mod_cases tactic

The mod_cases tactic does case disjunction on e % n, where e : ℤ, to yield a number of subgoals in which e ≡ 0 [ZMOD n], ..., e ≡ n-1 [ZMOD n] are assumed.

def Mathlib.Tactic.ModCases.OnModCases (n : ℕ) (a : ℤ) (lb : ℕ) (p : Sort u_1) : Sort (imax 1 u_1)

OnModCases n a lb p represents a partial proof by cases that there exists 0 ≤ z < n such that a ≡ z (mod n). It asserts that if ∃ z, lb ≤ z < n ∧ a ≡ z (mod n) holds, then p (where p is the current goal).

Equations

• Mathlib.Tactic.ModCases.OnModCases n a lb p = ((z : ℕ) → lb ≤ z ∧ z < n ∧ a ≡ ↑z [ZMOD ↑n] → p)

@[inline]

def Mathlib.Tactic.ModCases.onModCases_start (p : Sort u_1) (a : ℤ) (n : ℕ) (hn : Nat.ble 1 n = true) (H : Mathlib.Tactic.ModCases.OnModCases n a 0 p) : p

The first theorem we apply says that ∃ z, 0 ≤ z < n ∧ a ≡ z (mod n). The actual mathematical content of the proof is here.

Equations

• Mathlib.Tactic.ModCases.onModCases_start p a n hn H = H (Int.toNat (a % ↑n)) (_ : 0 ≤ Int.toNat (a % ↑n) ∧ Int.toNat (a % ↑n) < n ∧ a ≡ ↑(Int.toNat (a % ↑n)) [ZMOD ↑n])

@[inline]

def Mathlib.Tactic.ModCases.onModCases_stop (p : Sort u_1) (n : ℕ) (a : ℤ) : Mathlib.Tactic.ModCases.OnModCases n a n p

The end point is that once we have reduced to ∃ z, n ≤ z < n ∧ a ≡ z (mod n) there are no more cases to consider.

Equations

• Mathlib.Tactic.ModCases.onModCases_stop p n a x h = (_ : False).elim

@[inline]

def Mathlib.Tactic.ModCases.onModCases_succ {p : Sort u_1} {n : ℕ} {a : ℤ} (b : ℕ) (h : a ≡ OfNat.ofNat b [ZMOD OfNat.ofNat n] → p) (H : Mathlib.Tactic.ModCases.OnModCases n a (Nat.add b 1) p) : Mathlib.Tactic.ModCases.OnModCases n a b p

The successor case decomposes ∃ z, b ≤ z < n ∧ a ≡ z (mod n) into a ≡ b (mod n) ∨ ∃ z, b+1 ≤ z < n ∧ a ≡ z (mod n), and the a ≡ b (mod n) → p case becomes a subgoal.

Equations

• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.ModCases.«tacticMod_cases_:_%_» : Lean.ParserDescr

• The tactic mod_cases h : e % 3 will perform a case disjunction on e : ℤ and yield subgoals containing the assumptions h : e ≡ 0 [ZMOD 3], h : e ≡ 1 [ZMOD 3], h : e ≡ 2 [ZMOD 3] respectively.
• In general, mod_cases h : e % n works when n is a positive numeral and e is an expression of type ℤ.
• If h is omitted as in mod_cases e % n, it will be default-named H.

Equations

• One or more equations did not get rendered due to their size.