Nuprl Lemma : PZF_safe_functionality
∀[C:Type]. ∀[phi:Form(C)]. ∀[vs,ws:Atom List]. PZF_safe(phi;vs) = PZF_safe(phi;ws) supposing set-equal(Atom;vs;ws)
Proof
Definitions occuring in Statement :
PZF_safe: PZF_safe(phi;vs),
Form: Form(C),
set-equal: set-equal(T;x;y),
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
atom: Atom,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
PZF_safe: PZF_safe(phi;vs),
FormSafe2: FormSafe2(f),
FormVar: Vname,
Form_ind: Form_ind,
FormConst: Const(value),
FormSet: {var | phi},
FormEqual: left = right,
FormMember: element ∈ set,
FormAnd: left ∧ right),
FormOr: left ∨ right,
FormNot: ¬(body),
FormAll: ∀var. body,
FormExists: ∃var. body,
guard: {T},
subtype_rel: A ⊆r B,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_type: SQType(T),
let: let,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
cons: [a / b],
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
listp: A List+,
band: p ∧b q,
set-equal: set-equal(T;x;y),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
squash: ↓T,
ge: i ≥ j ,
sq_stable: SqStable(P),
has-value: (a)↓
Lemmas referenced :
Form-induction,
all_wf,
list_wf,
set-equal_wf,
equal_wf,
bool_wf,
PZF_safe_wf,
Form_wf,
bfalse_wf,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
null_wf3,
subtype_rel_list,
top_wf,
l_member_wf,
uall_wf,
not_wf,
nil-iff-no-member,
equal-wf-base,
list_subtype_base,
atom_subtype_base,
iff_wf,
assert_of_null,
assert_wf,
eqtt_to_assert,
btrue_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
list-diff_wf,
atom-deq_wf,
cons_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
length_wf,
bor_wf,
FormVar?_wf,
eq_atom_wf,
FormVar-name_wf,
assert_of_eq_atom,
bnot_wf,
deq-member_wf,
FormFvs_wf,
null_nil_lemma,
btrue_neq_bfalse,
null_cons_lemma,
member-list-diff,
member_singleton,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__le,
not-ge-2,
sq_stable__le,
add-swap,
le-add-cancel2,
hd_member,
assert_elim,
member-implies-null-eq-bfalse,
decidable__atom_equal,
value-type-has-value,
list-value-type,
band_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
assert-deq-member,
assert_functionality_wrt_uiff,
or_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
atomEquality,
hypothesis,
functionEquality,
cumulativity,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
applyEquality,
voidElimination,
voidEquality,
independent_pairFormation,
addLevel,
productElimination,
impliesFunctionality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
hypothesis_subsumption,
setElimination,
rename,
natural_numberEquality,
addEquality,
minusEquality,
dependent_set_memberEquality,
intEquality,
baseApply,
closedConclusion,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality,
productEquality,
imageElimination,
universeEquality,
imageMemberEquality,
callbyvalueReduce,
inlFormation,
inrFormation
Latex:
\mforall{}[C:Type]. \mforall{}[phi:Form(C)]. \mforall{}[vs,ws:Atom List].
PZF\_safe(phi;vs) = PZF\_safe(phi;ws) supposing set-equal(Atom;vs;ws)
Date html generated:
2018_05_21-PM-11_31_13
Last ObjectModification:
2017_10_12-PM-04_18_41
Theory : PZF
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