30 lines
947 B
Text
30 lines
947 B
Text
import data.finset
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open list
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namespace finset
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variable {A : Type}
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variable [h : decidable_eq A]
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include h
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set_option pp.implicit true
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set_option pp.notation false
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protected theorem induction₂ {P : finset A → Prop}
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(H1 : P empty)
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(H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
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∀s, P s :=
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take s,
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quot.induction_on s
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take u,
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subtype.destruct u
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take l,
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list.induction_on l
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(assume nodup_l, H1)
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(take a l',
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assume IH nodup_al',
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assert anl' : a ∉ l', from not_mem_of_nodup_cons nodup_al',
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assert H3 : list.insert a l' = a :: l', from insert_eq_of_not_mem anl',
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assert nodup_l' : nodup l', from nodup_of_nodup_cons nodup_al',
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assert P_l' : P (quot.mk (subtype.tag l' nodup_l')), from IH nodup_l',
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assert H4 : P (insert a (quot.mk (subtype.tag l' nodup_l'))), from H2 anl' P_l',
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begin rewrite [eq.symm H3], apply H4 end)
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