Nuprl Lemma : quot_rem_exists_n
∀a:ℕ. ∀b:ℕ+. ∃q:ℕ. ∃r:ℕb. (a = ((q * b) + r) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_wf,
int_term_value_add_lemma,
itermAdd_wf,
primrec-wf2,
less_than_wf,
set_wf,
decidable__lt,
all_wf,
equal_wf,
exists_wf,
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,
nat_properties,
nat_plus_properties,
int_seg_properties,
nat_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
thin,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
functionEquality,
introduction,
independent_functionElimination
Latex:
\mforall{}a:\mBbbN{}. \mforall{}b:\mBbbN{}\msupplus{}. \mexists{}q:\mBbbN{}. \mexists{}r:\mBbbN{}b. (a = ((q * b) + r))
Date html generated:
2016_05_14-PM-04_18_53
Last ObjectModification:
2016_01_14-PM-11_42_27
Theory : num_thy_1
Home
Index