Nuprl Lemma : mul-imin
∀[n:ℕ]. ∀[a,b:ℤ]. ((n * imin(a;b)) = imin(n * a;n * b) ∈ ℤ)
Proof
Definitions occuring in Statement :
imin: imin(a;b),
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
nat: ℕ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
imin_unfold,
iff_weakening_equal,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
nat_wf,
mul_preserves_le,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
multiplyEquality,
setElimination,
rename,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
voidElimination,
cumulativity,
intEquality,
isect_memberEquality,
axiomEquality,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbZ{}]. ((n * imin(a;b)) = imin(n * a;n * b))
Date html generated:
2017_04_14-AM-09_13_41
Last ObjectModification:
2017_02_27-PM-03_51_09
Theory : int_2
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