Nuprl Lemma : exp-rem-property
∀[m:ℕ+]. ∀[n,i:ℕ]. (exp-rem(i;n;m) ~ i^n rem m)
Proof
Definitions occuring in Statement :
exp-rem: exp-rem(i;n;m),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
remainder: n rem m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
nat_plus: ℕ+,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
exp-rem: exp-rem(i;n;m),
sq_type: SQType(T),
exp: i^n,
nequal: a ≠ b ∈ T ,
true: True,
less_than: a < b,
squash: ↓T,
int_nzero: ℤ-o,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
has-value: (a)↓,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
nat_plus_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
set_wf,
lelt_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
subtype_base_sq,
int_subtype_base,
exp0_lemma,
primrec1_lemma,
mul-commutes,
one-mul,
equal-wf-base,
true_wf,
div_bounds_1,
div_mono1,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
nat_plus_wf,
mul_bounds_1a,
divide_wf,
exp_add,
remainder_wf,
equal_wf,
squash_wf,
exp_wf2,
div_rem_sum,
nequal_wf,
multiply-is-int-iff,
add-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
two-mul,
iff_weakening_equal,
int_eq-as-ifthenelse,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
value-type-has-value,
int-value-type,
exp-rem_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
exp_wf4,
int_upper_properties,
rem_mul,
rem_bounds_1,
exp1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
because_Cache,
productElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
dependent_set_memberEquality,
instantiate,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
int_eqReduceFalseSq,
remainderEquality,
baseClosed,
divideEquality,
addLevel,
imageMemberEquality,
addEquality,
hyp_replacement,
imageElimination,
universeEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
multiplyEquality,
equalityEquality,
equalityElimination
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[n,i:\mBbbN{}]. (exp-rem(i;n;m) \msim{} i\^{}n rem m)
Date html generated:
2016_10_25-AM-11_00_19
Last ObjectModification:
2016_07_12-AM-07_08_01
Theory : general
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