Nuprl Lemma : divides_nchar
∀a,b:ℕ+. (a | b ⇐⇒ ∃c:ℕ+. (b = (a * c) ∈ ℕ+))
Proof
Definitions occuring in Statement :
divides: b | a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
equal: s = t ∈ T
Definitions unfolded in proof :
divides: b | a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
gt: i > j,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
guard: {T}
Lemmas referenced :
decidable__equal_int,
less_than_irreflexivity,
le_weakening2,
less_than_transitivity2,
less_than_wf,
pos_mul_arg_bounds,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
mul_nat_plus,
nat_plus_wf,
equal_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
multiplyEquality,
hypothesis,
because_Cache,
productElimination,
dependent_pairFormation,
dependent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
dependent_set_memberEquality,
applyEquality
Latex:
\mforall{}a,b:\mBbbN{}\msupplus{}. (a | b \mLeftarrow{}{}\mRightarrow{} \mexists{}c:\mBbbN{}\msupplus{}. (b = (a * c)))
Date html generated:
2016_05_14-PM-04_17_43
Last ObjectModification:
2016_01_14-PM-11_41_35
Theory : num_thy_1
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