Nuprl Lemma : member-residues-mod
∀n:ℕ. ∀x:residue(n). (x ∈ residues-mod(n))
Proof
Definitions occuring in Statement :
residues-mod: residues-mod(n),
residue: residue(n),
l_member: (x ∈ l),
nat: ℕ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
residue: residue(n),
residues-mod: residues-mod(n),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rev_uimplies: rev_uimplies(P;Q),
l_member: (x ∈ l),
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T)
Lemmas referenced :
residue_wf,
nat_wf,
filter_type,
int_seg_wf,
eq_int_wf,
gcd_wf,
upto_wf,
assert_wf,
coprime_wf,
subtype_rel_list_set,
assert_of_eq_int,
coprime_elim_a,
assoced_weakening,
member_filter,
subtype_rel_list,
int_seg_subtype_nat,
false_wf,
member_upto,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
assoced_nelim,
gcd-non-neg,
le_wf,
less_than_wf,
length_wf,
equal_wf,
select_wf,
sq_stable__coprime,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
natural_numberEquality,
setElimination,
rename,
lambdaEquality,
because_Cache,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
promote_hyp,
productEquality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x:residue(n). (x \mmember{} residues-mod(n))
Date html generated:
2018_05_21-PM-08_58_25
Last ObjectModification:
2017_07_26-PM-06_22_00
Theory : general
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