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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