Nuprl Lemma : fermat-little
∀p:ℕ. (prime(p) ⇒ (∀x:ℕ. (x^p ≡ x mod p)))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
prime: prime(a),
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
inject: Inj(A;B;f),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
false: False,
guard: {T},
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rotate: rot(n),
compose: f o g,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T),
le: A ≤ B,
subtract: n - m,
uiff: uiff(P;Q),
bfalse: ff,
equipollent: A ~ B,
prime: prime(a),
less_than': less_than'(a;b),
biject: Bij(A;B;f),
surject: Surj(A;B;f),
bool: 𝔹,
unit: Unit,
it: ⋅
Lemmas referenced :
eqmod-prime-order-fixedpoints,
exp_wf4,
prime_wf,
nat_wf,
int_seg_wf,
compose_wf,
rotate_wf,
equipollent-exp,
equipollent_wf,
exp_wf2,
inject_wf,
equal_wf,
fun_exp_wf,
istype-universe,
int_seg_properties,
nat_properties,
decidable__le,
le_wf,
less_than_wf,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
subtract_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
bool_wf,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
set_subtype_base,
int_subtype_base,
ifthenelse_wf,
subtype_base_sq,
eq_int_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base-T,
add-associates,
add-swap,
add-commutes,
zero-add,
bool_cases,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
biject_wf,
istype-false,
ge_wf,
subtract-1-ge-0,
equal-wf-T-base,
subtract-add-cancel,
uiff_transitivity,
int_seg_subtype_nat,
lelt_wf,
fun_exp_compose2,
rotate-order
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
Error :inhabitedIsType,
Error :universeIsType,
setElimination,
rename,
Error :dependent_pairFormation_alt,
functionEquality,
natural_numberEquality,
Error :lambdaEquality_alt,
because_Cache,
Error :functionIsType,
independent_pairFormation,
sqequalRule,
Error :productIsType,
setEquality,
applyEquality,
Error :equalityIsType1,
Error :functionExtensionality_alt,
Error :dependent_set_memberEquality_alt,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
Error :equalityIsType3,
baseApply,
closedConclusion,
intEquality,
addEquality,
Error :equalityIsType4,
cumulativity,
Error :setIsType,
intWeakElimination,
axiomEquality,
Error :functionIsTypeImplies,
equalityElimination,
hyp_replacement
Latex:
\mforall{}p:\mBbbN{}. (prime(p) {}\mRightarrow{} (\mforall{}x:\mBbbN{}. (x\^{}p \mequiv{} x mod p)))
Date html generated:
2019_06_20-PM-02_28_52
Last ObjectModification:
2018_10_05-PM-10_53_52
Theory : num_thy_1
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