a common pattern for recursive functions is
```
def countUp (n i acc : Nat) : Nat :=
if i < n then
countUp n (i+1) (acc + i)
else
acc
```
where we increase a value `i` until it hits an upper bound. This is
particularly common with array processing functions:
```
$ git grep 'termination_by.*size.*-' src/|wc -l
26
```
GuessLex now recognizes this pattern. The general approach is:
For every recursive call, check if the context contains hypotheses of
the form `e₁ < e₂` (or similar comparisions), and then consider `e₂ -
e₁` as a termination argument.
Currently, this only fires when `e₁` and `e₂` only depend on the
functions parameters, but not local let-bindings or variables bound in
local pattern matches.
Duplicates are removed.
In the table showing the termination argument failures, long termination
arguments are now given a number and abbreviated as e.g. `#4` in the
table headers.
More examples in the test file, here as some highlights:
```
def distinct (xs : Array Nat) : Bool :=
let rec loop (i j : Nat) : Bool :=
if _ : i < xs.size then
if _ : j < i then
if xs[j] = xs[i] then
false
else
loop i (j+1)
else
loop (i+1) 0
else
true
loop 0 0
```
infers
```
termination_by (Array.size xs - i, i - j)
```
and the weird functions where `i` goes up or down
```
def weird (xs : Array Nat) (i : Nat) : Bool :=
if _ : i < xs.size then
if _ : 0 < i then
if xs[i] = 42 then
weird xs.pop (i - 1)
else
weird xs (i+1)
else
weird xs (i+1)
else
true
decreasing_by all_goals simp_wf; omega
```
infers
```
termination_by (Array.size xs - i, i)
```
but unfortunately needs `decreasing_by` pending the “big
decreasing_tactic refactor” that
I expect we’ll want to do at some point.
closes#3022
With this commit, given the declaration
```
def foo : Nat → Nat
| 0 => 2
| n + 1 => foo n
```
when we unfold `foo (n+1)`, we now obtain `foo n` instead of `foo
(Nat.add n 0)`.
`nat?` checks if an expression is a "natural number in normal form",
i.e. of the form `OfNat n`, where `n` matches `.lit (.natVal n)` for
some `n`.
and if so returns `n`.
This is a widely used helper function in Std/Mathlib when matching on
expressions.
I've reordered some definitions to keep things together. This
introduces:
```
/-- Return the function (name) and arguments of an application. -/
def getAppFnArgs (e : Expr) : Name × Array Expr :=
withApp e λ e a => (e.constName, a)
```
and
```
/-- If the expression is a constant, return that name. Otherwise return `Name.anonymous`. -/
def constName (e : Expr) : Name :=
e.constName?.getD Name.anonymous
```
Give n-ary `Expr.app` constructors such as `mkApp2`, `mkApp3`, ...,
`mkApp10` the `@[match_pattern]` attribute so that it is easier to read
and write pattern matching for applications.
To handle delaborating notations that are functions that can be applied
to arguments, extracts the core function application delaborator as a
separate function that accepts the number of arguments to process and a
delaborator to apply to the "head" of the expression.
Defines `withOverApp`, which has the same interface as the combinator of
the same name from std4, but it uses this core function application
delaborator.
Uses `withOverApp` to improve a number of application delaborators,
notably projections. This means Mathlib can stop using `pp_dot` for
structure fields that have function types.
Incidentally fixes `getParamKinds` to specialize default values to use
supplied arguments, which impacts how default arguments are delaborated.
---------
Co-authored-by: Sebastian Ullrich <sebasti@nullri.ch>
Switches from encoding `let_fun` using an annotated `(fun x : t => b) v`
expression to a function application `letFun v (fun x : t => b)`.
---------
Co-authored-by: Sebastian Ullrich <sebasti@nullri.ch>
there were wrong italics, missing backticks, missing indentation and I
took the liberty to replace `[here]` links with link targets that better
tell the reader what to expect when clicking there.
```
open Lean Meta
-- Docs text:
-- The let-expression `let x : Nat := 2; Nat.succ x` is represented as
def old : Expr :=
Expr.letE `x (.const `Nat []) (.lit (.natVal 2)) (.bvar 0) true
elab "old" : term => return old
#check old -- let x := 2; x : Nat
#reduce old -- 2
def new : Expr :=
Expr.letE `x (.const `Nat []) (.lit (.natVal 2)) (.app (.const `Nat.succ []) (.bvar 0)) true
elab "new" : term => return new
#check new -- let x := 2; Nat.succ x : Nat
#reduce new -- 3
```
