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]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T)
Latex:
\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:
2016_05_16-AM-10_35_37
Last ObjectModification:
2016_01_17-PM-01_24_22
Theory : new!event-ordering
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