272dd5533f
We are considering removing `.` as an alternative for `·` in the lambda dot notation (e.g., `(·+·)`). Reasons: - `.` is not a perfect replacement for `·` (e.g., `(·.insert ·)`) - `.` is too overloaded: `(f.x)` and `(f .x)` and `(f . x)`. We want to keep the first two.
38 lines
1.5 KiB
Text
38 lines
1.5 KiB
Text
theorem ex1 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
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calc a + b = b + b := by rw [h₁]
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_ = 0 + c + b := rfl
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theorem ex2 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
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calc a + b = b + b := by rw [h₁]
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0 + c + b = 0 + c + b := rfl
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theorem ex3 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
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calc a + b = b + b := by rw [h₁]
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_ = 0 + b + b := by rw [Nat.zero_add]
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_ = 0 + c + b := by rw [h₂]
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theorem ex4 (p : Nat → Prop) (a b : Nat) (h₁ : p a) (h₂ : p b) : p c :=
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calc p a := h₁
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_ := h₂
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theorem ex5 (p : Nat → Nat → Prop) (a b : Nat) (h₁ : p a b) (h₂ : p b c) : p a c :=
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calc p a b := h₁
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p _ c := h₂
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infix:50 " ≅ " => HEq
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instance {α β γ} : Trans (· ≅ · : α → β → Prop) (· ≅ · : β → γ → Prop) (· ≅ · : α → γ → Prop) where
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trans h₁ h₂ := HEq.trans h₁ h₂
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theorem ex6 {a : α} {b : β} {c : γ} (h₁ : HEq a b) (h₂ : b ≅ c) : a ≅ c :=
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calc a ≅ b := h₁
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_ ≅ c := h₂ -- Error because the last two arguments of HEq are not explicit
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abbrev HEqRel {α β} (a : α) (b : β) := HEq a b
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infix:50 "===" => HEqRel
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instance {α β γ} : Trans (HEqRel : α → β → Prop) (HEqRel : β → γ → Prop) (HEqRel : α → γ → Prop) where
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trans h₁ h₂ := HEq.trans h₁ h₂
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theorem ex7 {a : α} {b : β} {c : γ} (h₁ : a ≅ b) (h₂ : b ≅ c) : a ≅ c :=
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calc a === b := h₁
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_ === c := h₂
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