Nuprl Lemma : p-mu-exists
∀P:ℕ ⟶ 𝔹. (Dec(∃n:ℕ. (↑(P n))) ⇒ (∃x:ℕ + Top. p-mu(P;x)))
Proof
Definitions occuring in Statement :
p-mu: p-mu(P;x),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
union: left + right
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
p-mu: p-mu(P;x),
cand: A c∧ B
Lemmas referenced :
unit_wf2,
it_wf,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
primrec-wf2,
less_than_wf,
set_wf,
decidable__lt,
p-mu_wf,
top_wf,
all_wf,
decidable__assert,
int_seg_subtype_nat,
decidable__exists_int_seg,
le_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
itermSubtract_wf,
intformnot_wf,
decidable__le,
lelt_wf,
false_wf,
int_seg_subtype,
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,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
bool_wf,
assert_wf,
nat_wf,
exists_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
functionEquality,
productElimination,
natural_numberEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
equalityTransitivity,
equalitySymmetry,
setEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
instantiate,
independent_functionElimination,
unionEquality,
introduction,
addEquality,
inlEquality,
inrEquality
Latex:
\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (Dec(\mexists{}n:\mBbbN{}. (\muparrow{}(P n))) {}\mRightarrow{} (\mexists{}x:\mBbbN{} + Top. p-mu(P;x)))
Date html generated:
2016_05_15-PM-03_32_48
Last ObjectModification:
2016_01_16-AM-10_51_06
Theory : general
Home
Index