Nuprl Lemma : wilson-theorem
∀n:{i:ℤ| 1 < i} . (prime(n) ⇐⇒ (n - 1)! ≡ (-1) mod n)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
fact: (n)!,
eqmod: a ≡ b mod m,
prime: prime(a),
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
subtract: n - m,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
cand: A c∧ B,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
label: ...$L... t,
sq_stable: SqStable(P),
guard: {T},
true: True,
subtract: n - m,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
inject: Inj(A;B;f),
cyclic-map: cyclic-map(T),
injection: A →⟶ B,
respects-equality: respects-equality(S;T),
compose: f o g,
rotate-by: rotate-by(n;i),
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
less_than: a < b,
equipollent: A ~ B,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
divides: b | a,
prime: prime(a),
eqmod: a ≡ b mod m,
pm_equal: i = ± j
Lemmas referenced :
prime_wf,
eqmod_wf,
fact_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
equal_wf,
rotate-by-rotate-by,
subtype_rel_sets_simple,
less_than_wf,
le_wf,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
sq_stable__less_than,
rotate-by_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
iff_weakening_equal,
add-commutes,
add-swap,
add-associates,
zero-add,
rotate-by-trivial-test,
int_seg_wf,
subtract-add-cancel,
rotate-by-trivial,
eqmod-prime-order-fixedpoints,
nat_plus_subtype_nat,
cyclic-map_wf,
cyclic-map-conjugate,
count-cyclic-map,
istype-false,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel,
equipollent_wf,
inject_wf,
fun_exp_wf,
compose_wf,
respects-equality-set,
injection_wf,
all_wf,
exists_wf,
nat_wf,
istype-nat,
respects-equality-set-trivial,
set_subtype_base,
lelt_wf,
int_subtype_base,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
nat_properties,
ge_wf,
add-comm,
rem_bounds_1,
rem_eq_args_z,
nequal_wf,
absval_wf,
add_functionality_wrt_eq,
subtract-1-ge-0,
subtype_base_sq,
rem_base_case,
less_than_transitivity2,
itermMinus_wf,
int_term_value_minus_lemma,
minus-add,
minus-minus,
minus-one-mul,
rem_addition,
le_weakening2,
rem_rec_case,
add-mul-special,
one-mul,
zero-mul,
iterate-rotate-by,
mul-commutes,
rem-zero,
subtract_nat_wf,
false_wf,
int_seg_subtype_nat,
subtract-is-int-iff,
rotate-by-cyclic-map,
gcd-prime,
add-member-int_seg2,
equal-wf-base-T,
compose-rotate-by,
biject_wf,
iterated-conjugate,
mul_bounds_1a,
itermMultiply_wf,
int_term_value_mul_lemma,
divides_reflexivity,
rotate-by-is-id,
eqmod-zero,
eqmod_functionality_wrt_eqmod,
eqmod_weakening,
subtract_functionality_wrt_eqmod,
primality-test,
divides_wf,
assoced_nelim,
mul_preserves_le,
mul_preserves_lt,
multiply-is-int-iff,
divides_add,
divides-fact,
divides_product,
only_pm_one_divs_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
minusEquality,
setIsType,
imageElimination,
intEquality,
productElimination,
imageMemberEquality,
baseClosed,
addEquality,
productIsType,
applyLambdaEquality,
setEquality,
functionIsType,
equalityIstype,
functionEquality,
sqequalBase,
functionExtensionality_alt,
instantiate,
universeEquality,
intWeakElimination,
axiomEquality,
functionIsTypeImplies,
remainderEquality,
closedConclusion,
cumulativity,
multiplyEquality,
hyp_replacement,
functionExtensionality,
pointwiseFunctionality,
promote_hyp,
baseApply,
inlFormation_alt
Latex:
\mforall{}n:\{i:\mBbbZ{}| 1 < i\} . (prime(n) \mLeftarrow{}{}\mRightarrow{} (n - 1)! \mequiv{} (-1) mod n)
Date html generated:
2019_10_15-AM-11_23_06
Last ObjectModification:
2018_12_08-AM-11_51_50
Theory : general
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