Nuprl Lemma : finite-Ramsey
∀k,n:ℕ.
∃N:ℕ+
∀g:ℕN ⟶ ℕN ⟶ ℕk
∃f:ℕn ⟶ ℕN. (Inj(ℕn;ℕN;f) ∧ (∀a,b,c,d:ℕn. (f a < f b ⇒ f c < f d ⇒ ((g (f a) (f b)) = (g (f c) (f d)) ∈ ℤ))))
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
finite-Ramsey1,
int_seg_wf,
less_than_wf,
inject_wf,
all_wf,
equal_wf,
exists_wf,
nat_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
productElimination,
dependent_pairFormation,
sqequalRule,
independent_pairFormation,
applyEquality,
functionExtensionality,
because_Cache,
productEquality,
functionEquality,
intEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}k,n:\mBbbN{}.
\mexists{}N:\mBbbN{}\msupplus{}
\mforall{}g:\mBbbN{}N {}\mrightarrow{} \mBbbN{}N {}\mrightarrow{} \mBbbN{}k
\mexists{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}N
(Inj(\mBbbN{}n;\mBbbN{}N;f)
\mwedge{} (\mforall{}a,b,c,d:\mBbbN{}n. (f a < f b {}\mRightarrow{} f c < f d {}\mRightarrow{} ((g (f a) (f b)) = (g (f c) (f d))))))
Date html generated:
2017_04_20-AM-07_26_05
Last ObjectModification:
2017_02_27-PM-05_59_29
Theory : continuity
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