Nuprl Lemma : fun_exp-rem
∀[T:Type]. ∀[f:T ⟶ T]. ∀[x:T]. ∀[n:ℕ+]. ∀[k:ℕ]. ((f^k x) = (f^k rem n x) ∈ T) supposing (f^n x) = x ∈ T
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
remainder: n rem m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
sq_type: SQType(T),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
label: ...$L... t,
fun_exp: f^n,
primrec: primrec(n;b;c)
Lemmas referenced :
div_rem_sum,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
fun_exp_wf,
nat_plus_subtype_nat,
nat_plus_wf,
remainder_wf,
nat_wf,
div_bounds_1,
mul_bounds_1a,
divide_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
intformnot_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
false_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
fun_exp_add-sq,
squash_wf,
true_wf,
fun_exp-mul,
iff_weakening_equal,
fun_exp-fixedpoint
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
axiomEquality,
cumulativity,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
addEquality,
multiplyEquality,
divideEquality,
instantiate,
productElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
dependent_set_memberEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[x:T]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[k:\mBbbN{}]. ((f\^{}k x) = (f\^{}k rem n x)) supposing (f\^{}n x) = x
Date html generated:
2017_04_14-AM-09_16_48
Last ObjectModification:
2017_02_27-PM-03_53_54
Theory : int_2
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