Nuprl Lemma : one_or_both-induction
∀[A,B:Type]. ∀[P:one_or_both(A;B) ⟶ ℙ].
((∀bval:A × B. P[oobboth(bval)])
⇒ (∀lval:A. P[oobleft(lval)])
⇒ (∀rval:B. P[oobright(rval)])
⇒ {∀x:one_or_both(A;B). P[x]})
Proof
Definitions occuring in Statement :
oobright: oobright(rval),
oobleft: oobleft(lval),
oobboth: oobboth(bval),
one_or_both: one_or_both(A;B),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
one_or_both: one_or_both(A;B),
oobboth: oobboth(bval),
oobleft: oobleft(lval),
oobright: oobright(rval),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
one_or_both_wf,
all_wf,
oobright_wf,
oobleft_wf,
oobboth_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
productEquality,
functionEquality,
cumulativity,
universeEquality,
independent_pairEquality,
dependent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:one\_or\_both(A;B) {}\mrightarrow{} \mBbbP{}].
((\mforall{}bval:A \mtimes{} B. P[oobboth(bval)])
{}\mRightarrow{} (\mforall{}lval:A. P[oobleft(lval)])
{}\mRightarrow{} (\mforall{}rval:B. P[oobright(rval)])
{}\mRightarrow{} \{\mforall{}x:one\_or\_both(A;B). P[x]\})
Date html generated:
2016_05_15-PM-05_32_43
Last ObjectModification:
2015_12_27-PM-02_10_37
Theory : general
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