Nuprl Lemma : fpf-inv-rename_wf
∀[A,C:Type]. ∀[B:A ⟶ Type]. ∀[D:C ⟶ Type]. ∀[rinv:C ⟶ (A?)]. ∀[r:A ⟶ C]. ∀[f:c:C fp-> D[c]].
  (fpf-inv-rename(r;rinv;f) ∈ a:A fp-> B[a]) supposing ((∀a:A. (D[r a] = B[a] ∈ Type)) and inv-rel(A;C;r;rinv))
Proof
Definitions occuring in Statement : 
fpf-inv-rename: fpf-inv-rename(r;rinv;f)
, 
fpf: a:A fp-> B[a]
, 
inv-rel: inv-rel(A;B;f;finv)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
unit: Unit
, 
member: t ∈ T
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
union: left + right
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
fpf-inv-rename: fpf-inv-rename(r;rinv;f)
, 
fpf: a:A fp-> B[a]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
prop: ℙ
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
all: ∀x:A. B[x]
, 
isl: isl(x)
, 
compose: f o g
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
inv-rel: inv-rel(A;B;f;finv)
, 
outl: outl(x)
, 
not: ¬A
, 
false: False
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bfalse: ff
Lemmas referenced : 
l_member_wf, 
all_wf, 
equal_wf, 
inv-rel_wf, 
fpf_wf, 
unit_wf2, 
mapfilter_wf, 
isl_wf, 
assert_wf, 
outl_wf, 
member_map_filter, 
assert_elim, 
and_wf, 
bfalse_wf, 
btrue_neq_bfalse, 
true_wf, 
false_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
sqequalRule, 
dependent_pairEquality, 
functionEquality, 
setEquality, 
hypothesisEquality, 
extract_by_obid, 
isectElimination, 
hypothesis, 
applyEquality, 
setElimination, 
rename, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
instantiate, 
cumulativity, 
lambdaEquality, 
universeEquality, 
isect_memberEquality, 
because_Cache, 
unionEquality, 
functionExtensionality, 
lambdaFormation, 
independent_isectElimination, 
dependent_functionElimination, 
independent_functionElimination, 
dependent_set_memberEquality, 
hyp_replacement, 
applyLambdaEquality, 
unionElimination, 
independent_pairFormation, 
voidElimination, 
inlEquality
Latex:
\mforall{}[A,C:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[D:C  {}\mrightarrow{}  Type].  \mforall{}[rinv:C  {}\mrightarrow{}  (A?)].  \mforall{}[r:A  {}\mrightarrow{}  C].  \mforall{}[f:c:C  fp->  D[c]].
    (fpf-inv-rename(r;rinv;f)  \mmember{}  a:A  fp->  B[a])  supposing 
          ((\mforall{}a:A.  (D[r  a]  =  B[a]))  and 
          inv-rel(A;C;r;rinv))
Date html generated:
2018_05_21-PM-09_27_28
Last ObjectModification:
2018_05_19-PM-04_38_21
Theory : finite!partial!functions
Home
Index