Nuprl Lemma : absval-imax-difference
∀[a,b,c,d:ℤ]. (|imax(a;b) - imax(c;d)| ≤ imax(|a - c|;|b - d|))
Proof
Definitions occuring in Statement :
imax: imax(a;b),
absval: |i|,
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
true: True,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b)
Lemmas referenced :
less_than'_wf,
imax_wf,
absval_wf,
subtract_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_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,
le_wf,
intformand_wf,
int_formula_prop_and_lemma,
squash_wf,
true_wf,
imax_unfold,
nat_wf,
subtype_rel_self,
iff_weakening_equal,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
itermMinus_wf,
int_term_value_minus_lemma,
absval_unfold,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
isect_memberEquality,
voidElimination,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
promote_hyp,
instantiate,
independent_pairFormation,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed,
setElimination,
rename,
universeEquality,
minusEquality,
lessCases,
axiomSqEquality
Latex:
\mforall{}[a,b,c,d:\mBbbZ{}]. (|imax(a;b) - imax(c;d)| \mleq{} imax(|a - c|;|b - d|))
Date html generated:
2019_06_20-PM-01_13_39
Last ObjectModification:
2018_08_20-PM-09_31_26
Theory : int_2
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