Nuprl Lemma : mod-eqmod
∀x:ℤ. ∀m:ℕ+. ((x mod m) ≡ x mod m)
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
modulus: a mod n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_plus_wf,
modulus-idempotent,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
nequal_wf,
less_than_wf,
subtype_rel_sets,
modulus_wf,
modulus-equal-iff-eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
setEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}m:\mBbbN{}\msupplus{}. ((x mod m) \mequiv{} x mod m)
Date html generated:
2016_05_14-PM-04_23_01
Last ObjectModification:
2016_01_14-PM-11_39_09
Theory : num_thy_1
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