FormalConjecturesForMathlib.Data.Int.IntermediateValue

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module

public import Mathlib.Data.Int.Order.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Data.Nat.Init

A discrete intermediate value theorem

An integer-valued sequence whose consecutive terms differ by at most 1 attains every integer value lying between two of its terms. This is the discrete analogue of the intermediate value theorem for a unit-step walk on ℤ.

@[expose] public section

Discrete intermediate value theorem. If an integer-valued sequence f : ℕ → ℤ has consecutive terms differing by at most 1, then it attains every value between f a and f b: for a ≤ b and f a ≤ t ≤ f b there is some index c ∈ [a, b] with f c = t.

theorem Int.exists_eq_of_natAbs_sub_le_one (f : ℕ → ℤ)
    (hf : ∀ n, (f (n + 1) - f n).natAbs ≤ 1) {a b : ℕ} (hab : a ≤ b) {t : ℤ}
    (hta : f a ≤ t) (htb : t ≤ f b) : ∃ c, a ≤ c ∧ c ≤ b ∧ f c = t := byf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb:ℕhab:a ≤ bt:ℤhta:f a ≤ thtb:t ≤ f b⊢ ∃ c, a ≤ c ∧ c ≤ b ∧ f c = t
  induction b, hab using Nat.le_induction with
  | base =>basef:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb:ℕt:ℤhta:f a ≤ thtb:t ≤ f a⊢ ∃ c, a ≤ c ∧ c ≤ a ∧ f c = t exact ⟨a, le_refl a, le_refl a, le_antisymm hta htb⟩All goals completed! 🐙
  | succ b hb ih =>succf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t
    by_cases h : t ≤ f bposf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:t ≤ f b⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = tnegf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:¬t ≤ f b⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t
    ·posf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:t ≤ f b⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t obtain ⟨c, hac, hcb, hfc⟩ := ih hposf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:t ≤ f bc:ℕhac:a ≤ chcb:c ≤ bhfc:f c = t⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t
      exact ⟨c, hac, byf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:t ≤ f bc:ℕhac:a ≤ chcb:c ≤ bhfc:f c = t⊢ c ≤ b + 1 omegaAll goals completed! 🐙, hfc⟩
    ·negf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:¬t ≤ f b⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t have hstep := hf bnegf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:¬t ≤ f bhstep:(f (b + 1) - f b).natAbs ≤ 1⊢ ∃ c, a ≤ c ∧ c ≤ b + 1 ∧ f c = t
      exact ⟨b + 1, byf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:¬t ≤ f bhstep:(f (b + 1) - f b).natAbs ≤ 1⊢ a ≤ b + 1 omegaAll goals completed! 🐙, le_refl _, byf:ℕ → ℤhf:∀ (n : ℕ), (f (n + 1) - f n).natAbs ≤ 1a:ℕb✝:ℕt:ℤhta:f a ≤ tb:ℕhb:a ≤ bih:t ≤ f b → ∃ c, a ≤ c ∧ c ≤ b ∧ f c = thtb:t ≤ f (b + 1)h:¬t ≤ f bhstep:(f (b + 1) - f b).natAbs ≤ 1⊢ f (b + 1) = t omegaAll goals completed! 🐙⟩