Nuprl Lemma : ifthenelse_functionality_wrt_rev_implies
∀b1,b2:𝔹.
∀[p1,q1,p2,q2:ℙ]. (b1 = b2
⇒ {q1
⇐ q2}
⇒ {p1
⇐ p2}
⇒ {if b1 then p1 else q1 fi
⇐ if b2 then p2 else q2 fi })
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi
,
bool: 𝔹
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
guard: {T}
,
all: ∀x:A. B[x]
,
rev_implies: P
⇐ Q
,
implies: P
⇒ Q
,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T}
,
all: ∀x:A. B[x]
,
uall: ∀[x:A]. B[x]
,
implies: P
⇒ Q
,
rev_implies: P
⇐ Q
,
member: t ∈ T
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
uimplies: b supposing a
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
iff: P
⇐⇒ Q
,
true: True
,
prop: ℙ
,
sq_type: SQType(T)
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
bnot: ¬bb
,
false: False
Lemmas referenced :
eqtt_to_assert,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
btrue_wf,
assert_wf,
true_wf,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
assert_of_bnot,
ifthenelse_wf,
rev_implies_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
cut,
hypothesisEquality,
thin,
because_Cache,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
productElimination,
independent_isectElimination,
independent_functionElimination,
instantiate,
independent_pairFormation,
natural_numberEquality,
equalitySymmetry,
dependent_functionElimination,
equalityTransitivity,
dependent_pairFormation,
promote_hyp,
voidElimination,
cumulativity,
universeEquality
Latex:
\mforall{}b1,b2:\mBbbB{}.
\mforall{}[p1,q1,p2,q2:\mBbbP{}].
(b1 = b2
{}\mRightarrow{} \{q1 \mLeftarrow{}{} q2\}
{}\mRightarrow{} \{p1 \mLeftarrow{}{} p2\}
{}\mRightarrow{} \{if b1 then p1 else q1 fi \mLeftarrow{}{} if b2 then p2 else q2 fi \})
Date html generated:
2017_04_14-AM-07_29_59
Last ObjectModification:
2017_02_27-PM-02_58_36
Theory : bool_1
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