Nuprl Lemma : ndiff_inv
∀[a:ℕ]. ∀[b:ℤ]. (((a + b) -- b) = a ∈ ℤ)
Proof
Definitions occuring in Statement :
ndiff: a -- b,
nat: ℕ,
uall: ∀[x:A]. B[x],
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
ndiff: a -- b,
imax: imax(a;b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
value-type-has-value,
int-value-type,
subtract_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
add-subtract-cancel,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
addEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[b:\mBbbZ{}]. (((a + b) -- b) = a)
Date html generated:
2017_04_14-AM-09_14_56
Last ObjectModification:
2017_02_27-PM-03_52_48
Theory : int_2
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