This PR changes the interface of the `ForIn`, `ForIn'`, and `ForM`
typeclasses to not take a `Monad m` parameter. This is a breaking change
for most downstream `instance`s, which will will now need to assume
`[Monad m]`.
The rationale is that if the provider of an instance requires `m` to be
a Monad, they should assume this up front. This makes it possible for
the instanve to assume `LawfulMonad m` or some other stronger
requirement, and also to provided a concrete instance for a particular
`m` without assuming a non-canonical `Monad` structure on it.
Zulip: [#lean4 > Monad assumptions in fields of other typeclasses @
💬](https://leanprover.zulipchat.com/#narrow/channel/270676-lean4/topic/Monad.20assumptions.20in.20fields.20of.20other.20typeclasses/near/537102158)
This PR adds the ability to define possibly non-terminating functions
and still be able to reason about them equationally, as long as they are
tail-recursive or monadic.
Typical uses of this feature are
```lean4
def ack : (n m : Nat) → Option Nat
| 0, y => some (y+1)
| x+1, 0 => ack x 1
| x+1, y+1 => do ack x (← ack (x+1) y)
partial_fixpiont
def whileSome (f : α → Option α) (x : α) : α :=
match f x with
| none => x
| some x' => whileSome f x'
partial_fixpiont
def computeLfp {α : Type u} [DecidableEq α] (f : α → α) (x : α) : α :=
let next := f x
if x ≠ next then
computeLfp f next
else
x
partial_fixpiont
noncomputable def geom : Distr Nat := do
let head ← coin
if head then
return 0
else
let n ← geom
return (n + 1)
partial_fixpiont
```
This PR contains
* The necessary fragment of domain theory, up to (a variant of)
Knaster–Tarski theorem (merged as
https://github.com/leanprover/lean4/pull/6477)
* A tactic to solve monotonicity goals compositionally (a bit like
mathlib’s `fun_prop`) (merged as
https://github.com/leanprover/lean4/pull/6506)
* An attribute to extend that tactic (merged as
https://github.com/leanprover/lean4/pull/6506)
* A “derecursifier” that uses that machinery to define recursive
function, including support for dependent functions and mutual
recursion.
* Fixed-point induction principles (technical, tedious to use)
* For `Option`-valued functions: Partial correctness induction theorems
that hide all the domain theory
This is heavily inspired by [Isabelle’s `partial_function`
command](https://isabelle.in.tum.de/doc/codegen.pdf).
