The simp_rw tactic
This module defines a tactic simp_rw which 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, ... and λ x, ....
Implementation notes
The tactic works by taking each rewrite rule in turn and applying simp only to
it. Arguments to simp_rw are of the format used by rw and are translated to
their equivalents for simp.
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, ... 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 ... hₙwill rewrite the given hypotheses.simp_rw [...] at ⊢ h₁ ... hₙrewrites the goal as well as 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 {α β : Type} {f : α → β} {t : set β} : (∀ s, f '' s ⊆ t) = ∀ s : set α, ∀ x ∈ s, x ∈ f ⁻¹' t :=
by simp_rw [set.image_subset_iff, set.subset_def]