Nuprl Lemma : cosh-inv-cosh
∀[x:{x:ℝ| r1 ≤ x} ]. (cosh(inv-cosh(x)) = x)
Proof
Definitions occuring in Statement :
inv-cosh: inv-cosh(x),
cosh: cosh(x),
rleq: x ≤ y,
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,
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
inv-cosh: inv-cosh(x),
cosh: cosh(x),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
guard: {T},
uiff: uiff(P;Q),
less_than: a < b,
true: True,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
top: Top,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rneq: x ≠ y,
or: P ∨ Q,
rdiv: (x/y)
Lemmas referenced :
sq_stable__req,
cosh_wf,
inv-cosh_wf,
rleq_wf,
int-to-real_wf,
rmul_preserves_rleq2,
less_than'_wf,
rsub_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rsqrt_nonneg,
ln_wf,
radd_wf,
rsqrt_wf,
req_wf,
rless_wf,
set_wf,
rlog_wf,
expr_wf,
rexp_wf,
rminus_wf,
equal_wf,
req_witness,
rleq-int,
false_wf,
rmul-identity1,
rleq_transitivity,
itermSubtract_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rleq_weakening_equal,
radd-zero,
rless-int,
rleq_functionality_wrt_implies,
rleq_functionality,
req_weakening,
rsub_functionality_wrt_rleq,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rdiv_wf,
rmul_preserves_req,
rinv_wf2,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
rmul_comm,
req_functionality,
int-rdiv-req,
req_transitivity,
radd_functionality,
rmul-rinv3,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
rexp-positive,
req_inversion,
rless_transitivity1,
rleq_weakening,
uiff_transitivity,
rexp_functionality,
rexp-rlog,
rexp-rminus,
rdiv_functionality,
radd_comm,
rmul-rinv,
rmul_functionality,
radd-preserves-req,
itermMinus_wf,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
voidElimination,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
productEquality,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
approximateComputation,
intEquality,
isect_memberEquality,
voidEquality,
addLevel,
instantiate,
cumulativity,
inrFormation,
int_eqEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r1 \mleq{} x\} ]. (cosh(inv-cosh(x)) = x)
Date html generated:
2017_10_04-PM-10_46_24
Last ObjectModification:
2017_06_21-PM-00_38_23
Theory : reals_2
Home
Index