Nuprl Lemma : decidable__exists-divisor
∀n:ℕ+. ∀P:ℕ ⟶ ℙ. ((∀d:ℕ. Dec(P[d])) ⇒ Dec(∃d:ℕ. ((d | n) ∧ P[d])))
Proof
Definitions occuring in Statement :
divides: b | a,
nat_plus: ℕ+,
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
lelt: i ≤ j < k,
guard: {T},
nat: ℕ,
cand: A c∧ B,
exists: ∃x:A. B[x],
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
int_seg: {i..j-},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
nat_plus_wf,
decidable_wf,
all_wf,
exists_wf,
not_wf,
decidable__divides_ext,
decidable__and2,
int_seg_wf,
false_wf,
int_seg_subtype_nat,
nat_wf,
divides_wf,
decidable__exists_int_seg,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
le_wf,
divisors_bound
Rules used in proof :
cumulativity,
functionEquality,
voidElimination,
dependent_set_memberEquality,
inrFormation,
dependent_pairFormation,
productElimination,
inlFormation,
unionElimination,
isect_memberEquality,
independent_functionElimination,
universeEquality,
independent_pairFormation,
independent_isectElimination,
functionExtensionality,
applyEquality,
because_Cache,
productEquality,
lambdaEquality,
sqequalRule,
isectElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
addEquality,
natural_numberEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
intEquality,
int_eqEquality,
approximateComputation
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}d:\mBbbN{}. Dec(P[d])) {}\mRightarrow{} Dec(\mexists{}d:\mBbbN{}. ((d | n) \mwedge{} P[d])))
Date html generated:
2018_05_21-PM-00_54_06
Last ObjectModification:
2018_01_01-PM-03_01_58
Theory : num_thy_1
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