Nuprl Lemma : div_unique2
∀[a:ℕ]. ∀[n:ℕ+]. ∀[p:ℕ]. uiff((a ÷ n) = p ∈ ℤ;Div(a;n;p))
Proof
Definitions occuring in Statement :
div_nrel: Div(a;n;q),
nat_plus: ℕ+,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
divide: n ÷ m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
true: True,
div_nrel: Div(a;n;q),
lelt: i ≤ j < k,
le: A ≤ B,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B
Lemmas referenced :
div_elim,
div_nrel_wf,
squash_wf,
true_wf,
nat_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_formula_prop_wf,
le_wf,
less_than'_wf,
member-less_than,
equal_wf,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
int_subtype_base,
nat_plus_wf,
div_unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
hyp_replacement,
equalitySymmetry,
sqequalRule,
equalityTransitivity,
applyEquality,
lambdaEquality,
imageElimination,
isectElimination,
because_Cache,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
multiplyEquality,
axiomEquality,
addEquality,
Error :universeIsType,
divideEquality,
lambdaFormation,
Error :inhabitedIsType
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[p:\mBbbN{}]. uiff((a \mdiv{} n) = p;Div(a;n;p))
Date html generated:
2019_06_20-PM-01_14_25
Last ObjectModification:
2018_09_26-PM-02_34_40
Theory : int_2
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