Nuprl Lemma : fpf-rename_wf
∀[A,C:Type]. ∀[B:A ⟶ Type]. ∀[D:C ⟶ Type]. ∀[eq:EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B[a]].
rename(r;f) ∈ c:C fp-> D[c] supposing ∀a:A. (D[r a] = B[a] ∈ Type)
Proof
Definitions occuring in Statement :
fpf-rename: rename(r;f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
fpf: a:A fp-> B[a],
fpf-rename: rename(r;f),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
deq: EqDecider(T),
exists: ∃x:A. B[x],
cand: A c∧ B,
eqof: eqof(d),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
set_wf,
l_member_wf,
map_wf,
all_wf,
equal_wf,
fpf_wf,
deq_wf,
subtype_rel-equal,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
hd-filter,
member_map,
safe-assert-deq,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
lambdaFormation,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
hypothesis,
dependent_pairEquality,
dependent_functionElimination,
setEquality,
functionEquality,
applyEquality,
setElimination,
rename,
instantiate,
cumulativity,
universeEquality,
dependent_set_memberEquality,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
because_Cache,
dependent_pairFormation,
independent_pairFormation,
productEquality
Latex:
\mforall{}[A,C:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[D:C {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(C)]. \mforall{}[r:A {}\mrightarrow{} C]. \mforall{}[f:a:A fp-> B[a]].
rename(r;f) \mmember{} c:C fp-> D[c] supposing \mforall{}a:A. (D[r a] = B[a])
Date html generated:
2018_05_21-PM-09_26_42
Last ObjectModification:
2018_05_19-PM-04_37_56
Theory : finite!partial!functions
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