Nuprl Lemma : fpf-restrict_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:A ⟶ 𝔹]. (fpf-restrict(f;P) ∈ x:{x:A| ↑(P x)} fp-> B[x])
Proof
Definitions occuring in Statement :
fpf-restrict: fpf-restrict(f;P),
fpf: a:A fp-> B[a],
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf: a:A fp-> B[a],
fpf-restrict: fpf-restrict(f;P),
pi2: snd(t),
fpf-domain: fpf-domain(f),
mk_fpf: mk_fpf(L;f),
pi1: fst(t),
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
filter_type,
assert_wf,
l_member_wf,
bool_wf,
fpf_wf,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
equal_wf,
less_than_wf,
length_wf,
filter_wf5,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
functionEquality,
setEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
lambdaEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
dependent_pairFormation,
promote_hyp,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
productEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. (fpf-restrict(f;P) \mmember{} x:\{x:A| \muparrow{}(P x)\} f\000Cp-> B[x])
Date html generated:
2018_05_21-PM-09_31_01
Last ObjectModification:
2018_02_09-AM-10_25_27
Theory : finite!partial!functions
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