Nuprl Lemma : rem_gen_base_case
∀[a:ℤ]. ∀[n:ℤ-o]. (a rem n) = a ∈ ℤ supposing |a| < |n|
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
true: True,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtract: n - m,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
le: A ≤ B
Lemmas referenced :
less_than_wf,
absval_wf,
nat_wf,
nat_plus_wf,
equal_wf,
rem_base_case,
iff_weakening_equal,
squash_wf,
true_wf,
absval_pos,
nat_plus_subtype_nat,
subtype_rel_self,
decidable__le,
le_wf,
rem_sym_1a,
subtype_rel_sets,
nequal_wf,
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,
nat_plus_properties,
minus_minus_cancel,
intformnot_wf,
intformle_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_minus_lemma,
absval_sym,
int_nzero_wf,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
add-zero,
zero-add,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
false_wf,
decidable__lt,
satisfiable-full-omega-tt,
int_nzero_properties,
rem_sym_2
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
sqequalRule,
intEquality,
because_Cache,
natural_numberEquality,
imageElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
instantiate,
universeEquality,
cumulativity,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality,
setEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
minusEquality,
remainderEquality,
axiomEquality,
addEquality,
isect_memberFormation,
computeAll
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (a rem n) = a supposing |a| < |n|
Date html generated:
2019_06_20-PM-01_14_54
Last ObjectModification:
2018_09_17-PM-05_56_08
Theory : int_2
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