Nuprl Lemma : rem_addition
∀[i,j:ℕ]. ∀[n:ℕ+]. (((i rem n) + (j rem n) rem n) = (i + j rem n) ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
remainder: n rem m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
nat: ℕ,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q)
Lemmas referenced :
nat_plus_wf,
nat_wf,
nequal_wf,
less_than_wf,
subtype_rel_sets,
int_subtype_base,
equal-wf-base,
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,
full-omega-unsat,
nat_properties,
nat_plus_properties,
equal_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
div_rem_sum,
subtype_rel_self,
iff_weakening_equal,
add-commutes,
add-swap,
mul-commutes,
mul-distributes-right,
add-associates,
add_nat_wf,
remainder_wf,
decidable__le,
add-is-int-iff,
intformnot_wf,
intformle_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
false_wf,
le_wf,
divide_wf,
rem_invariant
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
Error :inhabitedIsType,
setEquality,
divideEquality,
multiplyEquality,
baseClosed,
applyEquality,
independent_pairFormation,
voidEquality,
voidElimination,
dependent_functionElimination,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
rename,
setElimination,
addEquality,
remainderEquality,
intEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
instantiate,
productElimination,
dependent_set_memberEquality,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[i,j:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (((i rem n) + (j rem n) rem n) = (i + j rem n))
Date html generated:
2019_06_20-PM-01_15_02
Last ObjectModification:
2018_09_26-PM-02_36_46
Theory : int_2
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