Nuprl Lemma : fpf-rename-cap2
∀[A,C,B:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc,eqc':EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B]. ∀[a:A]. ∀[z:B].
rename(r;f)(r a)?z = f(a)?z ∈ B supposing Inj(A;C;r)
Proof
Definitions occuring in Statement :
fpf-rename: rename(r;f),
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
fpf-cap: f(x)?z,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
guard: {T},
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
inject: Inj(A;B;f),
rev_implies: P ⇐ Q
Lemmas referenced :
fpf-dom_wf,
subtype-fpf2,
top_wf,
istype-void,
fpf-rename-ap2,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
inject_wf,
fpf_wf,
deq_wf,
istype-universe,
equal_wf,
fpf-rename_wf,
fpf-dom_functionality2,
fpf-rename-dom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
equalityTransitivity,
hypothesis,
equalitySymmetry,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
independent_isectElimination,
lambdaFormation_alt,
isect_memberEquality_alt,
voidElimination,
because_Cache,
baseClosed,
isect_memberFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_functionElimination,
equalityIstype,
dependent_functionElimination,
universeIsType,
axiomEquality,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality,
voidEquality,
isect_memberEquality,
lambdaFormation,
functionExtensionality,
lambdaEquality,
cumulativity,
hyp_replacement,
applyLambdaEquality,
productEquality,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}[A,C,B:Type]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[eqc,eqc':EqDecider(C)]. \mforall{}[r:A {}\mrightarrow{} C]. \mforall{}[f:a:A fp-> B]. \mforall{}[a:A].
\mforall{}[z:B].
rename(r;f)(r a)?z = f(a)?z supposing Inj(A;C;r)
Date html generated:
2019_10_16-AM-11_26_15
Last ObjectModification:
2019_06_25-PM-03_26_11
Theory : finite!partial!functions
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