Nuprl Lemma : div_mul_add_cancel
∀[a:ℕ]. ∀[b:ℕ+]. ∀[r:ℕb]. ((((a * b) + r) ÷ b) = a ∈ ℤ)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
div_nrel: Div(a;n;q)
Lemmas referenced :
div_unique2,
add_nat_wf,
multiply_nat_wf,
nat_plus_subtype_nat,
int_seg_subtype_nat,
false_wf,
nat_wf,
nat_properties,
int_seg_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
mul_bounds_1a,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[b:\mBbbN{}\msupplus{}]. \mforall{}[r:\mBbbN{}b]. ((((a * b) + r) \mdiv{} b) = a)
Date html generated:
2017_04_14-AM-09_15_45
Last ObjectModification:
2017_02_27-PM-03_53_08
Theory : int_2
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