563d5e8bcf
This PR implements support for offset equality constraints in the
`grind` tactic and exhaustive equality propagation for them. The `grind`
tactic can now solve problems such as the following:
```lean
example (f : Nat → Nat) (a b c d e : Nat) :
f (a + 3) = b →
f (c + 1) = d →
c ≤ a + 2 →
a + 1 ≤ e →
e < c →
b = d := by
grind
```
153 lines
4.4 KiB
Text
153 lines
4.4 KiB
Text
opaque g : Nat → Nat
|
|
|
|
@[simp] def f (a : Nat) :=
|
|
match a with
|
|
| 0 => 10
|
|
| x+1 => g (f x)
|
|
|
|
set_option trace.grind.ematch.pattern true
|
|
set_option trace.grind.ematch.instance true
|
|
set_option trace.grind.assert true
|
|
|
|
/--
|
|
info: [grind.ematch.pattern] f.eq_2: [f (Lean.Grind.offset #0 (1))]
|
|
-/
|
|
#guard_msgs in
|
|
grind_pattern f.eq_2 => f (x + 1)
|
|
|
|
|
|
/--
|
|
info: [grind.assert] f (y + 1) = a
|
|
[grind.assert] ¬a = g (f y)
|
|
[grind.ematch.instance] f.eq_2: f y.succ = g (f y)
|
|
[grind.assert] f (y + 1) = g (f y)
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (y + 1) = a → a = g (f y):= by
|
|
grind
|
|
|
|
/--
|
|
info: [grind.assert] f 1 = a
|
|
[grind.assert] ¬a = g (f 0)
|
|
[grind.ematch.instance] f.eq_2: f (Nat.succ 0) = g (f 0)
|
|
[grind.assert] f 1 = g (f 0)
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f 1 = a → a = g (f 0) := by
|
|
grind
|
|
|
|
/--
|
|
info: [grind.assert] f 10 = a
|
|
[grind.assert] ¬a = g (f 9)
|
|
[grind.ematch.instance] f.eq_2: f (Nat.succ 8) = g (f 8)
|
|
[grind.ematch.instance] f.eq_2: f (Nat.succ 9) = g (f 9)
|
|
[grind.assert] f 9 = g (f 8)
|
|
[grind.assert] f 10 = g (f 9)
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f 10 = a → a = g (f 9) := by
|
|
grind
|
|
|
|
/--
|
|
info: [grind.assert] f (c + 2) = a
|
|
[grind.assert] ¬a = g (g (f c))
|
|
[grind.ematch.instance] f.eq_2: f (c + 1).succ = g (f (c + 1))
|
|
[grind.assert] f (c + 2) = g (f (c + 1))
|
|
[grind.ematch.instance] f.eq_2: f c.succ = g (f c)
|
|
[grind.assert] f (c + 1) = g (f c)
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (c + 2) = a → a = g (g (f c)) := by
|
|
grind
|
|
|
|
@[simp] def foo (a : Nat) :=
|
|
match a with
|
|
| 0 => 10
|
|
| 1 => 10
|
|
| a+2 => g (foo a)
|
|
|
|
/--
|
|
info: [grind.ematch.pattern] foo.eq_3: [foo (Lean.Grind.offset #0 (2))]
|
|
-/
|
|
#guard_msgs in
|
|
grind_pattern foo.eq_3 => foo (a_2 + 2)
|
|
|
|
-- The instance is correctly found in the following example.
|
|
-- TODO: to complete the proof, we need linear arithmetic support to prove that `b + 2 = c + 1`.
|
|
/--
|
|
info: [grind.assert] foo (c + 1) = a
|
|
[grind.assert] c = b + 1
|
|
[grind.assert] ¬a = g (foo b)
|
|
[grind.ematch.instance] foo.eq_3: foo b.succ.succ = g (foo b)
|
|
[grind.assert] foo (b + 2) = g (foo b)
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : foo (c + 1) = a → c = b + 1 → a = g (foo b) := by
|
|
grind
|
|
|
|
set_option trace.grind.assert false
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → a = b := by
|
|
fail_if_success grind (ematch := 2)
|
|
sorry
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
[grind.ematch.instance] f.eq_2: f (x + 97).succ = g (f (x + 97))
|
|
[grind.ematch.instance] f.eq_2: f (x + 96).succ = g (f (x + 96))
|
|
[grind.ematch.instance] f.eq_2: f (x + 95).succ = g (f (x + 95))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → a = b := by
|
|
fail_if_success grind (ematch := 5)
|
|
sorry
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
[grind.ematch.instance] f.eq_2: f (x + 97).succ = g (f (x + 97))
|
|
[grind.ematch.instance] f.eq_2: f (x + 96).succ = g (f (x + 96))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → a = b := by
|
|
fail_if_success grind (ematch := 100) (instances := 4)
|
|
sorry
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (y + 9).succ = g (f (y + 9))
|
|
[grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
[grind.ematch.instance] f.eq_2: f (y + 8).succ = g (f (y + 8))
|
|
[grind.ematch.instance] f.eq_2: f (y + 7).succ = g (f (y + 7))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → f (y + 10) = c → a = b := by
|
|
fail_if_success grind (ematch := 100) (instances := 5)
|
|
sorry
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → a = b := by
|
|
fail_if_success grind (gen := 2)
|
|
sorry
|
|
|
|
/--
|
|
info: [grind.ematch.instance] f.eq_2: f (x + 99).succ = g (f (x + 99))
|
|
[grind.ematch.instance] f.eq_2: f (x + 98).succ = g (f (x + 98))
|
|
[grind.ematch.instance] f.eq_2: f (x + 97).succ = g (f (x + 97))
|
|
[grind.ematch.instance] f.eq_2: f (x + 96).succ = g (f (x + 96))
|
|
[grind.ematch.instance] f.eq_2: f (x + 95).succ = g (f (x + 95))
|
|
-/
|
|
#guard_msgs (info) in
|
|
example : f (x + 100) = a → a = b := by
|
|
fail_if_success grind (gen := 5)
|
|
sorry
|