96 lines
4.3 KiB
Text
96 lines
4.3 KiB
Text
inductive term
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| const (c : string) : term
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| app (fn : string) (ts : list term) : term
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mutual inductive is_rename, is_rename_lst
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with is_rename : term → string → string → term → Prop
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| const_eq (c₁ c₂) : is_rename (term.const c₁) c₁ c₂ (term.const c₂)
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| const_ne (c₁ c₂ c₃) (hne : c₁ ≠ c₂) : is_rename (term.const c₁) c₂ c₃ (term.const c₁)
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| app (fn c₁ c₂ ts₁ ts₂) (h₁ : is_rename_lst ts₁ c₁ c₂ ts₂) : is_rename (term.app fn ts₁) c₁ c₂ (term.app fn ts₂)
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with is_rename_lst : list term → string → string → list term → Prop
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| nil (c₁ c₂) : is_rename_lst [] c₁ c₂ []
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| cons (t₁ ts₁ t₂ ts₂ c₁ c₂) (h₁ : is_rename t₁ c₁ c₂ t₂) (h₂ : is_rename_lst ts₁ c₁ c₂ ts₂) : is_rename_lst (t₁::ts₁) c₁ c₂ (t₂::ts₂)
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mutual def term.ind, term_list.ind
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(p : term → Prop) (ps : list term → Prop)
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(h₁ : ∀ c, p (term.const c))
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(h₂ : ∀ fn ts, ps ts → p (term.app fn ts))
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(h₃ : ps [])
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(h₄ : ∀ t ts, p t → ps ts → ps (t::ts))
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with term.ind : ∀ t : term, p t
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| (term.const c) := h₁ c
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| (term.app fn ts) := h₂ fn ts (term_list.ind ts)
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with term_list.ind : ∀ ts : list term, ps ts
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| [] := h₃
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| (t::ts) := h₄ t ts (term.ind t) (term_list.ind ts)
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lemma term.ind_on
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(p : term → Prop) (ps : list term → Prop)
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(t : term)
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(h₁ : ∀ c, p (term.const c))
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(h₂ : ∀ fn ts, ps ts → p (term.app fn ts))
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(h₃ : ps [])
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(h₄ : ∀ t ts, p t → ps ts → ps (t::ts)) : p t :=
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term.ind p ps h₁ h₂ h₃ h₄ t
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lemma is_rename_det : ∀ t₁ t₂ t₂' c₁ c₂, is_rename t₁ c₁ c₂ t₂ → is_rename t₁ c₁ c₂ t₂' → t₂ = t₂' :=
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begin
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intro t₁,
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apply term.ind_on
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(λ t₁ : term, ∀ t₂ t₂' c₁ c₂, is_rename t₁ c₁ c₂ t₂ → is_rename t₁ c₁ c₂ t₂' → t₂ = t₂')
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(λ ts₁ : list term, ∀ ts₂ ts₂' c₁ c₂, is_rename_lst ts₁ c₁ c₂ ts₂ → is_rename_lst ts₁ c₁ c₂ ts₂' → ts₂ = ts₂')
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t₁,
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{ intros c₁ t₂ t₂' c₁' c₂ h₁ h₂,
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cases h₁; cases h₂; trace_state; { refl <|> contradiction } },
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{ intros fn ts ih t₂ t₂' c₁ c₂ h₁ h₂, cases h₁; cases h₂, trace_state,
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have := ih _ _ _ _ h₁_h₁ h₂_h₁,
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simp [*] },
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{ intros ts₂ ts₂' c₁ c₂ h₁ h₂,
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cases h₁; cases h₂; refl },
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{ intros t ts ih₁ ih₂ ts₂ ts₂' c₁ c₂ h₁ h₂,
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cases h₁; cases h₂,
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have := ih₁ _ _ _ _ h₁_h₁ h₂_h₁,
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have := ih₂ _ _ _ _ h₁_h₂ h₂_h₂,
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simp [*] }
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end
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mutual def is_rename.ind, is_rename_lst.ind
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(p : ∀ {t₁ c₁ c₂ t₂}, is_rename t₁ c₁ c₂ t₂ → Prop)
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(ps : ∀ {ts₁ c₁ c₂ ts₂}, is_rename_lst ts₁ c₁ c₂ ts₂ → Prop)
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(h₁ : ∀ (c₁ c₂ : string), p (is_rename.const_eq c₁ c₂))
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(h₂ : ∀ (c₁ c₂ c₃ : string) (hne : c₁ ≠ c₂), p (is_rename.const_ne c₁ c₂ c₃ hne))
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(h₃ : ∀ (fn c₁ c₂ ts₁ ts₂ h) (ih : ps h), p (is_rename.app fn c₁ c₂ ts₁ ts₂ h))
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(h₄ : ∀ (c₁ c₂ : string), ps (is_rename_lst.nil c₁ c₂))
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(h₅ : ∀ (t₁ ts₁ t₂ ts₂ c₁ c₂ h₁ h₂) (ih₁ : p h₁) (ih₂ : ps h₂), ps (is_rename_lst.cons t₁ ts₁ t₂ ts₂ c₁ c₂ h₁ h₂))
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with is_rename.ind : ∀ (t₁ c₁ c₂ t₂) (h : is_rename t₁ c₁ c₂ t₂), p h
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| (term.const c) :=
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begin
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intros c₁ c₂ t₂ hr,
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cases t₂; cases hr,
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{ apply h₁ },
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{ apply h₂, assumption }
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end
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| (term.app fn ts₁) :=
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have ih : ∀ (c₁ c₂ ts₂) (h : is_rename_lst ts₁ c₁ c₂ ts₂), ps h, from is_rename_lst.ind ts₁,
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begin
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intros c₁ c₂ t₂ hr,
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cases t₂; cases hr,
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apply h₃, apply ih, assumption
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end
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with is_rename_lst.ind : ∀ (ts₁ c₁ c₂ ts₂) (h : is_rename_lst ts₁ c₁ c₂ ts₂), ps h
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| [] :=
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begin
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intros c₁ c₂ ts₂ hr,
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cases ts₂; cases hr, apply h₄
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end
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| (t₁::ts₁) :=
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have ih₁ : ∀ (c₁ c₂ t₂) (h : is_rename t₁ c₁ c₂ t₂), p h, from is_rename.ind t₁,
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have ih₂ : ∀ (c₁ c₂ ts₂) (h : is_rename_lst ts₁ c₁ c₂ ts₂), ps h, from is_rename_lst.ind ts₁,
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begin
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intros c₁ c₂ ts₂ hr,
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cases ts₂; cases hr,
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fapply h₅, exact hr_h₁, exact hr_h₂,
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exact ih₁ _ _ _ hr_h₁,
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exact ih₂ _ _ _ hr_h₂
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end
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