Nuprl Lemma : int-triangle-inequality
∀[a,b:ℤ]. (|a + b| ≤ (|a| + |b|))
Proof
Definitions occuring in Statement :
absval: |i|,
uall: ∀[x:A]. B[x],
le: A ≤ B,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
le: A ≤ B,
subtype_rel: A ⊆r B
Lemmas referenced :
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformand_wf,
itermMinus_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_term_value_minus_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
less_than'_wf,
absval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_pairEquality,
applyEquality,
axiomEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. (|a + b| \mleq{} (|a| + |b|))
Date html generated:
2017_04_14-AM-09_13_14
Last ObjectModification:
2017_02_27-PM-03_50_18
Theory : int_2
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