import MyNat.Definition

Tactic : induction

Assuming x is a variable in the local context with an inductive type, induction x applies induction on x to the main goal, producing one goal for each constructor of the inductive type, in which the target is replaced by a general instance of that constructor and an inductive hypothesis is added for each recursive argument to the constructor. If the type of an element in the local context depends on x, that element is reverted and reintroduced afterward, so that the inductive hypothesis incorporates that hypothesis as well.

For example, given n : Nat and a goal with a hypothesis h : P n and target Q n, induction n produces one goal with hypothesis h : P 0 and target Q 0, and one goal with hypotheses h : P (Nat.succ a) and ih₁ : P a → Q a and target Q (Nat.succ a). Here the names a and ih₁ are chosen automatically and are not accessible. You can use with to provide the variables names for each constructor.

induction e, where e is an expression instead of a variable, generalizes e in the goal, and then performs induction on the resulting variable. induction e using r allows the user to specify the principle of induction that should be used. Here r should be a theorem whose result type must be of the form C t, where C is a bound variable and t is a (possibly empty) sequence of bound variables induction e generalizing z₁ ... zₙ, where z₁ ... zₙ are variables in the local context, generalizes over z₁ ... zₙ before applying the induction but then introduces them in each goal. In other words, the net effect is that each inductive hypothesis is generalized. Given x : Nat,

induction x with | zero => tac₁ | succ x' ih => tac₂ uses tactic tac₁ for the zero case, and tac₂ for the succ case.