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refactor(complexity): use Mathlib list product lemmas in WPP and WPP.Lowness (#750) #758

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Merged
SamuelSchlesinger merged 1 commit into from
Oct 24, 2026
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refactor(complexity): use Mathlib list product lemmas in WPP and WPP.Lowness (#750) #758

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39 changes: 8 additions & 31 deletions Complexity/Computability/Complexity/Classes/WPP.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -339,34 +339,6 @@ theorem mem_LWPP_of_finset_targets {ι : Type*} (s : Finset ι) {g : α → ℤ}

/-! ### Unary-indexed target normalization and `LWPP` reduction closure -/

private theorem prod_map_ne_zero {γ : Type*} {f : γ → ℤ} {l : List γ}
(h : ∀ x ∈ l, f x ≠ 0) : (l.map f).prod ≠ 0 := by
induction l with
| nil => simp
| cons x l ih =>
simp only [List.map_cons, List.prod_cons]
exact mul_ne_zero (h x (by simp)) (ih fun y hy => h y (by simp [hy]))

private theorem mul_prod_filter_range_ne (f : ℕ → ℤ) {K m : ℕ} (hm : m < K) :
f m * (((List.range K).filter (fun i => decide (i ≠ m))).map f).prod =
((List.range K).map f).prod := by
induction K with
| zero => lia
| succ K ih =>
rw [List.range_succ, List.filter_append, List.map_append, List.prod_append,
List.map_append, List.prod_append]
rcases Nat.lt_succ_iff_lt_or_eq.mp hm with hlt | rfl
· have hne : decide (K ≠ m) = true := decide_eq_true (by lia)
simp only [List.filter_cons, hne, ↓reduceIte, List.filter_nil, List.map_cons, List.map_nil,
List.prod_cons, List.prod_nil, mul_one]
rw [← mul_assoc, ih hlt]
· have hfilt : (List.range m).filter (fun i => decide (i ≠ m)) = List.range m :=
List.filter_eq_self.mpr fun i hi => decide_eq_true (Nat.ne_of_lt (List.mem_range.mp hi))
simp only [hfilt, List.filter_cons, ne_eq, not_true_eq_false, decide_false,
Bool.false_eq_true, ↓reduceIte, List.filter_nil, List.map_nil, List.prod_nil, mul_one,
List.map_cons, List.prod_cons]
ring

/-- **Unary-indexed target normalization for `LWPP`:** if a `GapP` function takes `0` outside `A`
and `f (k a)` on `A`, where `f : ℕ → ℤ` is a nonzero unary-input polynomial-time target and
`k : α → ℕ` is computable in polynomial time in unary, then multiplying the gap by the
Expand All @@ -392,7 +364,7 @@ theorem mem_LWPP_of_polyTime_index {g : α → ℤ} {f : ℕ → ℤ} {k : α
ComputableTidilyInPolyTime.intBin_prod.comp
((ComputableTidilyInPolyTime.map hf).comp
(ComputableTidilyInPolyTime.range.comp hbound))
have hFn (n : ℕ) : F n ≠ 0 := prod_map_ne_zero fun _ _ => hn _
have hFn (n : ℕ) : F n ≠ 0 := List.prod_ne_zero (by simp [hn])
let h (a : α) : ℤ :=
(((List.range (bound (enc a).length)).filter (fun i => decide (i ≠ k a))).map f).prod
have hlist : ComputableTidilyInPolyTime
Expand All @@ -409,8 +381,13 @@ theorem mem_LWPP_of_polyTime_index {g : α → ℤ} {f : ℕ → ℤ} {k : α
ComputableTidilyInPolyTime.intBin_prod.comp
((ComputableTidilyInPolyTime.map hf).comp
(ComputableTidilyInPolyTime.filterWithInput hlist htest))
have hmul (a : α) : f (k a) * h a = F (enc a).length :=
mul_prod_filter_range_ne f (hlt a)
have hmul (a : α) : f (k a) * h a = F (enc a).length := by
have herase :
(List.range (bound (enc a).length)).filter (fun i => decide (i ≠ k a)) =
(List.range (bound (enc a).length)).erase (k a) := by
rw [List.nodup_range.erase_eq_filter]
exact List.filter_congr fun i _ => by simp [bne, beq_eq_decide]
simpa only [h, F, herase] using List.prod_map_erase f (List.mem_range.mpr (hlt a))
refine ⟨g * h, GapP.mul_mem hg (GapP.of_polyTime hh), F, .of_tidy hF, hFn, fun a => ?_,
fun a => ?_⟩
· rcases hr a with hz | hfk
Expand Down
39 changes: 8 additions & 31 deletions Complexity/Computability/Complexity/Classes/WPP/Lowness.lean
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Expand Up @@ -460,34 +460,6 @@ theorem PPWithOracle_eq_PP_of_mem_SPP (hA : A ∈ SPP encBits) (enc : α ↪ Lis

/-! ### `LWPP` scaling simulation and oracle lowness (FFK94, Section 5) -/

private theorem prod_map_ne_zero {γ : Type*} {f : γ → ℤ} {l : List γ}
(h : ∀ x ∈ l, f x ≠ 0) : (l.map f).prod ≠ 0 := by
induction l with
| nil => simp only [List.map_nil, List.prod_nil, ne_eq, one_ne_zero, not_false_eq_true]
| cons x l ih =>
simp only [List.map_cons, List.prod_cons]
exact mul_ne_zero (h x List.mem_cons_self) (ih fun y hy => h y (List.mem_cons_of_mem x hy))

private theorem mul_prod_filter_range_ne (f : ℕ → ℤ) {K m : ℕ} (hm : m < K) :
f m * (((List.range K).filter (fun i => decide (i ≠ m))).map f).prod =
((List.range K).map f).prod := by
induction K with
| zero => lia
| succ K ih =>
rw [List.range_succ, List.filter_append, List.map_append, List.prod_append,
List.map_append, List.prod_append]
rcases Nat.lt_succ_iff_lt_or_eq.mp hm with hlt | rfl
· have hne : decide (K ≠ m) = true := decide_eq_true (by lia)
simp only [List.filter_cons, hne, ↓reduceIte, List.filter_nil, List.map_cons, List.map_nil,
List.prod_cons, List.prod_nil, mul_one]
rw [← mul_assoc, ih hlt]
· have hfilt : (List.range m).filter (fun i => decide (i ≠ m)) = List.range m :=
List.filter_eq_self.mpr fun i hi => decide_eq_true (Nat.ne_of_lt (List.mem_range.mp hi))
simp only [hfilt, List.filter_cons, ne_eq, not_true_eq_false, decide_false,
Bool.false_eq_true, ↓reduceIte, List.filter_nil, List.map_nil, List.prod_nil, mul_one,
List.map_cons, List.prod_cons]
ring

private theorem exists_mul_mem_GapP_of_sharpPWithOracle_of_mem_LWPP (hA : A ∈ LWPP encBits)
{c : α → ℕ} (hc : c ∈ SharpPWithOracle A enc) :
∃ S : ℕ → ℤ, ComputableTidilyInPolyTime S encUnary encIntBin ∧
Expand All @@ -514,7 +486,7 @@ private theorem exists_mul_mem_GapP_of_sharpPWithOracle_of_mem_LWPP (hA : A ∈
ComputableTidilyInPolyTime.intBin_prod.comp
((ComputableTidilyInPolyTime.map hf).comp
(ComputableTidilyInPolyTime.range.comp hbound))
have hDn (n : ℕ) : D n ≠ 0 := prod_map_ne_zero fun _ _ => hfn _
have hDn (n : ℕ) : D n ≠ 0 := List.prod_ne_zero (by simp [hfn])
let S : ℕ → ℤ := fun n => D n ^ τ n
have hS : ComputableTidilyInPolyTime S encUnary encIntBin := hD.intBin_pow' hτ
have hSn (n : ℕ) : S n ≠ 0 := pow_ne_zero (τ n) (hDn n)
Expand Down Expand Up @@ -557,8 +529,13 @@ private theorem exists_mul_mem_GapP_of_sharpPWithOracle_of_mem_LWPP (hA : A ∈
have hH_exact (q : List Bool) (hq : q.length ≤ T a) :
H (a, q) = open Classical in if q ∈ A then D (enc a).length else 0 := by
have hlt : q.length < bound (enc a).length := Nat.lt_succ_of_le hq
have hmul : f q.length * K (a, q) = D (enc a).length :=
mul_prod_filter_range_ne f hlt
have hmul : f q.length * K (a, q) = D (enc a).length := by
have herase :
(List.range (bound (enc a).length)).filter (fun i => decide (i ≠ q.length)) =
(List.range (bound (enc a).length)).erase q.length := by
rw [List.nodup_range.erase_eq_filter]
exact List.filter_congr fun i _ => by simp [bne, beq_eq_decide]
simpa only [K, D, herase] using List.prod_map_erase f (List.mem_range.mpr hlt)
dsimp only [H]
rw [hg_eq q]
split_ifs
Expand Down
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