Nuprl Lemma : partial-arcsin_wf
∀a:{a:ℝ| ((r(-3)/r(4)) < a) ∧ (a < (r(3)/r(4)))} . (partial-arcsin(a) ∈ {x:ℝ| x = arcsine(a)} )
Proof
Definitions occuring in Statement :
partial-arcsin: partial-arcsin(a),
arcsine: arcsine(x),
rdiv: (x/y),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
partial-arcsin: partial-arcsin(a),
cand: A c∧ B,
guard: {T},
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_apply: x[s]
Lemmas referenced :
rleq-int-fractions3,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
rleq-int-fractions2,
real-from-approx_wf,
arcsine_wf,
member_rooint_lemma,
rless_transitivity2,
int-to-real_wf,
rdiv_wf,
rless-int,
rless_wf,
rless_transitivity1,
arcsine-approx_wf,
rneq-int,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_plus_wf,
real_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
hypothesisEquality,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
minusEquality,
setElimination,
rename,
closedConclusion,
inrFormation_alt,
because_Cache,
imageMemberEquality,
baseClosed,
productIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
setIsType
Latex:
\mforall{}a:\{a:\mBbbR{}| ((r(-3)/r(4)) < a) \mwedge{} (a < (r(3)/r(4)))\} . (partial-arcsin(a) \mmember{} \{x:\mBbbR{}| x = arcsine(a)\} )
Date html generated:
2019_10_31-AM-06_13_22
Last ObjectModification:
2019_05_21-PM-01_49_09
Theory : reals_2
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