4c57da4b0f
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.
19 lines
616 B
Text
19 lines
616 B
Text
wf1.lean:3:12-3:19: error: fail to show termination for
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g
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with errors
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argument #1 was not used for structural recursion
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failed to eliminate recursive application
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g (x - 1)
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argument #2 was not used for structural recursion
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insufficient number of parameters at recursive application
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g (x - 1)
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structural recursion cannot be used
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Could not find a decreasing measure.
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The arguments relate at each recursive call as follows:
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(<, ≤, =: relation proved, ? all proofs failed, _: no proof attempted)
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x y y - x
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1) 3:12-19 ≤ ? ?
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Please use `termination_by` to specify a decreasing measure.
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