Nuprl Lemma : rem_base_case_z
∀[a:ℤ]. ∀[b:ℤ-o]. (a rem b) = a ∈ ℤ supposing |a| < |b|
Proof
Definitions occuring in Statement :
absval: |i|,
int_nzero: ℤ-o,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
guard: {T},
le: A ≤ B,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
nat: ℕ,
prop: ℙ,
false: False,
not: ¬A,
squash: ↓T,
true: True,
top: Top,
less_than': less_than'(a;b),
less_than: a < b,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_nzero: ℤ-o,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtract: n - m,
int_lower: {...i}
Lemmas referenced :
nat_wf,
absval_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
iff_weakening_equal,
le-add-cancel,
zero-add,
add-commutes,
add-swap,
int_nzero_wf,
add-associates,
add_functionality_wrt_le,
less-iff-le,
not-lt-2,
false_wf,
decidable__lt,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_nzero_properties,
rem_base_case,
equal_wf,
less_than_wf,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
absval_unfold,
rem_sym,
int_subtype_base,
full-omega-unsat,
istype-int,
istype-le,
itermMinus_wf,
int_term_value_minus_lemma,
istype-less_than,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
add-zero,
minus-zero,
minus-add,
condition-implies-le,
not-equal-2,
rem_2_to_1,
rem_3_to_1
Rules used in proof :
axiomEquality,
cumulativity,
instantiate,
promote_hyp,
addEquality,
computeAll,
int_eqEquality,
dependent_pairFormation,
dependent_functionElimination,
dependent_set_memberEquality,
intEquality,
lambdaEquality,
applyEquality,
independent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
voidEquality,
voidElimination,
independent_pairFormation,
isect_memberEquality,
axiomSqEquality,
lessCases,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation,
natural_numberEquality,
minusEquality,
because_Cache,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
Error :memTop,
universeIsType,
universeEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. (a rem b) = a supposing |a| < |b|
Date html generated:
2020_05_19-PM-09_41_29
Last ObjectModification:
2019_12_28-PM-03_31_57
Theory : int_2
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