26 lines
1.3 KiB
Text
26 lines
1.3 KiB
Text
inductive Lex (ra : α → α → Prop) (rb : β → β → Prop) : α × β → α × β → Prop where
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| left {a₁} (b₁) {a₂} (b₂) (h : ra a₁ a₂) : Lex ra rb (a₁, b₁) (a₂, b₂)
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| right (a) {b₁ b₂} (h : rb b₁ b₂) : Lex ra rb (a, b₁) (a, b₂)
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def lexAccessible1 {ra : α → α → Prop} {rb : β → β → Prop} (aca : (a : α) → Acc ra a) (acb : (b : β) → Acc rb b) (a : α) (b : β) : Acc (Lex ra rb) (a, b) := by
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induction (aca a) generalizing b with
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| intro xa aca iha =>
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induction (acb b) with
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| intro xb acb ihb =>
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apply Acc.intro (xa, xb)
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intro p lt
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cases lt with
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| left b1 b2 h => apply iha _ h -- only explicit fields are provided by default
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| @right a b1 b2 h => apply ihb b1 h -- `@` allows us to provide names to implicit fields too
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def lexAccessible2 {ra : α → α → Prop} {rb : β → β → Prop} (aca : (a : α) → Acc ra a) (acb : (b : β) → Acc rb b) (a : α) (b : β) : Acc (Lex ra rb) (a, b) := by
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induction (aca a) generalizing b with
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| intro xa aca iha =>
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induction (acb b) with
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| intro xb acb ihb =>
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apply Acc.intro (xa, xb)
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intro p lt
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cases lt with
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| @left a1 b1 a2 b2 h => apply iha a1 h
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| right _ h => apply ihb _ h
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