Create Knaster_Tarski.lean #88
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johoelzl
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Mar 29, 2018
| variables (f : α → α) (M : monotone f) | ||
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| def fixed_points : set α := { x | f x = x } | ||
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I guess you want a lemma:
lemma fixed_points.eq {x} (h : x \in fixed_points f M) : f x = x := h
and use .2.eq this instead of .2 in the further proofs, then you can use simp and rw..
johoelzl
reviewed
Mar 29, 2018
| le_antisymm | ||
| (next.apply_le M) | ||
| (lfp_le $ sup_le (le_trans H (M next.le)) (M $ next.apply_le M)) | ||
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what about def next.fixed_point {x : : α} (H : x ≤ f x) : fixed_point (in similar for previous) then you can shorten some proofs below.
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Could you do the changes for me? I get strange errors when I do them. |
johoelzl
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Mar 30, 2018
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Closed by 162edc3 |
cipher1024
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May 11, 2019
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Knaster--Tarski theorem: if f is an order-preserving function from a complete lattice to itself, then the fixed points of f is a complete sublattice (in particular, this implies that a fixed point exists).