Nuprl Lemma : inv-sinh-sinh
∀[x:ℝ]. (inv-sinh(sinh(x)) = x)
Proof
Definitions occuring in Statement :
inv-sinh: inv-sinh(x),
sinh: sinh(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
inv-sinh: inv-sinh(x),
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
req_int_terms: t1 ≡ t2,
top: Top,
cosh: cosh(x),
sinh: sinh(x),
rneq: x ≠ y,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
true: True,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y)
Lemmas referenced :
inv-sinh-domain,
sinh_wf,
ln-req-iff,
radd_wf,
rsqrt_wf,
rmul_wf,
int-to-real_wf,
rleq_wf,
real_wf,
req_wf,
rless_wf,
req_witness,
inv-sinh_wf,
cosh-ge-1,
rsqrt-unique,
cosh_wf,
rleq-int,
false_wf,
rnexp_wf,
le_wf,
cosh2-sinh2,
req-implies-req,
rsub_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
expr_wf,
rexp_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
req_functionality,
req_inversion,
rnexp2,
radd_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
int-rdiv_wf,
rdiv_wf,
rless-int,
rminus_wf,
rmul_preserves_req,
rinv_wf2,
itermMultiply_wf,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
equal-wf-base,
true_wf,
nequal_wf,
rmul_comm,
int-rdiv-req,
req_transitivity,
int-rinv-cancel,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
productElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
sqequalRule,
independent_isectElimination,
independent_functionElimination,
because_Cache,
independent_pairFormation,
lambdaFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
unionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
addLevel
Latex:
\mforall{}[x:\mBbbR{}]. (inv-sinh(sinh(x)) = x)
Date html generated:
2017_10_04-PM-10_44_52
Last ObjectModification:
2017_06_24-AM-11_14_14
Theory : reals_2
Home
Index