# Documentation

## Init.Ext

Registers an extensionality theorem.

• When @[ext] is applied to a structure, it generates .ext and .ext_iff theorems and registers them for the ext tactic.

• When @[ext] is applied to a theorem, the theorem is registered for the ext tactic.

• An optional natural number argument, e.g. @[ext 9000], specifies a priority for the lemma. Higher-priority lemmas are chosen first, and the default is 1000.

• The flag @[ext (flat := false)] causes generated structure extensionality theorems to show inherited fields based on their representation, rather than flattening the parents' fields into the lemma's equality hypotheses. structures in the generated extensionality theorems.

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Creates the type of the extensionality theorem for the given structure, elaborating to x.1 = y.1 → x.2 = y.2 → x = y, for example.

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Creates the type of the iff-variant of the extensionality theorem for the given structure, elaborating to x = y ↔ x.1 = y.1 ∧ x.2 = y.2, for example.

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declare_ext_theorems_for A declares the extensionality theorems for the structure A.

These theorems state that two expressions with the structure type are equal if their fields are equal.

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Applies extensionality lemmas that are registered with the @[ext] attribute.

• ext pat* applies extensionality theorems as much as possible, using the patterns pat* to introduce the variables in extensionality theorems using rintro. For example, the patterns are used to name the variables introduced by lemmas such as funext.
• Without patterns,ext applies extensionality lemmas as much as possible but introduces anonymous hypotheses whenever needed.
• ext pat* : n applies ext theorems only up to depth n.

The ext1 pat* tactic is like ext pat* except that it only applies a single extensionality theorem.

Unused patterns will generate warning. Patterns that don't match the variables will typically result in the introduction of anonymous hypotheses.

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Apply a single extensionality theorem to the current goal.

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ext1 pat* is like ext pat* except that it only applies a single extensionality theorem rather than recursively applying as many extensionality theorems as possible.

The pat* patterns are processed using the rintro tactic. If no patterns are supplied, then variables are introduced anonymously using the intros tactic.

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theorem Prod.ext {α : Type u_1} {β : Type u_2} {x : α × β} {y : α × β} :
x.fst = y.fstx.snd = y.sndx = y
theorem PProd.ext {α : Sort u_1} {β : Sort u_2} {x : PProd α β} {y : PProd α β} :
x.fst = y.fstx.snd = y.sndx = y
theorem Sigma.ext :
∀ {α : Type u_1} {β : αType u_2} {x y : }, x.fst = y.fstHEq x.snd y.sndx = y
theorem PSigma.ext :
∀ {α : Sort u_1} {β : αSort u_2} {x y : }, x.fst = y.fstHEq x.snd y.sndx = y
theorem PUnit.ext (x : PUnit) (y : PUnit) :
x = y
theorem Unit.ext (x : Unit) (y : Unit) :
x = y