Nuprl Lemma : eqmod-prime-order-fixedpoints
∀n,k,p:ℕ.
(prime(p)
⇒ (∃T:Type. ∃f:T ⟶ T. (T ~ ℕn ∧ Inj(T;T;f) ∧ {x:T| (f x) = x ∈ T} ~ ℕk ∧ (∀x:T. ((f^p x) = x ∈ T))))
⇒ (n ≡ k mod p))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
prime: prime(a),
equipollent: A ~ B,
fun_exp: f^n,
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
orbit: orbit(T;f;L),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
or: P ∨ Q,
subtype_rel: A ⊆r B,
guard: {T},
assoced: a ~ b,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
eqmod-by-orbits,
member-less_than,
length_wf,
no_repeats_witness,
int_seg_wf,
orbit_wf,
list_wf,
inject_wf,
equipollent_wf,
equal_wf,
all_wf,
isect_wf,
or_wf,
equal-wf-T-base,
divides_wf,
exists_wf,
fun_exp_wf,
prime_wf,
nat_wf,
orbit-size-divides-order,
divides-prime,
assoced_nelim,
length_wf_nat,
false_wf,
le_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
hypothesis,
independent_pairFormation,
isect_memberFormation,
sqequalRule,
independent_pairEquality,
isectElimination,
natural_numberEquality,
cumulativity,
independent_isectElimination,
lambdaEquality,
axiomEquality,
rename,
functionExtensionality,
applyEquality,
productEquality,
because_Cache,
setEquality,
baseClosed,
setElimination,
functionEquality,
intEquality,
instantiate,
universeEquality,
unionElimination,
inrFormation,
inlFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}n,k,p:\mBbbN{}.
(prime(p)
{}\mRightarrow{} (\mexists{}T:Type. \mexists{}f:T {}\mrightarrow{} T. (T \msim{} \mBbbN{}n \mwedge{} Inj(T;T;f) \mwedge{} \{x:T| (f x) = x\} \msim{} \mBbbN{}k \mwedge{} (\mforall{}x:T. ((f\^{}p x) = x))))
{}\mRightarrow{} (n \mequiv{} k mod p))
Date html generated:
2017_04_17-AM-09_50_07
Last ObjectModification:
2017_02_27-PM-05_46_25
Theory : num_thy_1
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