Nuprl Lemma : div-cancel3
∀[x:ℕ]. ∀[y:ℕ+]. ∀[z:ℕy]. (((y * x) + z) ÷ y ~ x)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
nat_plus: ℕ+,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
div_nrel: Div(a;n;q),
sq_type: SQType(T)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
div_unique2,
add_nat_wf,
multiply_nat_wf,
nat_plus_subtype_nat,
int_seg_subtype_nat,
false_wf,
nat_wf,
nat_properties,
int_seg_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
because_Cache,
hypothesisEquality,
applyEquality,
sqequalRule,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
sqequalAxiom
Latex:
\mforall{}[x:\mBbbN{}]. \mforall{}[y:\mBbbN{}\msupplus{}]. \mforall{}[z:\mBbbN{}y]. (((y * x) + z) \mdiv{} y \msim{} x)
Date html generated:
2017_04_14-AM-09_15_41
Last ObjectModification:
2017_02_27-PM-03_53_10
Theory : int_2
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