Nuprl Lemma : finite-Ramsey1
∀c:ℕ. ∀s:ℕc ⟶ ℕ.
∃N:ℕ+
∀g:ℕN ⟶ ℕN ⟶ ℕc. ∃i:ℕc. ∃f:ℕs i ⟶ ℕN. (Inj(ℕs i;ℕN;f) ∧ (∀a,b:ℕs i. (f a < f b ⇒ ((g (f a) (f b)) = i ∈ 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 :
select: L[n],
cons: [a / b],
compose: f o g,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
eq_int: (i =z j),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
inject: Inj(A;B;f),
cand: A c∧ B,
true: True,
less_than': less_than'(a;b),
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
nat: ℕ,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
subtract-is-int-iff,
add-member-int_seg2,
compose_wf,
iff_imp_equal_bool,
iff_functionality_wrt_iff,
istype-true,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
le_int_wf,
equal-wf-T-base,
bool_cases,
int_upper_properties,
nequal-le-implies,
upper_subtype_nat,
ifthenelse_wf,
length_of_nil_lemma,
length_of_cons_lemma,
nil_wf,
cons_wf,
add_nat_wf,
int_formual_prop_imp_lemma,
intformimplies_wf,
iff_weakening_equal,
true_wf,
squash_wf,
equal_wf,
istype-assert,
assert_of_bnot,
iff_transitivity,
uiff_transitivity,
not_wf,
bnot_wf,
neg_assert_of_eq_int,
list_wf,
le_weakening2,
non_neg_length,
le_reflexive,
int_seg_subtype,
select-filter-from-upto-order-preserving,
filter_type,
add-is-int-iff,
subtype_rel_sets_simple,
select_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_or_lemma,
intformor_wf,
decidable__or,
assert_of_lt_int,
lt_int_wf,
btrue_wf,
int_seg_cases,
int_seg_subtype_special,
bfalse_wf,
bool_wf,
istype-universe,
length_wf,
l_member_wf,
filter_wf5,
length_wf_nat,
length-from-upto,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
subtype_rel_sets,
subtype_rel_list,
from-upto_wf,
filter-split-length,
false_wf,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
subtract_nat_wf,
identity-injection,
nat_plus_properties,
le_wf,
istype-false,
istype-nat,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
primrec-wf2,
lelt_wf,
equal-wf-base,
less_than_wf,
inject_wf,
nat_plus_wf,
nat_wf,
subtype_rel_self,
istype-less_than,
istype-le,
decidable__lt,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties
Rules used in proof :
inlFormation_alt,
inrFormation_alt,
isectIsType,
baseApply,
equalityIsType2,
equalityIsType4,
universeEquality,
minusEquality,
setEquality,
sqequalBase,
pointwiseFunctionality,
equalityElimination,
closedConclusion,
equalityIstype,
equalityIsType3,
equalityIsType1,
functionExtensionality,
baseClosed,
imageMemberEquality,
addEquality,
setIsType,
imageElimination,
productEquality,
functionEquality,
inhabitedIsType,
functionIsType,
intEquality,
cumulativity,
hypothesis_subsumption,
promote_hyp,
productIsType,
dependent_set_memberEquality_alt,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
instantiate,
applyEquality,
unionElimination,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
productElimination,
rename,
setElimination,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}c:\mBbbN{}. \mforall{}s:\mBbbN{}c {}\mrightarrow{} \mBbbN{}.
\mexists{}N:\mBbbN{}\msupplus{}
\mforall{}g:\mBbbN{}N {}\mrightarrow{} \mBbbN{}N {}\mrightarrow{} \mBbbN{}c
\mexists{}i:\mBbbN{}c. \mexists{}f:\mBbbN{}s i {}\mrightarrow{} \mBbbN{}N. (Inj(\mBbbN{}s i;\mBbbN{}N;f) \mwedge{} (\mforall{}a,b:\mBbbN{}s i. (f a < f b {}\mRightarrow{} ((g (f a) (f b)) = i))))
Date html generated:
2019_10_15-AM-10_27_36
Last ObjectModification:
2019_09_26-PM-04_40_18
Theory : continuity
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