107 lines
3.3 KiB
Plaintext
107 lines
3.3 KiB
Plaintext
/-
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Chapter 0
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Useful Facts About Sets
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-/
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import MathematicalIntroductionLogic.Tuple.Generic
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variable {k m n : Nat}
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variable (p : 1 ≤ m)
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variable (q : n + (m - 1) = m + k)
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private lemma n_eq_succ_k : n = k + 1 := by
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let ⟨m', h⟩ := Nat.exists_eq_succ_of_ne_zero $ show m ≠ 0 by
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intro h
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have ff : 1 ≤ 0 := h ▸ p
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ring_nf at ff
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exact ff.elim
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calc
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n = n + (m - 1) - (m - 1) := by rw [Nat.add_sub_cancel]
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_ = m' + 1 + k - (m' + 1 - 1) := by rw [q, h]
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_ = m' + 1 + k - m' := by simp
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_ = 1 + k + m' - m' := by rw [Nat.add_assoc, Nat.add_comm]
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_ = 1 + k := by simp
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_ = k + 1 := by rw [Nat.add_comm]
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private lemma n_pred_eq_k : n - 1 = k := by
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have h : k + 1 - 1 = k + 1 - 1 := rfl
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conv at h => lhs; rw [←n_eq_succ_k p q]
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simp at h
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exact h
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private lemma n_geq_one : 1 ≤ n := by
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rw [n_eq_succ_k p q]
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simp
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private lemma min_comm_succ_eq : min (m + k) (k + 1) = k + 1 :=
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Nat.recOn k
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(by simp; exact p)
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(fun k' ih => calc min (m + (k' + 1)) (k' + 1 + 1)
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_ = min (m + k' + 1) (k' + 1 + 1) := by conv => rw [Nat.add_assoc]
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_ = min (m + k') (k' + 1) + 1 := Nat.min_succ_succ (m + k') (k' + 1)
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_ = k' + 1 + 1 := by rw [ih])
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private lemma n_eq_min_comm_succ : n = min (m + k) (k + 1) := by
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rw [min_comm_succ_eq p]
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exact n_eq_succ_k p q
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private lemma n_pred_m_eq_m_k : n + (m - 1) = m + k := by
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rw [←Nat.add_sub_assoc p, Nat.add_comm, Nat.add_sub_assoc (n_geq_one p q)]
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conv => lhs; rw [n_pred_eq_k p q]
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private def cast_norm : GTuple α (n, m - 1) → Tuple α (m + k)
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| xs => cast (by rw [q]) xs.norm
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private def cast_fst : GTuple α (n, m - 1) → Tuple α (k + 1)
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| xs => cast (by rw [n_eq_succ_k p q]) xs.fst
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private def cast_take (ys : Tuple α (m + k)) :=
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cast (by rw [min_comm_succ_eq p]) (ys.take (k + 1))
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/--
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Lemma 0A
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Assume that `⟨x₁, ..., xₘ⟩ = ⟨y₁, ..., yₘ, ..., yₘ₊ₖ⟩`. Then
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`x₁ = ⟨y₁, ..., yₖ₊₁⟩`.
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-/
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theorem lemma_0a (xs : GTuple α (n, m - 1)) (ys : Tuple α (m + k))
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: (cast_norm q xs = ys) → (cast_fst p q xs = cast_take p ys) := by
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intro h
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suffices HEq
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(cast (_ : Tuple α n = Tuple α (k + 1)) xs.fst)
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(cast (_ : Tuple α (min (m + k) (k + 1)) = Tuple α (k + 1)) (Tuple.take ys (k + 1)))
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from eq_of_heq this
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congr
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· exact n_eq_min_comm_succ p q
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· rfl
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· exact n_eq_min_comm_succ p q
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· exact HEq.rfl
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· exact Eq.recOn
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(motive := fun _ h => HEq
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(_ : n + (n - 1) = n + k)
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(cast h (show n + (n - 1) = n + k by rw [n_pred_eq_k p q])))
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(show (n + (n - 1) = n + k) = (min (m + k) (k + 1) + (n - 1) = n + k) by
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rw [n_eq_min_comm_succ p q])
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HEq.rfl
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· exact n_geq_one p q
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· exact n_pred_eq_k p q
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· exact Eq.symm (n_eq_min_comm_succ p q)
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· exact n_pred_eq_k p q
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· rw [GTuple.self_fst_eq_norm_take]
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unfold cast_norm at h
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simp at h
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rw [←h, ←n_eq_succ_k p q]
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have h₂ := Eq.recOn
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(motive := fun x h => HEq
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(Tuple.take xs.norm n)
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(Tuple.take (cast (show Tuple α (n + (m - 1)) = Tuple α x by rw [h]) xs.norm) n))
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(show n + (m - 1) = m + k by rw [n_pred_m_eq_m_k p q])
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HEq.rfl
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exact Eq.recOn
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(motive := fun x h => HEq
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(cast h (Tuple.take xs.norm n))
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(Tuple.take (cast (_ : Tuple α (n + (m - 1)) = Tuple α (m + k)) xs.norm) n))
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(show Tuple α (min (n + (m - 1)) n) = Tuple α n by simp)
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h₂
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