Nuprl Lemma : rsqrt-rnexp-2
∀[x:{x:ℝ| r0 ≤ x} ]. (rsqrt(x)^2 = x)
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
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,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
true: True,
subtract: n - m,
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rnexp_wf,
false_wf,
le_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
real_wf,
req_wf,
rmul_wf,
set_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
subtract_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
req_functionality,
rnexp_unroll,
req_weakening,
rmul_functionality,
rsqrt_squared
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
setElimination,
rename,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
because_Cache,
independent_functionElimination,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
voidElimination,
intEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 \mleq{} x\} ]. (rsqrt(x)\^{}2 = x)
Date html generated:
2017_10_03-AM-10_42_46
Last ObjectModification:
2017_07_28-AM-08_18_00
Theory : reals
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