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 :
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
nat_plus: ℕ+,
top: Top,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rem_bounds_1,
exp_mul,
iff_weakening_equal,
exp2,
div_rem_sum,
remainder_wfa,
add-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
false_wf,
exp_wf2,
squash_wf,
true_wf,
istype-universe,
mul_bounds_1a,
divide_wf,
exp_add,
remainder_wf,
decidable__le,
nequal_wf,
divide_wfa,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformeq_wf,
itermVar_wf,
intformand_wf,
istype-le,
div_mono1,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
div_bounds_1,
istype-void,
exp0_lemma,
istype-int,
subtype_rel_self,
subtype_rel_function,
nat_wf,
le_wf,
equal-wf-base,
sq_exists_wf,
natrec_wf,
istype-nat,
lelt_wf,
set_subtype_base,
int_seg_wf,
exp1,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
int-value-type,
equal_wf,
set-value-type
Rules used in proof :
addEquality,
pointwiseFunctionality,
promote_hyp,
productElimination,
hyp_replacement,
imageElimination,
universeEquality,
multiplyEquality,
Error :productIsType,
imageMemberEquality,
int_eqEquality,
independent_pairFormation,
Error :dependent_pairFormation_alt,
approximateComputation,
voidElimination,
Error :isect_memberEquality_alt,
functionEquality,
functionExtensionality,
Error :setIsType,
Error :functionIsType,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
applyEquality,
Error :dependent_set_memberFormation_alt,
unionElimination,
natural_numberEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
instantiate,
rename,
setElimination,
independent_isectElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
sqequalRule,
equalitySymmetry,
equalityTransitivity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
Error :inhabitedIsType,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
introduction,
cutEval,
hypothesisEquality,
intEquality,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}n:\mBbbN{}. (\mexists{}j:\mBbbZ{} [(j = i\^{}n)])
Date html generated:
2019_06_20-PM-02_31_24
Last ObjectModification:
2019_06_19-PM-02_36_13
Theory : num_thy_1
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