Nuprl Lemma : cosh-gt-1
∀x:ℝ. (x ≠ r0 ⇒ (r1 < cosh(x)))
Proof
Definitions occuring in Statement :
cosh: cosh(x),
rneq: x ≠ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
cosh: cosh(x),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
nat: ℕ,
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rgt: x > y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
int_upper: {i...},
ge: i ≥ j ,
real: ℝ,
int-to-real: r(n),
rexp: e^x,
pi1: fst(t),
exp-exists-ext,
rsum: Σ{x[k] | n≤k≤m},
canonical-bound: canonical-bound(r),
absval: |i|,
rmul: a * b,
rabs: |x|,
accelerate: accelerate(k;f),
imax: imax(a;b),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
btrue: tt,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
int-rdiv: (a)/k1,
fact: (n)!,
primrec: primrec(n;b;c),
subtract: n - m,
rnexp: x^k1,
eq_int: (i =z j),
fastexp: i^n,
efficient-exp-ext,
rlessw: rlessw(x;y),
quick-find: quick-find(p;n),
radd: a + b,
rinv: rinv(x),
mu-ge: mu-ge(f;n),
reg-seq-inv: reg-seq-inv(x),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
reg-seq-adjust: reg-seq-adjust(n;x),
nil: [],
it: ⋅,
exp-ratio: exp-ratio(a;b;n;p;q),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
from-upto: [n, m),
radd-list: radd-list(L),
length: ||as||
Lemmas referenced :
rneq_wf,
int-to-real_wf,
real_wf,
int-rdiv_wf,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
istype-int,
int_formula_prop_eq_lemma,
istype-void,
int_term_value_constant_lemma,
int_formula_prop_wf,
nequal_wf,
radd_wf,
expr_wf,
rminus_wf,
rdiv_wf,
rexp_wf,
rless-int,
rless_wf,
rmul_preserves_rless,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
itermAdd_wf,
itermVar_wf,
rless_functionality,
req_weakening,
req_transitivity,
int-rdiv-req,
rdiv_functionality,
radd_functionality,
expr-req,
rmul-int,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
exp_wf2,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
istype-le,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
subtract_wf,
int_term_value_subtract_lemma,
istype-less_than,
primrec-wf2,
istype-nat,
exp0_lemma,
rexp_functionality,
rmul-identity1,
rminus_functionality,
square-rless-implies,
rnexp2,
trivial-rleq-radd,
rleq_weakening_equal,
itermMinus_wf,
radd-preserves-rless,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
rexp-positive,
req_inversion,
rexp-radd,
rmul_functionality,
squash_wf,
true_wf,
rminus-int,
real_term_value_minus_lemma,
rexp0,
nat_plus_properties,
req-int,
exp_step,
decidable__lt,
decidable__equal_int,
int_term_value_mul_lemma,
req_functionality,
rmul_assoc,
radd_comm_eq,
subtype_rel_self,
iff_weakening_equal,
rabs-neq-zero,
small-reciprocal-real,
rabs_wf,
nat_plus_subtype_nat,
rleq_wf,
exp-positive,
rleq-int-fractions2,
le_weakening2,
one-mul,
exp-greater,
nat_properties,
rexp-non-decreasing,
rminus_functionality_wrt_rleq,
rleq_weakening,
rmul_over_rminus,
req_wf,
rabs-rminus,
rabs-of-nonneg,
rleq_functionality,
radd_functionality_wrt_rless2,
rless_transitivity1,
exp-exists-ext,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality_alt,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
intEquality,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
because_Cache,
closedConclusion,
inrFormation_alt,
productElimination,
independent_pairFormation,
imageMemberEquality,
int_eqEquality,
unionElimination,
functionIsType,
setIsType,
functionEquality,
minusEquality,
imageElimination,
multiplyEquality,
instantiate,
universeEquality,
dependent_set_memberFormation_alt,
addEquality
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (r1 < cosh(x)))
Date html generated:
2019_10_31-AM-06_12_05
Last ObjectModification:
2018_12_14-AM-10_53_16
Theory : reals_2
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