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1 | | -def fibfib (n : Nat): Nat := |
2 | | - match n with |
3 | | - | 0 => 0 |
4 | | - | 1 => 0 |
5 | | - | 2 => 1 |
6 | | - | (n+3) => fibfib (n+2) + fibfib (n+1) + fibfib n |
7 | | - |
8 | | -theorem fibfib_of_three_le {n : Nat} (hn : 3 ≤ n) : |
9 | | - fibfib n = fibfib (n - 1) + fibfib (n -2) + fibfib (n - 3) := by |
10 | | - conv => |
11 | | - lhs |
12 | | - unfold fibfib |
13 | | - cases n with |
14 | | - | zero => simp at hn |
15 | | - | succ m => |
16 | | - cases m with |
17 | | - | zero => simp at hn |
18 | | - | succ n => |
19 | | - cases n with |
20 | | - | zero => simp at hn |
21 | | - | succ m => |
22 | | - simp |
23 | | - |
24 | | -def computeFibfib (n : Nat) : Nat := |
25 | | - match n with |
26 | | - | 0 => 0 |
27 | | - | 1 => 0 |
28 | | - | 2 => 1 |
29 | | - | (m + 3) => go (m + 3) 1 0 0 3 where |
30 | | - |
31 | | - go (m prev1 prev2 prev3 curr : Nat) : Nat := |
32 | | - if curr ≥ m |
33 | | - then prev1 + prev2 + prev3 |
34 | | - else go m (prev1 + prev2 + prev3) prev1 prev2 (curr + 1) |
35 | | - termination_by m - curr |
36 | | - |
37 | | -theorem testCase1 : computeFibfib 2 = 1 := by native_decide |
38 | | -theorem testCase2 : computeFibfib 1 = 0 := by native_decide |
39 | | -theorem testCase3 : computeFibfib 5 = 4 := by native_decide |
40 | | -theorem testCase4 : computeFibfib 8 = 24 := by native_decide |
41 | | -theorem testCase5 : computeFibfib 10 = 81 := by native_decide |
42 | | -theorem testCase6 : computeFibfib 12 = 274 := by native_decide |
43 | | -theorem testCase7 : computeFibfib 14 = 927 := by native_decide |
44 | | - |
45 | | -theorem computeFibfib_correct (n : Nat) : computeFibfib n = fibfib n := by |
46 | | - unfold computeFibfib |
47 | | - unfold fibfib |
48 | | - cases n with |
49 | | - | zero => simp[fibfib] |
50 | | - | succ m => |
51 | | - cases m with |
52 | | - | zero => simp[fibfib] |
53 | | - | succ n => |
54 | | - cases n with |
55 | | - | zero => simp[fibfib] |
56 | | - | succ m => |
57 | | - simp only |
58 | | - suffices ∀ (m curr prev1 prev2 prev3 : Nat), 3 ≤ curr → curr ≤ m → prev1 = fibfib (curr - 1) → |
59 | | - prev2 = fibfib (curr - 2) → prev3 = fibfib (curr - 3) → |
60 | | - computeFibfib.go m prev1 prev2 prev3 curr = fibfib (m - 1) + fibfib (m - 2) + fibfib (m-3) by |
61 | | - apply this |
62 | | - · omega |
63 | | - · omega |
64 | | - · simp [fibfib] |
65 | | - · simp [fibfib] |
66 | | - · simp [fibfib] |
67 | | - intro m curr |
68 | | - induction h : m - curr generalizing curr with |
69 | | - | zero => |
70 | | - simp [Nat.sub_eq_zero_iff_le] at h |
71 | | - intro prev1 prev2 prev3 hcurr1 hcurr2 hprev1 hprev2 hprev3 |
72 | | - have : curr = m := by omega |
73 | | - simp [computeFibfib.go, this, hprev1, hprev2, hprev3] |
74 | | - | succ n ih => |
75 | | - intro prev1 prev2 prev3 hcurr1 hcurr2 hprev1 hprev2 hprev3 |
76 | | - unfold computeFibfib.go |
77 | | - have : ¬ curr ≥ m := by omega |
78 | | - simp only [ge_iff_le, this, ↓reduceIte] |
79 | | - apply ih |
80 | | - · omega |
81 | | - · omega |
82 | | - · omega |
83 | | - · simp [fibfib_of_three_le, hcurr1, hprev1, hprev2, hprev3] |
84 | | - · simp [hprev1] |
85 | | - · simp [hprev2] |
86 | | - |
87 | 1 | /-! |
88 | 2 | ## Prompt |
89 | 3 |
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