Nuprl Lemma : rem_eq_args
∀[a:ℕ+]. ((a rem a) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
ge: i ≥ j ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
rem_rec_case,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtype_rel_self,
iff_weakening_equal,
minus-one-mul,
add-mul-special,
zero-mul,
rem-zero,
nat_plus_inc_int_nzero,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
applyEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
intEquality,
sqequalRule,
independent_isectElimination,
setElimination,
rename,
dependent_functionElimination,
because_Cache,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[a:\mBbbN{}\msupplus{}]. ((a rem a) = 0)
Date html generated:
2019_06_20-PM-01_15_07
Last ObjectModification:
2019_01_01-PM-01_15_22
Theory : int_2
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