Nuprl Lemma : can-apply-p-co-restrict
∀[A,B:Type]. ∀[f:A ⟶ (B + Top)]. ∀[P:A ⟶ ℙ]. ∀[p:∀x:A. Dec(P[x])]. ∀[x:A].
uiff(↑can-apply(p-co-restrict(f;p);x);(↑can-apply(f;x)) ∧ (¬P[x]))
Proof
Definitions occuring in Statement :
p-co-restrict: p-co-restrict(f;p),
can-apply: can-apply(f;x),
assert: ↑b,
decidable: Dec(P),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
p-co-restrict: p-co-restrict(f;p),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
top: Top,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
can-apply-compose-iff,
iff_weakening_uiff,
decidable_wf,
all_wf,
p-co-restrict_wf,
uiff_wf,
p-compose_wf,
not_wf,
and_wf,
subtype_rel_union,
subtype_rel_dep_function,
assert_wf,
assert_witness,
can-apply-p-co-filter,
do-apply-p-co-filter,
top_wf,
true_wf,
squash_wf,
p-co-filter_wf,
do-apply_wf,
can-apply_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
lambdaFormation,
independent_functionElimination,
voidElimination,
independent_pairEquality,
dependent_functionElimination,
productEquality,
cumulativity,
isect_memberEquality,
voidEquality,
universeEquality,
addLevel
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} (B + Top)]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[p:\mforall{}x:A. Dec(P[x])]. \mforall{}[x:A].
uiff(\muparrow{}can-apply(p-co-restrict(f;p);x);(\muparrow{}can-apply(f;x)) \mwedge{} (\mneg{}P[x]))
Date html generated:
2016_05_15-PM-03_31_29
Last ObjectModification:
2016_01_16-AM-10_49_04
Theory : general
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