Nuprl Lemma : rem_fun_sat_rem_nrel
∀a:ℕ. ∀n:ℕ+.  Rem(a;n;a rem n)
Proof
Definitions occuring in Statement : 
rem_nrel: Rem(a;n;r)
, 
nat_plus: ℕ+
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
remainder: n rem m
Definitions unfolded in proof : 
rem_nrel: Rem(a;n;r)
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
exists: ∃x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
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)
, 
false: False
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
int_nzero: ℤ-o
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
nat_plus_wf, 
nat_wf, 
divide_wf, 
div_nrel_wf, 
equal_wf, 
nat_properties, 
nat_plus_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, 
div_fun_sat_div_nrel, 
div_rem_sum, 
subtype_rel_sets, 
less_than_wf, 
nequal_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
hypothesis, 
dependent_pairFormation, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
productEquality, 
intEquality, 
addEquality, 
multiplyEquality, 
setElimination, 
rename, 
because_Cache, 
remainderEquality, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
lambdaEquality, 
int_eqEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
applyEquality, 
baseClosed, 
setEquality, 
equalitySymmetry
Latex:
\mforall{}a:\mBbbN{}.  \mforall{}n:\mBbbN{}\msupplus{}.    Rem(a;n;a  rem  n)
Date html generated:
2019_06_20-PM-01_14_42
Last ObjectModification:
2018_09_17-PM-05_47_56
Theory : int_2
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