Nuprl Lemma : qinv_wf
∀[r:ℚ]. 1/r ∈ ℚ supposing ¬↑qeq(r;0)
Proof
Definitions occuring in Statement :
qinv: 1/r,
rationals: ℚ,
qeq: qeq(r;s),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
rationals: ℚ,
all: ∀x:A. B[x],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
not: ¬A,
false: False,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
btrue: tt,
qinv: 1/r,
has-value: (a)↓,
has-valueall: has-valueall(a),
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
rationals_wf,
b-union_wf,
int_nzero_wf,
bool_wf,
qeq_wf,
qeq_refl,
qeq-functionality,
subtype_rel_b-union-left,
member_wf,
squash_wf,
true_wf,
istype-universe,
quotient-member-eq,
qeq-equiv,
qinv-wf,
subtype_base_sq,
bool_subtype_base,
equal-wf-T-base,
assert_wf,
istype-void,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
eqtt_to_assert,
assert_of_eq_int,
int_subtype_base,
product-valueall-type,
eq_int_wf,
set-valueall-type,
nequal_wf,
int_nzero_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformand_wf,
int_formula_prop_and_lemma,
equal_wf,
not_wf,
int-subtype-rationals,
qeq-wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
intEquality,
productEquality,
promote_hyp,
lambdaFormation_alt,
equalityIsType3,
hypothesisEquality,
baseClosed,
inhabitedIsType,
pointwiseFunctionality,
pertypeElimination,
productElimination,
natural_numberEquality,
applyEquality,
independent_isectElimination,
productIsType,
equalityIsType4,
dependent_functionElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
because_Cache,
instantiate,
cumulativity,
independent_functionElimination,
imageMemberEquality,
unionElimination,
equalityElimination,
functionIsType,
equalityIsType1,
callbyvalueReduce,
sqleReflexivity,
isintReduceTrue,
independent_pairEquality,
multiplyEquality,
setElimination,
rename,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
lambdaFormation
Latex:
\mforall{}[r:\mBbbQ{}]. 1/r \mmember{} \mBbbQ{} supposing \mneg{}\muparrow{}qeq(r;0)
Date html generated:
2019_10_16-AM-11_47_12
Last ObjectModification:
2018_10_11-PM-01_25_28
Theory : rationals
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