Nuprl Lemma : rem_invariant
∀[a,b:ℕ]. ∀[n:ℕ+]. ((a + (b * 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,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
uiff: uiff(P;Q),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
zero-mul,
add-zero,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_plus_wf,
nat_wf,
lt_to_le_rw,
mul_preserves_le,
nat_plus_subtype_nat,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
satisfiable-full-omega-tt,
add_functionality_wrt_le,
add-associates,
minus-one-mul,
add-commutes,
mul-distributes-right,
iff_weakening_equal,
le_wf,
rem_rec_case,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
remainderEquality,
because_Cache,
applyEquality,
baseClosed,
unionElimination,
productElimination,
addEquality,
computeAll,
multiplyEquality,
minusEquality,
imageMemberEquality,
dependent_set_memberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination
Latex:
\mforall{}[a,b:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a + (b * n) rem n) = (a rem n))
Date html generated:
2019_06_20-PM-01_14_58
Last ObjectModification:
2018_09_17-PM-05_47_10
Theory : int_2
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