Nuprl Lemma : rsqrt_functionality_wrt_rless
∀x:{x:ℝ| r0 ≤ x} . ∀y:ℝ. ((x < y) ⇒ (rsqrt(x) < rsqrt(y)))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
rge: x ≥ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
rless_wf,
real_wf,
set_wf,
rleq_wf,
int-to-real_wf,
rmul_wf,
radd-preserves-rless,
rminus_wf,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf,
rmul-is-positive,
rless-implies-rless,
itermConstant_wf,
rless_functionality,
req_transitivity,
radd_functionality,
rminus_functionality,
req_weakening,
rless_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
nat_plus_properties,
satisfiable-full-omega-tt,
intformless_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,
rsqrt_functionality_wrt_rleq,
rsqrt_nonneg,
rleq_weakening_rless,
rless_transitivity2,
req_wf,
rsqrt_wf,
rsqrt_squared
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
because_Cache,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
dependent_pairFormation,
productEquality,
setEquality,
applyEquality,
dependent_set_memberEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 \mleq{} x\} . \mforall{}y:\mBbbR{}. ((x < y) {}\mRightarrow{} (rsqrt(x) < rsqrt(y)))
Date html generated:
2017_10_03-AM-10_44_11
Last ObjectModification:
2017_07_28-AM-08_19_01
Theory : reals
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