Nuprl Lemma : qexp-eq-q-rng-nexp
∀[n:ℕ]. ∀[r:ℚ]. (r ↑ n = q-rng-nexp(r;n) ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
q-rng-nexp: q-rng-nexp(r;n),
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qexp: r ↑ n,
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
mk-rational: mk-rational(a;b),
rationals: ℚ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
q-rng-nexp: q-rng-nexp(r;n),
rng_nexp: e ↑r n,
mon_nat_op: n ⋅ e,
mul_mon_of_rng: r↓xmn,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
qrng: <ℚ+*>,
rng_times: *,
rng_one: 1,
nat_op: n x(op;id) e,
itop: Π(op,id) lb ≤ i < ub. E[i],
infix_ap: x f y,
ycomb: Y,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
rev_uimplies: rev_uimplies(P;Q),
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
eq_int: (i =z j),
qmul: r * s,
has-valueall: has-valueall(a)
Lemmas referenced :
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
qrep_wf,
nat_plus_wf,
equal_wf,
rationals_wf,
mk-rational_wf,
exp_wf2,
exp_wf3,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
q-rng-nexp_wf,
exp-fastexp,
equals-qrep,
subtype_rel_transitivity,
b-union_wf,
int_nzero_wf,
subtype_quotient,
equal-wf-T-base,
bool_wf,
qeq_wf,
qeq-equiv,
subtype_rel_b-union-right,
subtype_rel_b-union,
subtype_rel_self,
subtype_rel_product,
intformle_wf,
int_formula_prop_le_lemma,
ge_wf,
exp0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
qmul_wf,
iff_weakening_equal,
quotient-member-eq,
bfalse_wf,
ifthenelse_wf,
subtype_rel_b-union-left,
iff_imp_equal_bool,
qeq_wf2,
int-subtype-rationals,
btrue_wf,
assert-qeq,
assert_wf,
iff_wf,
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
exp_step,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_nzero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
productEquality,
lambdaFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
hyp_replacement,
applyLambdaEquality,
applyEquality,
setEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
intWeakElimination,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_pairEquality,
independent_pairEquality,
addLevel,
impliesFunctionality,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[r:\mBbbQ{}]. (r \muparrow{} n = q-rng-nexp(r;n))
Date html generated:
2018_05_21-PM-11_59_04
Last ObjectModification:
2017_07_26-PM-06_48_28
Theory : rationals
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