# Documentation

Mathlib.Tactic.SimpRw

simp_rw functions as a mix of simp and rw. Like rw, it applies each rewrite rule in the given order, but like simp it repeatedly applies these rules and also under binders like ∀ x, ...∀ x, ..., ∃ x, ...∃ x, ... and λ x, .... Usage:

• simp_rw [lemma_1, ..., lemma_n] will rewrite the goal by applying the lemmas in that order. A lemma preceded by ←← is applied in the reverse direction.
• simp_rw [lemma_1, ..., lemma_n] at h₁ ... hₙ will rewrite the given hypotheses.
• simp_rw [...] at * rewrites in the whole context: all hypotheses and the goal.

Lemmas passed to simp_rw must be expressions that are valid arguments to simp. For example, neither simp nor rw can solve the following, but simp_rw can:

example {a : ℕ}
(h1 : ∀ a b : ℕ, a - 1 ≤ b ↔ a ≤ b + 1)
(h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
∀ a b : ℕ, a - 1 ≤ b ↔ a ≤ b + 1)
(h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
≤ b ↔ a ≤ b + 1)
(h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
↔ a ≤ b + 1)
(h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
≤ b + 1)
(h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
≤ b ↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
↔ ∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
∀ c, c < a → c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
→ c < b) :
(∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
≤ b) = ∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
∀ b c : ℕ, c < a → c < b + 1 :=
by simp_rw [h1, h2]
→ c < b + 1 :=
by simp_rw [h1, h2]

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