply_iterated] rw [Function.iterate_add] conv => pattern f^[1 + (a + n + bb)] m rw [Function.iterate_add] simp [ih] have m1 (i k: ℕ) (h: i ≥ a): g i = g (i + bb * k) := by induction k generalizing i case zero => simp [h] case succ n ih => have ⟨ s, sh ⟩ := Nat.exists_eq_add_of_le h rw [sh] rw [Nat.mul_add] rw [ih (a + s) (by omega)] have y: a + s + bb * n = a + (s + bb * n) := by omega rw [y] rw [m1] apply congrArg omega exact s use bb constructor · omega unfold k' constructor · omega intro k by_cases c: k ≤ a · simp [c] simp only [c, ↓reduceIte] simp [Nat.mod_def] simp at c set x := (k - a) / bb have xx: a + (k - a - bb * x) ≥ a := by omega --have xx1: (k - a - bb * x ≥ 0) := sorry have xx2: k - a ≥ bb * x := by unfold x have := Nat.mul_le_of_le_div bb ((k-a)/bb) (k-a) (le_refl _) conv => pattern bb * _ rw [Nat.mul_comm] simp_all conv => arg 2 rw [m1 _ x xx] apply congrArg omega def repeating_function_of_iterate_fin_type { M } (h: Fintype M) { f: M → M } (m: M): RepeatingFunction (apply_iterated f m) := { k := h.card inv := by ext x constructor · intro h simp at h rcases h with ⟨y, hy⟩ simp have ⟨ a, b, hf ⟩ := apply_iterated_mod h f m use k' y a b constructor · have bounded := k'_bounded y a b (by simp [hf]) omega simp [←hy, hf.2.2 y] · intro h aesop } def repeating_function_of_composition { α β } { g: α → β } { f: ℕ → α } (h: RepeatingFunction f): RepeatingFunction (g ∘ f) := { k := h.k inv := by rw [image_comp, image_comp, h.inv] } lemma repeating_function_forall {α} { f: ℕ → α } (h: RepeatingFunction f) (p: α → Prop): (∀ i < h.k, p (f i)) ↔ (∀ i, p (f i)) := by have x := h.inv constructor · intro h1 intro k have : f k ∈ f '' univ := mem_image_of_mem f (mem_univ k) rw [x] at this obtain ⟨k', hk', eqfk⟩ := this rw [←eqfk] exact h1 k' hk' · intro h1 intro k intro h2 exact (h1 k) lemma find_some_bounded_idx_eq_find_some_idxd_of_repeating_function { α } { f: ℕ → Option α } (h: RepeatingFunction f): find_some_bounded_idx f h.k = find_some_idxd f := by cases c: find_some_bounded_idx f h.k case some val => simp [find_some_bounded_idx_eq_some_iff] at c apply Eq.symm simp [find_some_idxd_eq_some_iff] simp [c] exact c.2.2 case none => simp [find_some_bounded_eq_none_iff] at c apply Eq.symm rw [find_some_idxd_eq_none_iff] simp_all [repeating_function_forall h (fun val => val = none)] lemma find_some_bounded_eq_find_some_of_repeating_function { α } { f: ℕ → Option α } (h: RepeatingFunction f): find_some_bounded f h.k = find_some f := by simp [find_some_bounded, find_some, find_some_bounded_idx_eq_find_some_idxd_of_repeating_function]