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.
95 lines
2.4 KiB
Text
95 lines
2.4 KiB
Text
Inferred termination argument:
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termination_by n - i
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Inferred termination argument:
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termination_by Array.size xs - i
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Inferred termination argument:
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termination_by Array.size xs - i
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Inferred termination argument:
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termination_by (Array.size xs - i, Array.size ys - j)
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Inferred termination argument:
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termination_by (Array.size xs - i, i - j)
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Inferred termination argument:
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termination_by (Array.size xs - i, i)
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guessLexDiff.lean:85:26-85:38: error: fail to show termination for
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failure
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with errors
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argument #2 was not used for structural recursion
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failed to eliminate recursive application
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_root_.failure xs i
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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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i #1 #2 i + i
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1) 85:26-38 = = = =
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2) 85:58-76 ? < _ _
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3) 85:26-38 = = = =
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4) 88:26-42 _ < _ _
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5) 88:26-42 ? ≤ ≤ ?
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6) 88:26-42 _ < _ _
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7) 88:26-42 _ < _ _
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8) 88:26-42 _ < _ _
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9) 97:8-20 _ < _ _
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#1: Array.size xs - i
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#2: Array.size xs - (i + 1)
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Please use `termination_by` to specify a decreasing measure.
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guessLexDiff.lean:102:4-102:18: error: fail to show termination for
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mutual_failure
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mutual_failure2
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with errors
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structural recursion does not handle mutually recursive functions
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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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Call from mutual_failure to mutual_failure2 at 104:4-24:
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i #1 #2
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i = ? ?
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#3 ? = ?
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Call from mutual_failure to mutual_failure2 at 104:52-78:
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i #1 #2
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i ? _ _
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#3 _ < _
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Call from mutual_failure to mutual_failure2 at 104:4-24:
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i #1 #2
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i _ _ _
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#3 _ = _
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Call from mutual_failure to mutual_failure2 at 111:4-28:
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i #1 #2
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i _ _ _
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#3 _ < _
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Call from mutual_failure to mutual_failure2 at 117:8-28:
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i #1 #2
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i _ _ _
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#3 _ ? _
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Call from mutual_failure2 to mutual_failure at 123:4-23:
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i #3
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i _ _
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#1 _ _
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#2 _ _
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Call from mutual_failure2 to mutual_failure at 123:50-75:
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i #3
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i _ _
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#1 _ _
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#2 _ _
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Call from mutual_failure2 to mutual_failure at 127:4-27:
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i #3
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i _ _
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#1 _ _
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#2 _ _
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Call from mutual_failure2 to mutual_failure at 133:8-27:
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i #3
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i _ _
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#1 _ _
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#2 _ _
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#1: Array.size xs - i
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#2: Array.size xs - (i + 1)
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#3: Array.size xs - i
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Please use `termination_by` to specify a decreasing measure.
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