Nuprl Lemma : absval_eq
∀x,y:ℤ. (|x| = |y| ∈ ℤ ⇐⇒ x = ± y)
Proof
Definitions occuring in Statement :
pm_equal: i = ± j,
absval: |i|,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
pm_equal: i = ± j,
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal-wf-base,
itermMinus_wf,
intformless_wf,
itermConstant_wf,
int_term_value_minus_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_subtype_base,
or_wf,
intformor_wf,
int_formula_prop_or_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
inlFormation,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
baseApply,
closedConclusion,
applyEquality,
promote_hyp,
instantiate,
cumulativity,
inrFormation
Latex:
\mforall{}x,y:\mBbbZ{}. (|x| = |y| \mLeftarrow{}{}\mRightarrow{} x = \mpm{} y)
Date html generated:
2017_04_14-AM-09_13_16
Last ObjectModification:
2017_02_27-PM-03_50_55
Theory : int_2
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