Nuprl Lemma : fpf-normalize-ap
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[g:x:A fp-> B[x]]. ∀[x:A].
fpf-normalize(eq;g)(x) = g(x) ∈ B[x] supposing ↑x ∈ dom(g)
Proof
Definitions occuring in Statement :
fpf-normalize: fpf-normalize(eq;g),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf: a:A fp-> B[a],
fpf-ap: f(x),
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_lambda3,
so_apply: x[s1;s2;s3],
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
top: Top,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
it: ⋅,
nil: [],
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
or: P ∨ Q,
guard: {T},
and: P ∧ Q,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
false: False,
nat: ℕ,
le: A ≤ B,
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
deq-member: x ∈b L,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
uiff: uiff(P;Q),
bor: p ∨bq,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
fpf_ap_pair_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
istype-assert,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf_wf,
deq_wf,
istype-universe,
equal_wf,
reduce_cons_lemma,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
product_subtype_list,
reduce_nil_lemma,
list-cases,
list_wf,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
list-subtype,
full-omega-unsat,
istype-int,
istype-less_than,
set_wf,
l_member_wf,
colength-cons-not-zero,
istype-void,
istype-le,
subtract-1-ge-0,
istype-nat,
false_wf,
uiff_transitivity,
bool_wf,
assert_wf,
eqtt_to_assert,
safe-assert-deq,
squash_wf,
true_wf,
member_wf,
subtype_rel-equal,
trivial-equal,
iff_weakening_equal,
subtype_rel_self,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
deq-member_wf,
cons_wf,
subtype_rel_list,
assert-deq-member,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
independent_isectElimination,
lambdaFormation_alt,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
imageElimination,
baseClosed,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
hypothesis_subsumption,
promote_hyp,
unionElimination,
because_Cache,
cumulativity,
axiomSqEquality,
computeAll,
independent_pairFormation,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
approximateComputation,
dependent_pairFormation_alt,
functionIsTypeImplies,
setEquality,
equalityIstype,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
sqequalBase,
equalityElimination,
hyp_replacement,
imageMemberEquality,
productIsType
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[g:x:A fp-> B[x]]. \mforall{}[x:A].
fpf-normalize(eq;g)(x) = g(x) supposing \muparrow{}x \mmember{} dom(g)
Date html generated:
2020_05_20-AM-09_03_30
Last ObjectModification:
2020_01_04-PM-11_11_10
Theory : finite!partial!functions
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