Nuprl Lemma : FormSafe1-Fvs-subset
∀C:Type. ∀phi:Form(C). ∀vs:Atom List. ((FormSafe1(phi) vs) ⇒ l_subset(Atom;vs;FormFvs(phi)))
Proof
Definitions occuring in Statement :
FormSafe1: FormSafe1(f),
FormFvs: FormFvs(f),
Form: Form(C),
l_subset: l_subset(T;as;bs),
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
atom: Atom,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
FormFvs: FormFvs(f),
FormSafe1: FormSafe1(f),
FormVar: Vname,
Form_ind: Form_ind,
false: False,
FormConst: Const(value),
FormSet: {var | phi},
FormEqual: left = right,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
and: P ∧ Q,
or: P ∨ Q,
FormMember: element ∈ set,
FormAnd: left ∧ right),
FormOr: left ∨ right,
FormNot: ¬(body),
FormAll: ∀var. body,
FormExists: ∃var. body,
guard: {T},
uiff: uiff(P;Q),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
l_subset: l_subset(T;as;bs),
sq_type: SQType(T),
set-equal: set-equal(T;x;y),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
FormVar-name: FormVar-name(v),
pi2: snd(t),
FormVar?: FormVar?(v),
pi1: fst(t),
assert: ↑b,
bfalse: ff,
bnot: ¬bb,
not: ¬A
Lemmas referenced :
Form-induction,
all_wf,
list_wf,
FormSafe1_wf,
l_subset_wf,
FormFvs_wf,
Form_wf,
false_wf,
or_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
exists_wf,
set-equal_wf,
cons_wf,
nil_wf,
FormVar?_wf,
equal-wf-T-base,
FormVar-name_wf,
atom_subtype_base,
not_wf,
l_member_wf,
append_wf,
l_disjoint_wf,
assert_of_null,
squash_wf,
true_wf,
l-union_wf,
atom-deq_wf,
iff_weakening_equal,
l_subset_nil_left,
subtype_base_sq,
member_singleton,
member-union,
Form-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
cons_member,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
member_append,
equal-wf-base,
member_filter,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
atomEquality,
functionEquality,
applyEquality,
cumulativity,
independent_functionElimination,
voidElimination,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidEquality,
productEquality,
productElimination,
universeEquality,
unionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
dependent_functionElimination,
inlFormation,
inrFormation,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
equalityElimination,
dependent_pairFormation,
independent_pairFormation,
impliesFunctionality,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}C:Type. \mforall{}phi:Form(C). \mforall{}vs:Atom List. ((FormSafe1(phi) vs) {}\mRightarrow{} l\_subset(Atom;vs;FormFvs(phi)))
Date html generated:
2018_05_21-PM-11_28_44
Last ObjectModification:
2017_10_12-AM-00_45_16
Theory : PZF
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