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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