Nuprl Lemma : fan_theorem
∀[X:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ]
(∀n:ℕ. ∀s:ℕn ⟶ 𝔹. Dec(X[n;s])) ⇒ (∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. X[n;f]) supposing ∀f:ℕ ⟶ 𝔹. (↓∃n:ℕ. X[n;f])
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s1;s2],
le: A ≤ B,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
less_than: a < b,
outl: outl(x)
Lemmas referenced :
simple_fan_theorem'-ext,
set-value-type,
equal_wf,
int-value-type,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtype_rel_function,
nat_wf,
bool_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
istype-nat,
decidable_wf,
squash_wf,
int_seg_decide_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
independent_isectElimination,
Error :lambdaFormation_alt,
independent_functionElimination,
setElimination,
intEquality,
cutEval,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
Error :universeIsType,
Error :dependent_pairFormation_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productElimination,
Error :productIsType,
because_Cache,
applyEquality,
Error :functionIsType,
instantiate,
universeEquality,
productEquality,
functionExtensionality,
functionEquality
Latex:
\mforall{}[X:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbP{}]
(\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. Dec(X[n;s])) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}k. X[n;f])
supposing \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f])
Date html generated:
2019_06_20-PM-01_15_29
Last ObjectModification:
2019_01_27-PM-01_53_25
Theory : int_2
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