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 :
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
exp: i^n,
remainder: n rem m,
divide: n ÷ m,
exp-rem: exp-rem(i;n;m),
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_upper: {i...}
Lemmas referenced :
nat_plus_wf,
istype-nat,
int_term_value_add_lemma,
itermAdd_wf,
div_mono1,
div_bounds_1,
nequal_wf,
divide_wfa,
nat_plus_inc_int_nzero,
remainder_wfa,
one-mul,
mul-commutes,
primrec1_lemma,
exp0_lemma,
subtype_rel_self,
istype-le,
decidable__lt,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract_wf,
decidable__equal_int,
subtract-1-ge-0,
int_seg_wf,
nat_plus_properties,
int_seg_properties,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
remainder_wf,
exp_add,
divide_wf,
mul_bounds_1a,
istype-universe,
true_wf,
squash_wf,
equal_wf,
nat_wf,
exp_wf2,
le_wf,
false_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
multiply-is-int-iff,
equal-wf-base,
div_rem_sum,
exp2,
iff_weakening_equal,
exp_mul,
int_upper_properties,
exp_wf4,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
less_than_wf,
exp-rem_wf,
int-value-type,
value-type-has-value,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
zero-add,
nequal-le-implies,
int_upper_subtype_nat,
int_eq-as-ifthenelse,
rem_mul,
rem_bounds_1,
exp1
Rules used in proof :
addEquality,
imageMemberEquality,
sqequalBase,
baseClosed,
Error :equalityIstype,
int_eqReduceFalseSq,
sqleReflexivity,
callbyvalueReduce,
intEquality,
cumulativity,
hypothesis_subsumption,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
instantiate,
applyEquality,
unionElimination,
productElimination,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
axiomSqEquality,
Error :universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
Error :lambdaFormation_alt,
thin,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
closedConclusion,
multiplyEquality,
universeEquality,
imageElimination,
hyp_replacement,
lambdaEquality,
computeAll,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
baseApply,
promote_hyp,
pointwiseFunctionality,
remainderEquality,
lambdaFormation,
addLevel,
dependent_set_memberEquality,
divideEquality,
equalityElimination
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[n,i:\mBbbN{}]. (exp-rem(i;n;m) \msim{} i\^{}n rem m)
Date html generated:
2019_06_20-PM-02_32_20
Last ObjectModification:
2019_03_10-AM-10_28_06
Theory : num_thy_1
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