Nuprl Lemma : div-one
∀[x:ℤ]. (x ÷ 1 ~ x)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
divide: n ÷ m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
false_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
multiply-is-int-iff,
add-is-int-iff,
decidable__equal_int,
rem-one,
nequal_wf,
true_wf,
equal_wf,
div_rem_sum,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
addLevel,
lambdaFormation,
cumulativity,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
sqequalRule,
unionElimination,
pointwiseFunctionality,
rename,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
sqequalAxiom
Latex:
\mforall{}[x:\mBbbZ{}]. (x \mdiv{} 1 \msim{} x)
Date html generated:
2016_05_15-PM-06_13_19
Last ObjectModification:
2016_01_16-PM-00_47_45
Theory : general
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