Nuprl Lemma : pi-irrational-instance
∃k:ℕ. (((r1)/k + 2 < |(r(22))/7 - 4 * MachinPi4()|) ∧ (|(r(22))/7 - 4 * MachinPi4()| < (r1)/k + 1))
Proof
Definitions occuring in Statement :
MachinPi4: MachinPi4(),
rless: x < y,
rabs: |x|,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
rsub: x - y,
int-to-real: r(n),
nat: ℕ,
exists: ∃x:A. B[x],
and: P ∧ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
int-rdiv: (a)/k1,
divide: n ÷ m,
int-to-real: r(n),
rabs: |x|,
absval: |i|,
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
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],
rminus: -(x),
int-rmul: k1 * a,
MachinPi4: MachinPi4(),
atan: atan(a;x),
atan_approx: atan_approx(a;x;M),
atan-log: atan-log(a;M),
gen_log_aux: gen_log_aux(p;c;x;i;n;M),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
btrue: tt,
bfalse: ff,
atan-approx: atan-approx(k;x;N),
poly-approx: poly-approx(a;x;k;N),
rmul: a * b,
imax: imax(a;b),
reg-seq-mul: reg-seq-mul(x;y),
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
eq_int: (i =z j),
remainder: n rem m,
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
istype-void,
istype-le,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
int-rmul_wf,
MachinPi4_wf,
rless_wf,
nat_properties,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
nat_plus_properties,
set_subtype_base,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
sqequalHypSubstitution,
voidElimination,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberFormation_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
universeIsType,
imageMemberEquality,
baseClosed,
addEquality,
applyEquality,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
sqequalBase,
closedConclusion,
because_Cache,
productIsType,
setElimination,
rename,
int_eqEquality,
inhabitedIsType,
baseApply
Latex:
\mexists{}k:\mBbbN{}. (((r1)/k + 2 < |(r(22))/7 - 4 * MachinPi4()|) \mwedge{} (|(r(22))/7 - 4 * MachinPi4()| < (r1)/k + 1))
Date html generated:
2019_10_31-AM-06_06_11
Last ObjectModification:
2019_05_17-PM-02_57_27
Theory : reals_2
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