Nuprl Lemma : less-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},
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
top: Top,
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_wf,
set_subtype_base,
lelt_wf,
istype-nat,
natrec_wf,
sq_exists_wf,
equal-wf-base,
le_wf,
nat_wf,
subtype_rel_function,
subtype_rel_self,
istype-int,
exp1,
exp0_lemma,
istype-void,
div_bounds_1,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
div_mono1,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
intformand_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
divide_wfa,
nequal_wf,
mul_bounds_1a,
divide_wf,
exp_add,
remainder_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
exp_wf2,
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_mul,
iff_weakening_equal,
exp2,
rem_bounds_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
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,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
Error :setIsType,
Error :inhabitedIsType,
Error :equalityIstype,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
Error :lambdaEquality_alt,
sqequalBase,
equalitySymmetry,
functionExtensionality,
functionEquality,
Error :dependent_set_memberFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
imageMemberEquality,
equalityTransitivity,
Error :productIsType,
multiplyEquality,
hyp_replacement,
imageElimination,
universeEquality,
addEquality,
productElimination,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}n:\mBbbN{}. (\mexists{}j:\mBbbZ{} [(j = i\^{}n)])
Date html generated:
2019_06_20-PM-02_31_34
Last ObjectModification:
2019_03_06-AM-10_53_39
Theory : num_thy_1
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