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lean4-htt/tests/elab/funInduction.lean
Garmelon 08eb78a5b2
chore: switch to new test/bench suite (#12590) 
This PR sets up the new integrated test/bench suite. It then migrates
all benchmarks and some related tests to the new suite. There's also
some documentation and some linting.

For now, a lot of the old tests are left alone so this PR doesn't become
even larger than it already is. Eventually, all tests should be migrated
to the new suite though so there isn't a confusing mix of two systems.
2026-02-25 13:51:53 +00:00

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namespace Ex1
variable (P : Nat → Prop)
def ackermann : (Nat × Nat) → Nat
| (0, m) => m + 1
| (n+1, 0) => ackermann (n, 1)
| (n+1, m+1) => ackermann (n, ackermann (n + 1, m))
termination_by p => p
/--
error: Failed: `fail` tactic was invoked
case case1
P : Nat → Prop
m✝ : Nat
⊢ P (m✝ + 1)
case case2
P : Nat → Prop
n✝ : Nat
ih1✝ : P (ackermann (n✝, 1))
⊢ P (ackermann (n✝, 1))
case case3
P : Nat → Prop
n✝ m✝ : Nat
ih2✝ : P (ackermann (n✝ + 1, m✝))
ih1✝ : P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
⊢ P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
-/
#guard_msgs in
example : P (ackermann p) := by
fun_induction ackermann p
fail
/--
error: Failed: `fail` tactic was invoked
case case1
P : Nat → Prop
m✝ : Nat
⊢ P (m✝ + 1)
case case2
P : Nat → Prop
n✝ : Nat
⊢ P (ackermann (n✝, 1))
case case3
P : Nat → Prop
n✝ m✝ : Nat
⊢ P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
-/
#guard_msgs in
example : P (ackermann p) := by
fun_cases ackermann p
fail
/--
error: unsolved goals
case case1
P : Nat → Prop
n m m✝ : Nat
⊢ P (m✝ + 1)
case case2
P : Nat → Prop
n m n✝ : Nat
ih1✝ : P (ackermann (n✝, 1))
⊢ P (ackermann (n✝, 1))
case case3
P : Nat → Prop
n m n✝ m✝ : Nat
ih2✝ : P (ackermann (n✝ + 1, m✝))
ih1✝ : P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
⊢ P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
-/
#guard_msgs in
example : P (ackermann (n, m)) := by
fun_induction ackermann (n, m)
/--
error: unsolved goals
case case1
P : Nat → Prop
n m m✝ : Nat
⊢ P (m✝ + 1)
case case2
P : Nat → Prop
n m n✝ : Nat
⊢ P (ackermann (n✝, 1))
case case3
P : Nat → Prop
n m n✝ m✝ : Nat
⊢ P (ackermann (n✝, ackermann (n✝ + 1, m✝)))
-/
#guard_msgs in
example : P (ackermann (n, m)) := by
fun_cases ackermann (n, m)
-- Testing Generalization:
/--
error: unsolved goals
case case1
P : Nat → Prop
n m m✝ : Nat
⊢ P (ackermann (n, m))
case case2
P : Nat → Prop
n m n✝ : Nat
⊢ P (ackermann (n, m))
case case3
P : Nat → Prop
n m n✝ m✝ : Nat
⊢ P (ackermann (n, m))
-/
#guard_msgs in
example : P (ackermann (n, m)) := by
fun_cases ackermann (n+n, m)
end Ex1
namespace Ex2
variable (P : Nat → Prop)
def ackermann : Nat → Nat → Nat
| 0, m => m + 1
| n+1, 0 => ackermann n 1
| n+1, m+1 => ackermann n (ackermann (n + 1) m)
termination_by n m => (n, m)
/--
error: unsolved goals
case case1
P : Nat → Prop
m✝ : Nat
⊢ P (m✝ + 1)
case case2
P : Nat → Prop
n✝ : Nat
ih1✝ : P (ackermann n✝ 1)
⊢ P (ackermann n✝ 1)
case case3
P : Nat → Prop
n✝ m✝ : Nat
ih2✝ : P (ackermann (n✝ + 1) m✝)
ih1✝ : P (ackermann n✝ (ackermann (n✝ + 1) m✝))
⊢ P (ackermann n✝ (ackermann (n✝ + 1) m✝))
-/
#guard_msgs in
example : P (ackermann n m) := by
fun_induction ackermann n m
/--
error: Expected fully applied application of `ackermann` with 2 arguments, but found 1 arguments
-/
#guard_msgs in
example : P (ackermann n m) := by
fun_induction ackermann n
end Ex2
namespace Ex3
variable (P : List α → Prop)
def ackermann {α} (inc : List α) : List α → List α → List α
| [], ms => ms ++ inc
| _::ns, [] => ackermann inc ns inc
| n::ns, _::ms => ackermann inc ns (ackermann inc (n::ns) ms)
termination_by ns ms => (ns, ms)
/--
error: unsolved goals
case case1
α : Type u_1
P : List α → Prop
inc ms✝ : List α
⊢ P (ms✝ ++ inc)
case case2
α : Type u_1
P : List α → Prop
inc : List α
head✝ : α
ns✝ : List α
ih1✝ : P (ackermann inc ns✝ inc)
⊢ P (ackermann inc ns✝ inc)
case case3
α : Type u_1
P : List α → Prop
inc : List α
n✝ : α
ns✝ : List α
head✝ : α
ms✝ : List α
ih2✝ : P (ackermann inc (n✝ :: ns✝) ms✝)
ih1✝ : P (ackermann inc ns✝ (ackermann inc (n✝ :: ns✝) ms✝))
⊢ P (ackermann inc ns✝ (ackermann inc (n✝ :: ns✝) ms✝))
-/
#guard_msgs in
example : P (ackermann inc n m) := by
fun_induction ackermann inc n m
/--
error: Expected fully applied application of `ackermann` with 4 arguments, but found 3 arguments
-/
#guard_msgs in
example : P (ackermann inc n m) := by
fun_induction ackermann inc n
/--
error: Expected fully applied application of `ackermann` with 4 arguments, but found 2 arguments
-/
#guard_msgs in
example : P (ackermann inc n m) := by
fun_induction ackermann inc
end Ex3
namespace Structural
variable (P : Nat → Prop)
def fib : Nat → Nat
| 0 => 0
| 1 => 1
| n+2 => fib n + fib (n+1)
termination_by structural x => x
/--
error: Failed: `fail` tactic was invoked
case case1
P : Nat → Prop
⊢ P 0
case case2
P : Nat → Prop
⊢ P 1
case case3
P : Nat → Prop
n✝ : Nat
ih2✝ : P (fib n✝)
ih1✝ : P (fib (n✝ + 1))
⊢ P (fib n✝ + fib (n✝ + 1))
-/
#guard_msgs in
example : P (fib n) := by
fun_induction fib n
fail
example : n ≤ fib (n + 2) := by
fun_induction fib n
case case1 => simp [fib]
case case2 => simp [fib]
case case3 n ih1 ih2 => simp_all [fib]; omega
example : n ≤ fib (n + 2) := by
fun_induction fib n with
| case1 | case2 => simp [fib]
| case3 => simp_all [fib]; omega
end Structural
namespace StructuralWithOmittedParam
variable (P : Nat → Prop)
variable (inc : Nat)
def fib : Nat → Nat
| 0 => 0
| 1 => inc
| n+2 => fib n + fib (n+1)
termination_by structural x => x
/--
info: StructuralWithOmittedParam.fib.induct (motive : Nat → Prop) (case1 : motive 0) (case2 : motive 1)
(case3 : ∀ (n : Nat), motive n → motive (n + 1) → motive n.succ.succ) (a✝ : Nat) : motive a✝
-/
#guard_msgs in
#check fib.induct -- NB: No inc showing up
/--
error: Failed: `fail` tactic was invoked
case case1
P : Nat → Prop
inc : Nat
⊢ P (fib 2 0)
case case2
P : Nat → Prop
inc : Nat
⊢ P (fib 2 1)
case case3
P : Nat → Prop
inc n✝ : Nat
ih2✝ : P (fib 2 n✝)
ih1✝ : P (fib 2 (n✝ + 1))
⊢ P (fib 2 n✝.succ.succ)
-/
#guard_msgs in
example : P (fib 2 n) := by
fun_induction fib 3 n
fail
/--
error: Failed: `fail` tactic was invoked
case case1
P : Nat → Prop
inc : Nat
⊢ P 0
case case2
P : Nat → Prop
inc : Nat
⊢ P 2
case case3
P : Nat → Prop
inc n✝ : Nat
ih2✝ : P (fib 2 n✝)
ih1✝ : P (fib 2 (n✝ + 1))
⊢ P (fib 2 n✝ + fib 2 (n✝ + 1))
-/
#guard_msgs in
example : P (fib 2 n) := by
fun_induction fib _ n
fail
end StructuralWithOmittedParam
namespace StructuralIndices
-- Testing recursion on an indexed data type
inductive Finn : Nat → Type where
| fzero : {n : Nat} → Finn n
| fsucc : {n : Nat} → Finn n → Finn (n+1)
def Finn.min (x : Bool) {n : Nat} (m : Nat) : Finn n → (f : Finn n) → Finn n
| fzero, _ => fzero
| _, fzero => fzero
| fsucc i, fsucc j => fsucc (Finn.min (not x) (m + 1) i j)
termination_by structural i => i
def Finn.min' (x : Bool) {n : Nat} (m : Nat) : Finn n → (f : Finn n) → Finn n
| fzero, _ => fzero
| _, fzero => fzero
| fsucc i, fsucc j => fsucc (Finn.min' (not x) (m + 1) i j)
termination_by structural _ j => j
def Finn.min'' (x : Bool) {n : Nat} (m : Nat) : Finn n → (f : Finn n) → Finn n
| fzero, _ => fzero
| _, fzero => fzero
| fsucc i, fsucc j => fsucc (Finn.min'' (not x) (m + 1) i j)
termination_by structural n
def Finn.le : Finn n → Finn n → Bool
| fzero, _ => true
| _, fzero => false
| fsucc i, fsucc j => Finn.le i j
theorem Finn.min_le_right₀ : (Finn.min x m i j).le j := by
induction x, m, i, j using @Finn.min.induct <;> simp_all [Finn.min, Finn.le]
theorem Finn.min_le_right : (Finn.min x m i j).le j := by
fun_induction Finn.min x m i j <;> simp_all [Finn.min, Finn.le]
theorem Finn.min_le_right' : (Finn.min' x m i j).le j := by
fun_induction Finn.min' x m i j <;> simp_all [Finn.min', Finn.le]
theorem Finn.min_le_right'' : (Finn.min'' x m i j).le j := by
fun_induction Finn.min'' x m i j <;> simp_all [Finn.min'', Finn.le]
end StructuralIndices
namespace Nonrec
def foo := 1
/-- error: No functional induction theorem for `foo`, or function is mutually recursive -/
#guard_msgs in
example : True := by
fun_induction foo
end Nonrec
namespace Mutual
inductive Tree (α : Type u) : Type u where
| node : α → (Bool → List (Tree α)) → Tree α
-- Recursion over nested inductive
mutual
def Tree.size : Tree α → Nat
| .node _ tsf => 1 + size_aux (tsf true) + size_aux (tsf false)
termination_by structural t => t
def Tree.size_aux : List (Tree α) → Nat
| [] => 0
| t :: ts => size t + size_aux ts
end
/-- error: No functional induction theorem for `Tree.size`, or function is mutually recursive -/
#guard_msgs in
example (t : Tree α) : True := by
fun_induction Tree.size t
end Mutual
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