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
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