03ce7cb17c
closes #1279 Originally reported at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/case.20in.20dependent.20match.20not.20triggering.20.28.3F.29/near/288328239
87 lines
2.6 KiB
Text
87 lines
2.6 KiB
Text
inductive O
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| int
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| real
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| bool
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| unit
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deriving Inhabited, BEq, Repr
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-- only `Arrow.id` and `Arrow.comp` really matter for my problem
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inductive Arrow : (_dom _cod : O) → Type
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-- identity
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| id : {α : O} → Arrow α α
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-- `α → α` arrows
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| unit : Arrow O.unit O.unit
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| not : Arrow O.bool O.bool
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| succᵢ : Arrow O.int O.int
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| succᵣ : Arrow O.real O.real
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| comp {α β γ} : Arrow β γ → Arrow α β → Arrow α γ
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-- `unit → bool`
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| tru : Arrow O.unit O.bool
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| fls : Arrow O.unit O.bool
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-- `unit → int`
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| zero : Arrow O.unit O.int
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-- `int → bool`
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| isZero : Arrow O.int O.bool
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-- `int → real`
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| toReal : Arrow O.int O.real
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deriving Repr
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def Arrow.compose₁ {α β γ : O} :
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Arrow β γ
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→ Arrow α β
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→ Arrow α γ
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-- id.compose₁ g = g
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| id, g => g
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-- f.compose₁ id = f
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| f, id => f
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-- else
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| comp f₁ f₂, g => comp f₁ (comp f₂ g)
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| f, g => comp f g
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#print Arrow.compose₁
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-- def Arrow.compose₁ : {α β γ : O} → Arrow β γ → Arrow α β → Arrow α γ :=
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-- fun {α β γ} x x_1 =>
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-- match β, γ, x, x_1 with
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-- | β, .(β), Arrow.id, g => g
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-- | .(α), γ, f, Arrow.id => f
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-- | β, γ, Arrow.comp f₁ f₂, g => Arrow.comp f₁ (Arrow.comp f₂ g)
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-- | β, γ, f, g => Arrow.comp f g
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#eval Arrow.compose₁ Arrow.unit Arrow.id
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-- Arrow.comp (Arrow.unit) (Arrow.id) -- Error: it should be `Arrow.unit`
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example : Arrow.compose₁ .id .id = (.id (α := o)) := rfl
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example : Arrow.compose₁ .id .unit = .unit := rfl
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example : Arrow.compose₁ .id (.comp f g) = .comp f g := rfl
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example : Arrow.compose₁ .unit .id = .unit := rfl
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example : Arrow.compose₁ (.comp f g) .id = .comp f g := rfl
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example : Arrow.compose₁ .unit .unit = .comp .unit .unit := rfl
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example : Arrow.compose₁ (.comp f g) .unit = .comp f (.comp g .unit) := rfl
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example : Arrow.compose₁ .unit (.comp f g) = .comp .unit (.comp f g) := rfl
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theorem ex_1 : Arrow.compose₁ f .id = f := by
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cases f <;> simp!
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theorem ex_2 : Arrow.compose₁ f .id = f := by
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cases f <;> simp!
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theorem ex_3 : Arrow.compose₁ .id f = f := by
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cases f <;> simp!
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theorem ex_4 : h ≠ .id → Arrow.compose₁ (.comp f g) h = .comp f (.comp g h) := by
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intros
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cases h <;> simp_all!
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def Arrow.isId : Arrow dom com → Prop
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| .id => True
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| _ => False
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def Arrow.isComp : Arrow dom com → Prop
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| .comp .. => True
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| _ => False
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theorem ex_5 (f : Arrow β γ) (g : Arrow α β) : ¬ f.isId → ¬ g.isId → ¬ f.isComp → Arrow.compose₁ f g = .comp f g := by
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intros
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cases f <;> cases g <;> simp_all!
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