Nuprl Lemma : q-triangle-inequality
∀[r,s:ℚ]. (|r + s| ≤ (|r| + |s|))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qabs: |r|,
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
squash: ↓T,
and: P ∧ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
isl: isl(x),
bnot: ¬bb,
assert: ↑b,
q-constraints: q-constraints(k;A;y),
top: Top,
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
sq_type: SQType(T),
select: L[n],
cons: [a / b],
pi2: snd(t),
pi1: fst(t),
q-rel: q-rel(r;x),
eq_int: (i =z j),
nat_plus: ℕ+,
less_than: a < b,
subtract: n - m,
select?: as[i]?a,
lt_int: i <z j,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
qadd: r + s,
evalall: evalall(t)
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
qadd_wf,
evalall-reduce,
qle_witness,
qabs_wf,
qpositive_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qless_wf,
int-subtype-rationals,
qle_reflexivity,
bnot_wf,
not_wf,
decidable__qle,
qmul_wf,
qle_wf,
squash_wf,
true_wf,
qmul_over_plus_qrng,
subtype_rel_self,
iff_weakening_equal,
uiff_transitivity,
eqtt_to_assert,
assert-qpositive,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
decidable__q-constraints-opt,
false_wf,
le_wf,
cons_wf,
nat_wf,
select?_wf,
nil_wf,
outr_wf,
sq_exists_wf,
list_wf,
q-constraints_wf,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
q-linear-unroll,
less_than_wf,
q-linear-base,
int_seg_subtype,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
qadd_preserves_qless,
mon_ident_q,
qadd_preserves_qle,
qless_complement_qorder,
qle_complement_qorder,
qmul_one_qrng,
mon_assoc_q,
qadd_comm_q,
qmul_zero_qrng,
qinverse_q,
qadd_ac_1_q,
q_distrib,
qadd_inv_assoc_q,
qmul_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
natural_numberEquality,
applyEquality,
dependent_functionElimination,
minusEquality,
unionElimination,
voidElimination,
lambdaEquality,
imageElimination,
productElimination,
imageMemberEquality,
instantiate,
universeEquality,
lambdaFormation,
equalityElimination,
independent_pairFormation,
impliesFunctionality,
dependent_set_memberEquality,
productEquality,
functionEquality,
intEquality,
independent_pairEquality,
computeAll,
dependent_set_memberFormation,
voidEquality,
setElimination,
rename,
cumulativity,
hypothesis_subsumption,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality
Latex:
\mforall{}[r,s:\mBbbQ{}]. (|r + s| \mleq{} (|r| + |s|))
Date html generated:
2018_05_22-AM-00_25_59
Last ObjectModification:
2018_05_19-PM-04_08_06
Theory : rationals
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