# Documentation

Std.Tactic.NormCast

# The norm_cast family of tactics. #

Prove a = b using the given simp set.

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Prove a = b by simplifying using move and squash lemmas.

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Construct the expression (e : ty).

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• = do let __discr ← match __discr with | => pure e' | x => failure
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Check if an expression is the coercion of some other expression, and if so return that expression.

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Check if an expression is a numeral in some type, and if so return that type and the natural number.

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This is the main heuristic used alongside the elim and move lemmas. The goal is to help casts move past operators by adding intermediate casts. An expression of the shape: op (↑(x : α) : γ) (↑(y : β) : γ) is rewritten to: op (↑(↑(x : α) : β) : γ) (↑(y : β) : γ) when (↑(↑(x : α) : β) : γ) = (↑(x : α) : γ) can be proven with a squash lemma

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Discharging function used during simplification in the "squash" step.

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Core rewriting function used in the "squash" step, which moves casts upwards and eliminates them.

It tries to rewrite an expression using the elim and move lemmas. On failure, it calls the splitting procedure heuristic.

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If possible, rewrite (n : α) to (Nat.cast n : α) where n is a numeral and α ≠ ℕ. Returns a pair of the new expression and proof that they are equal.

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The core simplification routine of normCast.

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Term elaborator which uses the expected type to insert coercions.

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Implementation of the norm_cast tactic when operating on the main goal.

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Implementation of the norm_cast tactic when operating on a hypothesis.

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Implementation of norm_cast (the full norm_cast calls trivial afterwards).

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assumption_mod_cast runs norm_cast on the goal. For each local hypothesis h, it also normalizes h and tries to use that to close the goal.

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The norm_cast family of tactics is used to normalize casts inside expressions. It is basically a simp tactic with a specific set of lemmas to move casts upwards in the expression. Therefore it can be used more safely as a non-terminating tactic. It also has special handling of numerals.

For instance, given an assumption

a b : ℤ
h : ↑a + ↑b < (10 : ℚ)


writing norm_cast at h will turn h into

h : a + b < 10


You can also use exact_mod_cast, apply_mod_cast, rw_mod_cast or assumption_mod_cast. Writing exact_mod_cast h and apply_mod_cast h will normalize the goal and h before using exact h or apply h. Writing assumption_mod_cast will normalize the goal and for every expression h in the context it will try to normalize h and use exact h. rw_mod_cast acts like the rw tactic but it applies norm_cast between steps.

See also push_cast, for move casts inwards.

The implementation and behavior of the norm_cast family is described in detail at .

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Rewrite with the given rules and normalize casts between steps.

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Normalize the goal and the given expression, then close the goal with exact.

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Normalize the goal and the given expression, then apply the expression to the goal.

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norm_cast tactic in conv mode.

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norm_cast tactic in conv mode.

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push_cast rewrites the expression to move casts toward the leaf nodes. This uses norm_cast lemmas in the forward direction. For example, ↑(a + b) will be written to ↑a + ↑b. It is equivalent to simp only with push_cast. It can also be used at hypotheses with push_cast at h and with extra simp lemmas with push_cast [int.add_zero].

example (a b : ℕ) (h1 : ((a + b : ℕ) : ℤ) = 10) (h2 : ((a + b + 0 : ℕ) : ℤ) = 10) :
((a + b : ℕ) : ℤ) = 10 :=
begin
push_cast,
push_cast at h1,
end


The implementation and behavior of the norm_cast family is described in detail at .

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push_cast rewrites the expression to move casts toward the leaf nodes. This uses norm_cast lemmas in the forward direction. For example, ↑(a + b) will be written to ↑a + ↑b. It is equivalent to simp only with push_cast. It can also be used at hypotheses with push_cast at h and with extra simp lemmas with push_cast [int.add_zero].

example (a b : ℕ) (h1 : ((a + b : ℕ) : ℤ) = 10) (h2 : ((a + b + 0 : ℕ) : ℤ) = 10) :
((a + b : ℕ) : ℤ) = 10 :=
begin
push_cast,
push_cast at h1,

The implementation and behavior of the norm_cast family is described in detail at .