Nuprl Lemma : arcsin1
arcsin(r1) = π/2
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
halfpi: π/2,
req: x = y,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
rless: x < y,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
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,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-to-real: r(n),
halfpi: π/2,
divide: n ÷ m,
cubic_converge: cubic_converge(b;m),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
fastpi: fastpi(n),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
true: True,
and: P ∧ Q,
subtype_rel: A ⊆r B,
real: ℝ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
req_int_terms: t1 ≡ t2
Lemmas referenced :
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,
int-to-real_wf,
halfpi_wf,
arcsin-unique,
member_rccint_lemma,
rleq-int,
istype-false,
rleq_weakening_equal,
rleq_wf,
rminus_wf,
rsin-halfpi,
radd-preserves-rleq,
radd_wf,
rmul_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
itermMultiply_wf,
rmul_preserves_rless,
rless-int,
rleq_weakening_rless,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
rless_functionality
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberFormation_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
unionElimination,
isectElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
hypothesisEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addEquality,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
minusEquality,
productElimination,
lambdaFormation_alt,
because_Cache,
productIsType,
int_eqEquality
Latex:
arcsin(r1) = \mpi{}/2
Date html generated:
2019_10_31-AM-06_15_22
Last ObjectModification:
2019_05_24-PM-04_37_49
Theory : reals_2
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