Nuprl Lemma : qabs-qinv
∀[s:{s:ℚ| ¬(s = 0 ∈ ℚ)} ]. (|1/s| = 1/|s| ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qinv: 1/r,
rationals: ℚ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
false: False,
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
qabs: |r|,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
qdiv: (r/s),
qmul: r * s,
qpositive: qpositive(r),
qinv: 1/r,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
bor: p ∨bq,
mk-rational: mk-rational(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T
Lemmas referenced :
rationals_wf,
int-subtype-rationals,
istype-void,
qabs_wf,
equal-wf-T-base,
qabs-zero,
q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
equal-wf-base,
int_subtype_base,
assert-qeq,
istype-assert,
not_wf,
equal_wf,
qinv_wf,
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
int-valueall-type,
product-valueall-type,
mul-one,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
less_than_wf,
istype-less_than,
intformnot_wf,
int_formula_prop_not_lemma,
mk-rational_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
nequal_wf,
mk-rational-qdiv,
decidable__equal_int,
qmul-preserves-eq,
qdiv_wf,
qmul_wf,
squash_wf,
true_wf,
istype-universe,
qmul_zero_qrng,
subtype_rel_self,
qmul-qdiv-cancel,
iff_weakening_equal,
int-equal-in-rationals,
qmul-mul,
itermMultiply_wf,
int_term_value_mul_lemma,
qmul_over_minus_qrng,
qmul_assoc,
qmul_one_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesis,
setIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalRule,
functionIsType,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
natural_numberEquality,
applyEquality,
thin,
sqequalHypSubstitution,
independent_pairFormation,
because_Cache,
baseClosed,
isectElimination,
voidElimination,
independent_functionElimination,
lambdaFormation,
rename,
setElimination,
independent_isectElimination,
productElimination,
dependent_functionElimination,
lambdaFormation_alt,
closedConclusion,
baseApply,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
hyp_replacement,
applyLambdaEquality,
functionEquality,
callbyvalueReduce,
sqleReflexivity,
isintReduceTrue,
minusEquality,
intEquality,
productEquality,
lambdaEquality_alt,
independent_pairEquality,
unionElimination,
equalityElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality_alt,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[s:\{s:\mBbbQ{}| \mneg{}(s = 0)\} ]. (|1/s| = 1/|s|)
Date html generated:
2020_05_20-AM-09_16_25
Last ObjectModification:
2019_12_31-PM-06_30_43
Theory : rationals
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