Nuprl Lemma : rsqrt-unique
∀[x,s:{x:ℝ| r0 ≤ x} ]. s = rsqrt(x) supposing (s * s) = x
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
false: False,
nat_plus: ℕ+,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
true: True
Lemmas referenced :
sq_stable__req,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
real_wf,
req_wf,
rmul_wf,
rsqrt_squared,
rsqrt_nonneg,
equal_wf,
req_witness,
set_wf,
req-iff-not-rneq,
rneq_wf,
rsub_wf,
radd_wf,
rminus_wf,
req_weakening,
req_functionality,
rsub_functionality,
radd-rminus-both,
req_inversion,
uiff_transitivity,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rmul_comm,
radd-assoc,
radd-rminus-assoc,
radd-preserves-rless,
rless_wf,
rless_functionality,
radd-zero-both,
radd-ac,
radd_comm,
rmul-neq-zero,
nat_plus_properties,
satisfiable-full-omega-tt,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rneq_functionality,
radd-preserves-req,
rleq_transitivity,
rleq_weakening,
radd-preserves-rleq,
rleq_antisymmetry,
rleq_functionality,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
minus-zero,
rminus_functionality,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality,
hypothesis,
natural_numberEquality,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
sqequalRule,
independent_functionElimination,
independent_pairFormation,
because_Cache,
lambdaFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
independent_isectElimination,
unionElimination,
inlFormation,
inrFormation,
addLevel,
levelHypothesis,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
addEquality,
universeEquality
Latex:
\mforall{}[x,s:\{x:\mBbbR{}| r0 \mleq{} x\} ]. s = rsqrt(x) supposing (s * s) = x
Date html generated:
2017_10_03-AM-10_43_04
Last ObjectModification:
2017_07_28-AM-08_18_16
Theory : reals
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