Nuprl Lemma : FormSafe1_functionality
∀C:Type. ∀phi:Form(C). ∀vs,ws:Atom List. (l_subset(Atom;ws;vs) ⇒ {(FormSafe1(phi) vs) ⇒ (FormSafe1(phi) ws)})
Proof
Definitions occuring in Statement :
FormSafe1: FormSafe1(f),
Form: Form(C),
l_subset: l_subset(T;as;bs),
list: T List,
guard: {T},
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,
guard: {T},
so_apply: x[s],
FormSafe1: FormSafe1(f),
FormVar: Vname,
Form_ind: Form_ind,
false: False,
FormConst: Const(value),
FormSet: {var | phi},
FormEqual: left = right,
FormMember: element ∈ set,
FormAnd: left ∧ right),
FormOr: left ∨ right,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
FormNot: ¬(body),
FormAll: ∀var. body,
FormExists: ∃var. body,
or: P ∨ Q,
uimplies: b supposing a,
top: Top,
uiff: uiff(P;Q),
sq_type: SQType(T),
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
exists: ∃x:A. B[x],
cons: [a / b],
bfalse: ff,
set-equal: set-equal(T;x;y),
l_subset: l_subset(T;as;bs),
rev_implies: P ⇐ Q,
l_disjoint: l_disjoint(T;l1;l2),
not: ¬A,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Form-induction,
all_wf,
list_wf,
l_subset_wf,
FormSafe1_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,
FormFvs_wf,
assert_of_null,
subtype_base_sq,
list_subtype_base,
l_subset_nil_right,
and_wf,
equal_wf,
null_nil_lemma,
btrue_wf,
bool_wf,
bool_subtype_base,
list-cases,
product_subtype_list,
null_cons_lemma,
l_subset_cons,
member_singleton,
cons_member,
append_wf,
l_disjoint_wf,
l_intersection_wf,
atom-deq_wf,
member_append,
member-intersection,
iff_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
atomEquality,
hypothesis,
functionEquality,
applyEquality,
cumulativity,
independent_functionElimination,
voidElimination,
because_Cache,
productElimination,
independent_pairFormation,
productEquality,
universeEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
inlFormation,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
inrFormation,
dependent_pairFormation,
equalityUniverse,
levelHypothesis,
addLevel,
impliesFunctionality,
orFunctionality,
orLevelFunctionality
Latex:
\mforall{}C:Type. \mforall{}phi:Form(C). \mforall{}vs,ws:Atom List.
(l\_subset(Atom;ws;vs) {}\mRightarrow{} \{(FormSafe1(phi) vs) {}\mRightarrow{} (FormSafe1(phi) ws)\})
Date html generated:
2018_05_21-PM-11_27_47
Last ObjectModification:
2017_10_12-AM-00_12_02
Theory : PZF
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