8655f7706f
This PR changes how Lean proves the equational theorems for structural recursion. The core idea is to let-bind the `f` argument to `brecOn` and rewriting `.brecOn` with an unfolding theorem. This means no extra case split for the `.rec` in `.brecOn` is needed, and `simp` doesn't change the `f` argument which can break the definitional equality with the defined function. With this, we can prove the unfolding theorem first, and derive the equational theorems from that, like for all other ways of defining recursive functions. Backs out the changes from #10415, the old strategy works well with the new goals. Fixes #5667 Fixes #10431 Fixes #10195 Fixes #2962
131 lines
3.3 KiB
Text
131 lines
3.3 KiB
Text
inductive Expr where
|
|
| const (i : BitVec 32)
|
|
| op (op : Unit) (e1 : Expr)
|
|
|
|
def optimize : Expr → Expr
|
|
| .const i => .const i
|
|
| .op bop e1 =>
|
|
match bop, optimize e1 with
|
|
| _, .const _ => .op bop (.const 0)
|
|
| _, _ => .const 0
|
|
|
|
/--
|
|
info: optimize.eq_def (x✝ : Expr) :
|
|
optimize x✝ =
|
|
match x✝ with
|
|
| Expr.const i => Expr.const i
|
|
| Expr.op bop e1 =>
|
|
match bop, optimize e1 with
|
|
| x, Expr.const i => Expr.op bop (Expr.const 0)
|
|
| x, x_1 => Expr.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#check optimize.eq_def
|
|
|
|
/--
|
|
info: equations:
|
|
@[defeq] theorem optimize.eq_1 : ∀ (i : BitVec 32), optimize (Expr.const i) = Expr.const i
|
|
theorem optimize.eq_2 : ∀ (e1 : Expr) (bop : Unit) (i : BitVec 32),
|
|
optimize e1 = Expr.const i → optimize (Expr.op bop e1) = Expr.op bop (Expr.const 0)
|
|
theorem optimize.eq_3 : ∀ (e1 : Expr) (bop : Unit),
|
|
(∀ (i : BitVec 32), optimize e1 = Expr.const i → False) → optimize (Expr.op bop e1) = Expr.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#print equations optimize
|
|
|
|
-- works:
|
|
|
|
def optimize2 : Expr → Expr
|
|
| .const i => .const i
|
|
| .op bop e1 =>
|
|
match optimize2 e1 with
|
|
| .const _ => .op bop (.const 0)
|
|
| _ => .const 0
|
|
|
|
/--
|
|
info: equations:
|
|
@[defeq] theorem optimize2.eq_1 : ∀ (i : BitVec 32), optimize2 (Expr.const i) = Expr.const i
|
|
@[defeq] theorem optimize2.eq_2 : ∀ (bop : Unit) (e1 : Expr),
|
|
optimize2 (Expr.op bop e1) =
|
|
match optimize2 e1 with
|
|
| Expr.const i => Expr.op bop (Expr.const 0)
|
|
| x => Expr.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#print equations optimize2
|
|
|
|
-- also works:
|
|
|
|
def optimize3 : Expr → Expr
|
|
| .const i => .const i
|
|
| .op bop e1 =>
|
|
match bop, e1 with
|
|
| _, .const _ => .op bop (optimize3 e1)
|
|
| _, _ => .const 0
|
|
|
|
/--
|
|
info: equations:
|
|
@[defeq] theorem optimize3.eq_1 : ∀ (i : BitVec 32), optimize3 (Expr.const i) = Expr.const i
|
|
@[defeq] theorem optimize3.eq_2 : ∀ (bop : Unit) (i : BitVec 32),
|
|
optimize3 (Expr.op bop (Expr.const i)) = Expr.op bop (optimize3 (Expr.const i))
|
|
theorem optimize3.eq_3 : ∀ (bop : Unit) (e1 : Expr),
|
|
(∀ (i : BitVec 32), e1 = Expr.const i → False) → optimize3 (Expr.op bop e1) = Expr.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#print equations optimize3
|
|
|
|
|
|
-- Mutual
|
|
|
|
namespace Mutual
|
|
|
|
mutual
|
|
inductive Expr where
|
|
| const (i : BitVec 32)
|
|
| op (op : Unit) (e1 : Expr')
|
|
inductive Expr' where
|
|
| const (i : BitVec 32)
|
|
| op (op : Unit) (e1 : Expr)
|
|
end
|
|
|
|
mutual
|
|
def optimize : Expr → Expr
|
|
| .const i => .const i
|
|
| .op bop e1 =>
|
|
match bop, optimize' e1 with
|
|
| _, .const _ => .op bop (.const 0)
|
|
| _, _ => .const 0
|
|
def optimize' : Expr' → Expr'
|
|
| .const i => .const i
|
|
| .op bop e1 =>
|
|
match bop, optimize e1 with
|
|
| _, .const _ => .op bop (.const 0)
|
|
| _, _ => .const 0
|
|
end
|
|
|
|
/--
|
|
info: Mutual.optimize.eq_def (x✝ : Expr) :
|
|
optimize x✝ =
|
|
match x✝ with
|
|
| Expr.const i => Expr.const i
|
|
| Expr.op bop e1 =>
|
|
match bop, optimize' e1 with
|
|
| x, Expr'.const i => Expr.op bop (Expr'.const 0)
|
|
| x, x_1 => Expr.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#check optimize.eq_def
|
|
/--
|
|
info: Mutual.optimize'.eq_def (x✝ : Expr') :
|
|
optimize' x✝ =
|
|
match x✝ with
|
|
| Expr'.const i => Expr'.const i
|
|
| Expr'.op bop e1 =>
|
|
match bop, optimize e1 with
|
|
| x, Expr.const i => Expr'.op bop (Expr.const 0)
|
|
| x, x_1 => Expr'.const 0
|
|
-/
|
|
#guard_msgs in
|
|
#check optimize'.eq_def
|
|
|
|
end Mutual
|