Nuprl Lemma : fpf-normalize-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[g:x:A fp-> B[x]]. ∀[x:A]. (x ∈ dom(fpf-normalize(eq;g)) ~ x ∈ dom(g))
Proof
Definitions occuring in Statement :
fpf-normalize: fpf-normalize(eq;g),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf: a:A fp-> B[a],
fpf-dom: x ∈ dom(f),
fpf-normalize: fpf-normalize(eq;g),
pi2: snd(t),
pi1: fst(t),
fpf-empty: ⊗,
fpf-single: x : v,
fpf-join: f ⊕ g,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
member: t ∈ T,
top: Top,
so_apply: x[s1;s2;s3],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
uiff: uiff(P;Q),
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
top_wf,
equal_wf,
fpf_wf,
deq_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
reduce_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
bool_wf,
uiff_transitivity,
assert_wf,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
bool_subtype_base,
iff_imp_equal_bool,
deq-member_wf,
filter_wf5,
bor_wf,
bfalse_wf,
l_member_wf,
member_filter_2,
eqof_wf,
member_filter,
or_wf,
false_wf,
assert_of_bor,
or_false_r,
assert-deq-member,
iff_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaFormation,
isectElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
independent_functionElimination,
because_Cache,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
setEquality,
orFunctionality,
andLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[g:x:A fp-> B[x]]. \mforall{}[x:A].
(x \mmember{} dom(fpf-normalize(eq;g)) \msim{} x \mmember{} dom(g))
Date html generated:
2018_05_21-PM-09_32_18
Last ObjectModification:
2018_02_09-AM-10_26_58
Theory : finite!partial!functions
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