3a408e0e54
This PR changes the signature of `Array.get` to take a Nat and a proof, rather than a `Fin`, for consistency with the rest of the (planned) Array API. Note that because of bootstrapping issues we can't provide `get_elem_tactic` as an autoparameter for the proof. As users will mostly use the `xs[i]` notation provided by `GetElem`, this hopefully isn't a problem. We may restore `Fin` based versions, either here or downstream, as needed, but they won't be the "main" functions. --------- Co-authored-by: David Thrane Christiansen <david@davidchristiansen.dk>
32 lines
1.1 KiB
Text
32 lines
1.1 KiB
Text
@[specialize]
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def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
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if h : i < a.size then
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have : i < b.size := hsz ▸ h
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p a[i] b[i] && isEqvAux a b hsz p (i+1)
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else
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true
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termination_by a.size - i
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decreasing_by simp_wf; decreasing_trivial_pre_omega
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theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size) (heqv : isEqvAux a b hsz (fun x y => x = y) i) : ∀ (j : Nat) (hl : i ≤ j) (hj : j < a.size), a[j] = b[j] := by
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intro j low high
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by_cases h : i < a.size
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· unfold isEqvAux at heqv
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simp [h] at heqv
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have hind := eq_of_isEqvAux a b hsz (i+1) (Nat.succ_le_of_lt h) heqv.2
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by_cases heq : i = j
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· subst heq; exact heqv.1
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· exact hind j (Nat.succ_le_of_lt (Nat.lt_of_le_of_ne low heq)) high
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· have heq : i = a.size := Nat.le_antisymm hi (Nat.ge_of_not_lt h)
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subst heq
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exact absurd (Nat.lt_of_lt_of_le high low) (Nat.lt_irrefl j)
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termination_by _ _ _ => a.size - i
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@[simp] def f (x y : Nat) : Nat → Nat :=
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if h : x > 0 then
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fun z => f (x - 1) (y + 1) z + 1
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else
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(· + y)
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termination_by x
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#check f.eq_1
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