Nuprl Lemma : rem_rec_case
∀[a:ℕ]. ∀[n:ℕ+]. (a rem n) = (a - n rem n) ∈ ℤ supposing a ≥ n
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
remainder: n rem m,
subtract: n - m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
ge: i ≥ j ,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
ge_wf,
nat_plus_wf,
nat_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
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,
subtract_wf,
equal_wf,
squash_wf,
true_wf,
rem_to_div,
iff_weakening_equal,
decidable__equal_int,
intformnot_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
div_rec_case
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intEquality,
applyEquality,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
imageElimination,
universeEquality,
imageMemberEquality,
productElimination,
multiplyEquality,
divideEquality,
unionElimination
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (a rem n) = (a - n rem n) supposing a \mgeq{} n
Date html generated:
2017_04_14-AM-09_16_08
Last ObjectModification:
2017_02_27-PM-03_53_30
Theory : int_2
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