Nuprl Lemma : efficient-exp
∀i:ℤ. ∀n:ℕ. (∃j:{ℤ| (j = i^n ∈ ℤ)})
Proof
Definitions occuring in Statement :
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
all_wf,
int_seg_wf,
sq_exists_wf,
equal-wf-base-T,
exp_wf2,
int_seg_subtype_nat,
false_wf,
nat_wf,
natrec_wf,
exp1,
iff_weakening_equal,
equal-wf-base,
exp0_lemma,
div_bounds_1,
less_than_wf,
div_mono1,
le_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
set-value-type,
equal_wf,
int-value-type,
decidable__le,
lelt_wf,
mul_bounds_1a,
divide_wf,
exp_add,
remainder_wf,
squash_wf,
true_wf,
div_rem_sum,
nequal_wf,
add-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
multiply-is-int-iff,
two-mul,
mul-commutes,
rem_bounds_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_pairFormation,
functionExtensionality,
functionEquality,
dependent_set_memberFormation,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
baseApply,
closedConclusion,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
imageMemberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
cutEval,
hyp_replacement,
imageElimination,
universeEquality,
addLevel,
addEquality,
remainderEquality,
pointwiseFunctionality,
promote_hyp,
multiplyEquality
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}n:\mBbbN{}. (\mexists{}j:\{\mBbbZ{}| (j = i\^{}n)\})
Date html generated:
2016_10_25-AM-10_59_52
Last ObjectModification:
2016_07_12-AM-07_07_01
Theory : general
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