# mathlibdocumentation

tactic.generalizes

# The generalizes tactic #

This module defines the tactic.generalizes' tactic and its interactive version tactic.interactive.generalizes. These work like generalize, but they can generalize over multiple expressions at once. This is particularly handy when there are dependencies between the expressions, in which case generalize will usually fail but generalizes may succeed.

## Implementation notes #

To generalize the target T over expressions j₁ : J₁, ..., jₙ : Jₙ, we first create the new target type

T' = ∀ (k₁ : J₁) ... (kₙ : Jₙ) (k₁_eq : k₁ = j₁) ... (kₙ_eq : kₙ == jₙ), U


where U is T with any occurrences of the jᵢ replaced by the corresponding kᵢ. Note that some of the kᵢ_eq may be heterogeneous; this happens when there are dependencies between the jᵢ. The construction of T' is performed by generalizes.step1 and generalizes.step2.

Having constructed T', we can assert it and use it to construct a proof of the original target by instantiating the binders with

j₁ ... jₙ (eq.refl j₁) ... (heq.refl jₙ).


This leaves us with a generalized goal. This construction is performed by generalizes.step3.

meta def tactic.generalizes.step1 (md : tactic.transparency) (unify : bool) (e : expr) (to_generalize : list ) :

Input:

• Target expression e.
• List of expressions jᵢ to be generalised, along with a name for the local const that will replace them. The jᵢ must be in dependency order: [n, fin n] is okay but [fin n, n] is not.

Output:

• List of new local constants kᵢ, one for each jᵢ.
• e with the jᵢ replaced by the kᵢ, i.e. e[jᵢ := kᵢ]...[j₀ := k₀].

Note that the substitution also affects the types of the kᵢ: If jᵢ : Jᵢ then kᵢ : Jᵢ[jᵢ₋₁ := kᵢ₋₁]...[j₀ := k₀].

The transparency md and the boolean unify are passed to kabstract when we abstract over occurrences of the jᵢ in e.

meta def tactic.generalizes.step2 (md : tactic.transparency) (to_generalize : list (name × ) :

Input: for each equation that should be generated: the equation name, the argument jᵢ and the corresponding local constant kᵢ from step 1.

Output: for each element of the input list a new local constant of type jᵢ = kᵢ or jᵢ == kᵢ and a proof of jᵢ = jᵢ or jᵢ == jᵢ.

The transparency md is used when determining whether the type of jᵢ is defeq to the type of kᵢ (and thus whether to generate a homogeneous or heterogeneous equation).

meta def tactic.generalizes.step3 (e : expr) (js ks eqs eq_proofs : list expr) :

Input: The jᵢ; the local constants kᵢ from step 1; the equations and their proofs from step 2.

This step is the first one that changes the goal (and also the last one overall). It asserts the generalized goal, then derives the current goal from it. This leaves us with the generalized goal.

meta def tactic.generalizes' (args : list (name × ) (unify : bool := bool.tt) :

Generalizes the target over each of the expressions in args. Given args = [(a₁, h₁, arg₁), ...], this changes the target to

∀ (a₁ : T₁) ... (h₁ : a₁ = arg₁) ..., U


where U is the current target with every occurrence of argᵢ replaced by aᵢ. A similar effect can be achieved by using generalize once for each of the args, but if there are dependencies between the args, this may fail to perform some generalizations.

The replacement is performed using keyed matching/unification with transparency md. unify determines whether matching or unification is used. See kabstract.

The args must be given in dependency order, so [n, fin n] is okay but [fin n, n] will result in an error.

After generalizing the args, the target type may no longer type check. generalizes' will then raise an error.

meta def tactic.generalizes_intro (args : list (name × ) (unify : bool := bool.tt) :

Like generalizes', but also introduces the generalized constants and their associated equations into the context.

meta def tactic.interactive.generalizes (args : interactive.parse generalizes_args_parser) :

Generalizes the target over multiple expressions. For example, given the goal

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
⊢ P (nat.succ n) (fin.succ f)


you can use generalizes [n'_eq : nat.succ n = n', f'_eq : fin.succ f == f'] to get

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
n' : ℕ
n'_eq : n' = nat.succ n
f' : fin n'
f'_eq : f' == fin.succ f
⊢ P n' f'


The expressions must be given in dependency order, so [f'_eq : fin.succ f == f', n'_eq : nat.succ n = n'] would fail since the type of fin.succ f is nat.succ n.

You can choose to omit some or all of the generated equations. For the above example, generalizes [nat.succ n = n', fin.succ f == f'] gets you

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
n' : ℕ
f' : fin n'
⊢ P n' f'


After generalization, the target type may no longer type check. generalizes will then raise an error.

Generalizes the target over multiple expressions. For example, given the goal

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
⊢ P (nat.succ n) (fin.succ f)


you can use generalizes [n'_eq : nat.succ n = n', f'_eq : fin.succ f == f'] to get

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
n' : ℕ
n'_eq : n' = nat.succ n
f' : fin n'
f'_eq : f' == fin.succ f
⊢ P n' f'


The expressions must be given in dependency order, so [f'_eq : fin.succ f == f', n'_eq : nat.succ n = n'] would fail since the type of fin.succ f is nat.succ n.

You can choose to omit some or all of the generated equations. For the above example, generalizes [nat.succ n = n', fin.succ f == f'] gets you

P : ∀ n, fin n → Prop
n : ℕ
f : fin n
n' : ℕ
f' : fin n'
⊢ P n' f'


After generalization, the target type may no longer type check. generalizes will then raise an error.