Nuprl Lemma : ireal-approx_functionality
∀[j:ℕ]. ∀[M:ℕ+]. ∀[z:ℤ]. ∀[x,y:ℝ]. j-approx(x;M;z) ⇐⇒ j-approx(y;M;z) supposing x = y
Proof
Definitions occuring in Statement :
ireal-approx: j-approx(x;M;z),
req: x = y,
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
ireal-approx: j-approx(x;M;z),
prop: ℙ,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
nat: ℕ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ireal-approx_wf,
less_than'_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rabs_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
nat_plus_wf,
req_wf,
real_wf,
nat_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
setElimination,
rename,
independent_isectElimination,
inrFormation,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
multiplyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[j:\mBbbN{}]. \mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[z:\mBbbZ{}]. \mforall{}[x,y:\mBbbR{}]. j-approx(x;M;z) \mLeftarrow{}{}\mRightarrow{} j-approx(y;M;z) supposing x = y
Date html generated:
2018_05_22-PM-01_58_53
Last ObjectModification:
2017_10_25-PM-03_33_59
Theory : reals
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