Nuprl Lemma : rem_rem_to_rem
∀[a:ℕ]. ∀[n:ℕ+].  ((a rem n rem n) = (a rem n) ∈ ℤ)
Proof
Definitions occuring in Statement : 
nat_plus: ℕ+
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
remainder: n rem m
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
and: P ∧ Q
, 
nat: ℕ
, 
nat_plus: ℕ+
, 
nequal: a ≠ b ∈ T 
, 
ge: i ≥ j 
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
all: ∀x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
true: True
, 
squash: ↓T
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
nat_plus_wf, 
nat_wf, 
remainder_wf, 
rem_bounds_1, 
nat_plus_properties, 
nat_properties, 
full-omega-unsat, 
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, 
equal-wf-base, 
int_subtype_base, 
equal_wf, 
squash_wf, 
true_wf, 
rem_base_case, 
subtype_rel_self, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
hypothesis, 
extract_by_obid, 
sqequalRule, 
sqequalHypSubstitution, 
isect_memberEquality, 
isectElimination, 
thin, 
hypothesisEquality, 
axiomEquality, 
because_Cache, 
intEquality, 
productElimination, 
remainderEquality, 
setElimination, 
rename, 
lambdaFormation, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
dependent_functionElimination, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
applyEquality, 
baseClosed, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
imageMemberEquality, 
instantiate
Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    ((a  rem  n  rem  n)  =  (a  rem  n))
Date html generated:
2019_06_20-PM-01_15_05
Last ObjectModification:
2018_09_17-PM-05_44_27
Theory : int_2
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