Nuprl Lemma : max-of-intset
∀[P:ℕ ─→ ℙ]
((∀n:ℕ. Dec(P[n]))
⇒ (∀n:ℕ. ((∀y:ℕ. ¬P[y] supposing y ≤ n) ∨ (∃y:ℕ. ((y ≤ n) ∧ P[y] ∧ (∀x:ℕ. ¬P[x] supposing y < x ∧ (x ≤ n)))))))
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ─→ B[x]
Lemmas :
false_wf,
le_wf,
less_than_transitivity1,
less_than_irreflexivity,
less_than_wf,
all_wf,
isect_wf,
not_wf,
nat_wf,
le_antisymmetry,
subtype_base_sq,
int_subtype_base,
exists_wf,
le_weakening2,
sq_stable__le,
le_weakening,
decidable__le,
subtract_wf,
equal-wf-T-base,
not-le-2,
decidable__equal_int,
not-equal-2,
condition-implies-le,
minus-one-mul,
add-commutes,
minus-add,
minus-minus,
add-associates,
add-swap,
zero-add,
add_functionality_wrt_le,
le-add-cancel,
or_wf,
le-add-cancel2
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]
((\mforall{}n:\mBbbN{}. Dec(P[n]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
((\mforall{}y:\mBbbN{}. \mneg{}P[y] supposing y \mleq{} n)
\mvee{} (\mexists{}y:\mBbbN{}. ((y \mleq{} n) \mwedge{} P[y] \mwedge{} (\mforall{}x:\mBbbN{}. \mneg{}P[x] supposing y < x \mwedge{} (x \mleq{} n)))))))
Date html generated:
2015_07_17-AM-09_09_59
Last ObjectModification:
2015_01_27-PM-00_49_21
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