Nuprl Lemma : free-from-atom-fpf-ap
∀[a:Atom1]. ∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ 𝕌']. ∀[f:x:A fp-> B[x]].
∀[x:A]. (a#f(x):B[x]) supposing ((↑x ∈ dom(f)) and a#x:A) supposing a#f:x:A fp-> B[x]
Proof
Definitions occuring in Statement :
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
free-from-atom: a#x:T,
atom: Atom$n,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
fpf-ap: f(x),
fpf-domain: fpf-domain(f),
fpf: a:A fp-> B[a]
Lemmas referenced :
sq_stable__free-from-atom,
fpf-ap_wf,
member-fpf-domain,
subtype-fpf2,
top_wf,
assert_wf,
fpf-dom_wf,
free-from-atom_wf,
fpf_wf,
deq_wf,
l_member_wf,
fpf-domain_wf,
set_wf,
equal_wf,
pi2_wf,
list_wf,
pi1_wf_top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
hypothesisEquality,
cumulativity,
sqequalRule,
lambdaEquality,
independent_isectElimination,
hypothesis,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
universeEquality,
atomnEquality,
dependent_set_memberEquality,
freeFromAtomApplication,
freeFromAtomSet,
hyp_replacement,
equalitySymmetry,
setElimination,
rename,
setEquality,
natural_numberEquality,
equalityTransitivity,
freeFromAtomTriviality,
independent_pairEquality,
productEquality
Latex:
\mforall{}[a:Atom1]. \mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} \mBbbU{}']. \mforall{}[f:x:A fp-> B[x]].
\mforall{}[x:A]. (a\#f(x):B[x]) supposing ((\muparrow{}x \mmember{} dom(f)) and a\#x:A) supposing a\#f:x:A fp-> B[x]
Date html generated:
2019_10_16-AM-11_26_47
Last ObjectModification:
2018_09_18-PM-10_17_58
Theory : finite!partial!functions
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