Nuprl Lemma : not-not-Ramsey
∀[R:ℕ ⟶ ℕ ⟶ ℙ]. (¬(∀s:StrictInc. ∃n:ℕ. (¬homogeneous(R;n;s))))
Proof
Definitions occuring in Statement :
strict-inc: StrictInc,
homogeneous: homogeneous(R;n;s),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬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,
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
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,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
strict-inc: StrictInc,
le: A ≤ B,
less_than': less_than'(a;b),
weakly-safe-seq: weakly-safe-seq(R;n;s),
weakly-infinite: w∃∞p.S[p],
homogeneous: homogeneous(R;n;s),
strictly-increasing-seq: strictly-increasing-seq(n;s),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
seq-add: s.x@n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
guard: {T},
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
cand: A c∧ B
Lemmas referenced :
monotone-bar-induction-strict,
strictly-increasing-seq_wf,
int_seg_wf,
nat_wf,
not_wf,
homogeneous_wf,
weakly-safe-seq_wf,
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,
seq-add_wf,
set_wf,
all_wf,
squash_wf,
strict-inc_wf,
exists_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
homogeneous-extension-implies,
less_than_wf,
no-weakly-safe-extensions,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
true_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
le2-homogeneous
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
because_Cache,
natural_numberEquality,
setElimination,
rename,
functionEquality,
setEquality,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
cumulativity,
universeEquality,
productElimination,
productEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
int_eqReduceFalseSq
Latex:
\mforall{}[R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]. (\mneg{}(\mforall{}s:StrictInc. \mexists{}n:\mBbbN{}. (\mneg{}homogeneous(R;n;s))))
Date html generated:
2017_04_20-AM-07_23_51
Last ObjectModification:
2017_02_27-PM-05_59_15
Theory : continuity
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