Nuprl Lemma : fpf-as-apply-alist
∀[A,B:Type]. ∀[f:a:A fp-> B]. ∀[eq:EqDecider(A)].
(f = <fpf-domain(f), λx.outl(apply-alist(eq;map(λx.<x, f(x)>;fpf-domain(f));x))> ∈ a:A fp-> B)
Proof
Definitions occuring in Statement :
fpf-ap: f(x),
fpf-domain: fpf-domain(f),
fpf: a:A fp-> B[a],
apply-alist: apply-alist(eq;L;x),
map: map(f;as),
deq: EqDecider(T),
outl: outl(x),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
pair: <a, b>,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf: a:A fp-> B[a],
fpf-domain: fpf-domain(f),
fpf-ap: f(x),
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
outl: outl(x),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
squash: ↓T
Lemmas referenced :
pi2_wf,
pi1_wf_top,
equal_wf,
and_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
select_wf,
map_select,
length_wf,
lelt_wf,
map-length,
set_wf,
subtype_rel_dep_function,
alist-domain-first,
subtype_rel_product,
top_wf,
subtype_rel_list,
apply-alist-cases,
fpf_wf,
deq_wf,
list-subtype,
l_member_wf,
map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
lemma_by_obid,
isectElimination,
setEquality,
hypothesis,
productEquality,
lambdaEquality,
independent_pairEquality,
setElimination,
rename,
applyEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
functionExtensionality,
independent_isectElimination,
lambdaFormation,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:a:A fp-> B]. \mforall{}[eq:EqDecider(A)].
(f = <fpf-domain(f), \mlambda{}x.outl(apply-alist(eq;map(\mlambda{}x.<x, f(x)>fpf-domain(f));x))>)
Date html generated:
2018_05_21-PM-09_18_06
Last ObjectModification:
2018_02_09-AM-10_16_52
Theory : finite!partial!functions
Home
Index