f9dc77673b
This PR lets recursive functions defined by well-founded recursion use a different `fix` function when the termination measure is of type `Nat`. This fix-point operator use structural recursion on “fuel”, initialized by the given measure, and is thus reasonable to reduce, e.g. in `by decide` proofs. Extra provisions are in place that the fixpoint operator only starts reducing when the fuel is fully known, to prevent “accidential” defeqs when the remaining fuel for the recursive calls match the initial fuel for that recursive argument. To opt-out, the idiom `termination_by (n,0)` can be used. We still use `@[irreducible]` as the default for such recursive definitions, to avoid unexpected `defeq` lemmas. Making these functions `@[semireducible]` by default showed performance regressions in lean. When the measure is of type `Nat`, the system will accept an explicit `@[semireducible]` without the usual warning. Fixes #5234. Fixes: #11181.
84 lines
2.5 KiB
Text
84 lines
2.5 KiB
Text
import Module.Basic
|
|
import Lean
|
|
|
|
/-- info: @[defeq] theorem f.eq_def : f = 1 -/
|
|
#guard_msgs in #print sig f.eq_def
|
|
|
|
/-- info: @[defeq] theorem f.eq_unfold : f = 1 -/
|
|
#guard_msgs in #print sig f.eq_unfold
|
|
|
|
/-- info: @[defeq] theorem f_struct.eq_1 : f_struct 0 = 0 -/
|
|
#guard_msgs in #print sig f_struct.eq_1
|
|
/--
|
|
info: theorem f_struct.eq_def : ∀ (x : Nat),
|
|
f_struct x =
|
|
match x with
|
|
| 0 => 0
|
|
| n.succ => f_struct n
|
|
-/
|
|
#guard_msgs in #print sig f_struct.eq_def
|
|
|
|
/--
|
|
info: theorem f_struct.eq_unfold : f_struct = fun x =>
|
|
match x with
|
|
| 0 => 0
|
|
| n.succ => f_struct n
|
|
-/
|
|
#guard_msgs in #print sig f_struct.eq_unfold
|
|
|
|
/-- info: theorem f_wfrec.eq_1 : ∀ (x : Nat), f_wfrec 0 x = x -/
|
|
#guard_msgs(pass trace, all) in #print sig f_wfrec.eq_1
|
|
|
|
/--
|
|
info: theorem f_wfrec.eq_def : ∀ (x x_1 : Nat),
|
|
f_wfrec x x_1 =
|
|
match x, x_1 with
|
|
| 0, acc => acc
|
|
| n.succ, acc => f_wfrec n (acc + 1)
|
|
-/
|
|
#guard_msgs(pass trace, all) in #print sig f_wfrec.eq_def
|
|
|
|
/--
|
|
info: theorem f_wfrec.eq_unfold : f_wfrec = fun x x_1 =>
|
|
match x, x_1 with
|
|
| 0, acc => acc
|
|
| n.succ, acc => f_wfrec n (acc + 1)
|
|
-/
|
|
#guard_msgs(pass trace, all) in #print sig f_wfrec.eq_unfold
|
|
|
|
/--
|
|
info: theorem f_wfrec.induct_unfolding : ∀ (motive : Nat → Nat → Nat → Prop),
|
|
(∀ (acc : Nat), motive 0 acc acc) →
|
|
(∀ (n acc : Nat), motive n (acc + 1) (f_wfrec n (acc + 1)) → motive n.succ acc (f_wfrec n (acc + 1))) →
|
|
∀ (a a_1 : Nat), motive a a_1 (f_wfrec a a_1)
|
|
-/
|
|
#guard_msgs(pass trace, all) in #print sig f_wfrec.induct_unfolding
|
|
|
|
/-- info: @[defeq] theorem f_exp_wfrec.eq_1 : ∀ (x : Nat), f_exp_wfrec 0 x = x -/
|
|
#guard_msgs(pass trace, all) in
|
|
#print sig f_exp_wfrec.eq_1
|
|
|
|
/--
|
|
info: theorem f_exp_wfrec.eq_def : ∀ (x x_1 : Nat),
|
|
f_exp_wfrec x x_1 =
|
|
match x, x_1 with
|
|
| 0, acc => acc
|
|
| n.succ, acc => f_exp_wfrec n (acc + 1)
|
|
-/
|
|
#guard_msgs in #print sig f_exp_wfrec.eq_def
|
|
|
|
/--
|
|
info: theorem f_exp_wfrec.eq_unfold : f_exp_wfrec = fun x x_1 =>
|
|
match x, x_1 with
|
|
| 0, acc => acc
|
|
| n.succ, acc => f_exp_wfrec n (acc + 1)
|
|
-/
|
|
#guard_msgs(pass trace, all) in #print sig f_exp_wfrec.eq_unfold
|
|
|
|
/--
|
|
info: theorem f_exp_wfrec.induct_unfolding : ∀ (motive : Nat → Nat → Nat → Prop),
|
|
(∀ (acc : Nat), motive 0 acc acc) →
|
|
(∀ (n acc : Nat), motive n (acc + 1) (f_exp_wfrec n (acc + 1)) → motive n.succ acc (f_exp_wfrec n (acc + 1))) →
|
|
∀ (a a_1 : Nat), motive a a_1 (f_exp_wfrec a a_1)
|
|
-/
|
|
#guard_msgs(pass trace, all) in #print sig f_exp_wfrec.induct_unfolding
|