Nuprl Lemma : Ramsey-n-3
∀n:ℕ. ∃N:ℕ+. ∀g:ℕN ⟶ ℕN ⟶ ℕn. ∃i,j,k:ℕN. (i < j ∧ j < k ∧ ((g i j) = (g i k) ∈ ℤ) ∧ ((g i k) = (g j k) ∈ ℤ))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
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,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_apply: x[s],
inject: Inj(A;B;f),
guard: {T},
lelt: i ≤ j < k,
uimplies: b supposing a,
sq_type: SQType(T),
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
finite-Ramsey,
false_wf,
le_wf,
int_seg_wf,
all_wf,
exists_wf,
less_than_wf,
equal_wf,
nat_wf,
int_seg_properties,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__or,
decidable__lt,
intformand_wf,
intformnot_wf,
intformor_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
not_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
isectElimination,
productElimination,
dependent_pairFormation,
functionEquality,
setElimination,
rename,
because_Cache,
lambdaEquality,
productEquality,
intEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
instantiate,
cumulativity,
independent_isectElimination,
voidElimination,
promote_hyp,
isect_memberEquality,
voidEquality,
computeAll,
imageMemberEquality,
baseClosed,
unionElimination,
int_eqEquality,
imageElimination
Latex:
\mforall{}n:\mBbbN{}
\mexists{}N:\mBbbN{}\msupplus{}. \mforall{}g:\mBbbN{}N {}\mrightarrow{} \mBbbN{}N {}\mrightarrow{} \mBbbN{}n. \mexists{}i,j,k:\mBbbN{}N. (i < j \mwedge{} j < k \mwedge{} ((g i j) = (g i k)) \mwedge{} ((g i k) = (g j k)))
Date html generated:
2017_04_20-AM-07_26_16
Last ObjectModification:
2017_02_27-PM-05_59_59
Theory : continuity
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