Nuprl Lemma : fpf-rename-ap
∀[A,C:Type]. ∀[B:A ⟶ Type]. ∀[eqa:EqDecider(A)]. ∀[eqc:EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B[a]]. ∀[a:A].
(rename(r;f)(r a) = f(a) ∈ B[a]) supposing ((↑a ∈ dom(f)) and Inj(A;C;r))
Proof
Definitions occuring in Statement :
fpf-rename: rename(r;f),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
inject: Inj(A;B;f),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fpf-ap: f(x),
fpf-rename: rename(r;f),
fpf: a:A fp-> B[a],
pi2: snd(t),
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
deq: EqDecider(T),
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
eqof: eqof(d),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
fpf-dom: x ∈ dom(f),
iff: P ⇐⇒ Q,
guard: {T},
inject: Inj(A;B;f),
squash: ↓T,
true: True,
rev_implies: P ⇐ Q
Lemmas referenced :
hd-filter,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
inject_wf,
fpf_wf,
deq_wf,
safe-assert-deq,
l_member_wf,
assert-deq-member,
equal_wf,
squash_wf,
true_wf,
subtype_rel-equal,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
applyEquality,
setElimination,
rename,
hypothesis,
functionExtensionality,
cumulativity,
independent_functionElimination,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
independent_pairFormation,
productEquality,
dependent_set_memberEquality,
imageElimination,
instantiate,
setEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,C:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[eqc:EqDecider(C)]. \mforall{}[r:A {}\mrightarrow{} C].
\mforall{}[f:a:A fp-> B[a]]. \mforall{}[a:A].
(rename(r;f)(r a) = f(a)) supposing ((\muparrow{}a \mmember{} dom(f)) and Inj(A;C;r))
Date html generated:
2018_05_21-PM-09_26_56
Last ObjectModification:
2018_02_09-AM-10_22_14
Theory : finite!partial!functions
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