Nuprl Lemma : better-not-not-Ramsey
∀[R:ℕ ⟶ ℕ ⟶ ℙ]. (¬(∀[s:StrictInc]. ⇃(∃n,m,p,q:ℕ. ((n < m ∧ R[s n;s m]) ∧ p < q ∧ (¬R[s p;s q])))))
Proof
Definitions occuring in Statement :
strict-inc: StrictInc,
quotient: x,y:A//B[x; y],
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
b-almost-full: b-almost-full(n,m.R[n; m]),
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
nat: ℕ,
strict-inc: StrictInc,
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
homogeneous: homogeneous(R;n;s),
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
b-almost-full-intersection-lemma,
nat_wf,
not_wf,
implies-quotient-true,
exists_wf,
less_than_wf,
int_upper_wf,
int_upper_subtype_nat,
int_upper_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
intformless_wf,
int_formula_prop_less_lemma,
uall_wf,
strict-inc_wf,
quotient_wf,
true_wf,
equiv_rel_true,
not-not-Ramsey,
all-quotient-true,
trivial-quotient-true,
canonicalizable_wf,
canonicalizable-set,
all_wf,
int_seg_wf,
canonicalizable-nat-to-nat,
not-quotient-true,
imax_wf,
imax_nat,
add_nat_wf,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
homogeneous_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
subtype_rel_self,
lelt_wf,
decidable__lt,
imax_strict_ub,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
isectElimination,
independent_functionElimination,
because_Cache,
productEquality,
setElimination,
rename,
addEquality,
natural_numberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
universeEquality,
functionEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
inlFormation,
inrFormation,
addLevel,
orFunctionality
Latex:
\mforall{}[R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]. (\mneg{}(\mforall{}[s:StrictInc]. \00D9(\mexists{}n,m,p,q:\mBbbN{}. ((n < m \mwedge{} R[s n;s m]) \mwedge{} p < q \mwedge{} (\mneg{}R[s p;s q])))))
Date html generated:
2017_04_20-AM-07_26_24
Last ObjectModification:
2017_02_27-PM-06_00_06
Theory : continuity
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