Nuprl Lemma : equipollent-union-sum
∀[A,B:Type]. ∀[C:A ⟶ Type]. ∀[D:B ⟶ Type].
a:A × C[a] + (b:B × D[b]) ~ d:A + B × case d of inl(a) => C[a] | inr(b) => D[b]
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
so_lambda: λ2x.t[x],
pi1: fst(t),
pi2: snd(t),
outl: outl(x),
uimplies: b supposing a,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
sq_type: SQType(T),
guard: {T},
false: False,
outr: outr(x),
bnot: ¬bb,
bfalse: ff
Lemmas referenced :
equal_wf,
biject_wf,
pi2_wf,
pi1_wf,
and_wf,
outl_wf,
assert_wf,
isl_wf,
subtype_base_sq,
int_subtype_base,
outr_wf,
bnot_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_pairFormation,
lambdaEquality,
cut,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
thin,
unionEquality,
productEquality,
applyEquality,
lambdaFormation,
unionElimination,
sqequalRule,
spreadEquality,
dependent_pairEquality,
inlEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
independent_functionElimination,
inrEquality,
independent_pairFormation,
functionEquality,
cumulativity,
universeEquality,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
setElimination,
rename,
independent_isectElimination,
promote_hyp,
hyp_replacement,
natural_numberEquality,
instantiate,
intEquality,
voidElimination,
because_Cache
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} Type]. \mforall{}[D:B {}\mrightarrow{} Type].
a:A \mtimes{} C[a] + (b:B \mtimes{} D[b]) \msim{} d:A + B \mtimes{} case d of inl(a) => C[a] | inr(b) => D[b]
Date html generated:
2019_06_20-PM-02_17_51
Last ObjectModification:
2018_08_21-PM-01_56_01
Theory : equipollence!!cardinality!
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