Nuprl Lemma : isl-ite
∀[x:𝔹]. ∀[a,b:Top + Top]. (isl(if x then a else b fi ) ~ (isl(a) ∧b x) ∨b(isl(b) ∧b (¬bx)))
Proof
Definitions occuring in Statement :
bor: p ∨bq,
band: p ∧b q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
isl: isl(x),
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
union: left + right,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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 ,
bnot: ¬bb,
squash: ↓T,
prop: ℙ,
band: p ∧b q,
bfalse: ff,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
exists: ∃x:A. B[x],
or: P ∨ Q,
assert: ↑b,
false: False,
isl: isl(x),
bor: p ∨bq
Lemmas referenced :
bool_wf,
eqtt_to_assert,
subtype_base_sq,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
bor_wf,
isl_wf,
top_wf,
band_ff_simp,
iff_weakening_equal,
band_tt_simp,
bfalse_wf,
bor_ff_simp,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
sqequalRule,
instantiate,
cumulativity,
equalitySymmetry,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
sqequalAxiom,
unionEquality,
isect_memberEquality,
dependent_pairFormation,
promote_hyp,
voidElimination
Latex:
\mforall{}[x:\mBbbB{}]. \mforall{}[a,b:Top + Top]. (isl(if x then a else b fi ) \msim{} (isl(a) \mwedge{}\msubb{} x) \mvee{}\msubb{}(isl(b) \mwedge{}\msubb{} (\mneg{}\msubb{}x)))
Date html generated:
2017_10_01-AM-09_12_39
Last ObjectModification:
2017_07_26-PM-04_48_17
Theory : general
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