Nuprl Lemma : type-separation
∀x,y:Base.
(((x)↓ ∨ is-exception(x))
⇒ ((y)↓ ∨ is-exception(y))
⇒ (∀n,m:ℤ. ∀T:Type. ((x = n ∈ T) ⇒ (y = m ∈ T) ⇒ (x = y ∈ T)))
⇒ (x = y ∈ Base))
Proof
Definitions occuring in Statement :
has-value: (a)↓,
is-exception: is-exception(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
int: ℤ,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
label: ...$L... t,
has-value: (a)↓
Lemmas referenced :
all_wf,
equal-wf-base,
int_subtype_base,
or_wf,
has-value_wf_base,
is-exception_wf,
base_wf,
has-value-implies-dec-isint,
imax_wf,
less_than_wf,
ifthenelse_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_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,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imax_unfold,
subtype_rel_self,
iff_weakening_equal,
intformeq_wf,
int_formula_prop_eq_lemma,
not_wf,
exists_wf,
and_wf,
exception-not-value,
value-type-has-value,
int-value-type,
EquatePairs_wf,
EquatePairs-equality,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
sqequalRule,
lambdaEquality,
universeEquality,
functionEquality,
hypothesisEquality,
applyEquality,
because_Cache,
unionElimination,
dependent_functionElimination,
baseClosed,
independent_functionElimination,
dependent_pairFormation,
addEquality,
natural_numberEquality,
independent_pairFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
productElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp,
imageElimination,
imageMemberEquality,
baseApply,
closedConclusion,
sqequalIntensionalEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
isintReduceTrue,
inrFormation,
inlFormation
Latex:
\mforall{}x,y:Base.
(((x)\mdownarrow{} \mvee{} is-exception(x))
{}\mRightarrow{} ((y)\mdownarrow{} \mvee{} is-exception(y))
{}\mRightarrow{} (\mforall{}n,m:\mBbbZ{}. \mforall{}T:Type. ((x = n) {}\mRightarrow{} (y = m) {}\mRightarrow{} (x = y)))
{}\mRightarrow{} (x = y))
Date html generated:
2018_05_21-PM-01_13_20
Last ObjectModification:
2018_05_01-PM-04_37_20
Theory : num_thy_1
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