Nuprl Lemma : free-vars_functionality
∀[opr:Type]. ∀t1,t2:term(opr). (alpha-eq-terms(opr;t1;t2) ⇒ (free-vars(t1) = free-vars(t2) ∈ (varname() List)))
Proof
Definitions occuring in Statement :
free-vars: free-vars(t),
alpha-eq-terms: alpha-eq-terms(opr;a;b),
term: term(opr),
varname: varname(),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
free-vars: free-vars(t),
alpha-eq-terms: alpha-eq-terms(opr;a;b),
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
false: False,
so_apply: x[s],
alpha-aux: alpha-aux(opr;vs;ws;a;b),
varterm: varterm(v),
mkterm: mkterm(opr;bts),
bound-term: bound-term(opr),
pi2: snd(t),
respects-equality: respects-equality(S;T),
guard: {T},
free-vars-aux: free-vars-aux(bnds;t),
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
or: P ∨ Q,
same-binding: same-binding(vs;ws;v;w),
nil: [],
it: ⋅,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
colength: colength(L),
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
assert: ↑b,
decidable: Dec(P),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
var-deq: VarDeq,
bool: 𝔹,
unit: Unit,
btrue: tt,
bnot: ¬bb,
bor: p ∨bq,
band: p ∧b q,
int_seg: {i..j-},
lelt: i ≤ j < k,
pi1: fst(t),
select: L[n],
l-union-list: l-union-list(eq;ll),
list_ind: list_ind,
map: map(f;as),
label: ...$L... t,
subtract: n - m
Lemmas referenced :
nil_wf,
varname_wf,
term_wf,
istype-universe,
term-induction,
list_wf,
alpha-aux_wf,
equal_wf,
free-vars-aux_wf,
subtype_rel_list,
not_wf,
equal-wf-T-base,
nullvar_wf,
istype-void,
varterm_wf,
mkterm_wf,
bound-term_wf,
l_member_wf,
respects-equality-list,
respects-equality-set-trivial2,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
list-cases,
deq_member_nil_lemma,
assert-eq_var,
squash_wf,
true_wf,
cons_wf,
subtype_rel_self,
iff_weakening_equal,
istype-assert,
eq_var_wf,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
deq_member_cons_lemma,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
istype-nat,
eqtt_to_assert,
testxxx_lemma,
istype-true,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
same-binding_wf,
alpha-aux-mkterm,
select_wf,
int_seg_properties,
decidable__lt,
select_member,
rev-append_wf,
int_seg_wf,
length_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
length_of_cons_lemma,
le_weakening2,
non_neg_length,
add-is-int-iff,
false_wf,
map_cons_lemma,
l-union-list_wf,
deq_wf,
var-deq_wf,
list_extensionality,
map_wf,
map-length,
select-map,
top_wf,
map_length,
select_cons_tl_sq2,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
universeIsType,
instantiate,
universeEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
applyEquality,
setEquality,
baseClosed,
independent_isectElimination,
setElimination,
rename,
setIsType,
functionIsType,
equalityIstype,
independent_functionElimination,
voidElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
axiomEquality,
functionIsTypeImplies,
unionElimination,
imageElimination,
imageMemberEquality,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
equalityElimination,
cumulativity,
productEquality,
addEquality,
pointwiseFunctionality,
productIsType,
spreadEquality,
independent_pairEquality
Latex:
\mforall{}[opr:Type]. \mforall{}t1,t2:term(opr). (alpha-eq-terms(opr;t1;t2) {}\mRightarrow{} (free-vars(t1) = free-vars(t2)))
Date html generated:
2020_05_19-PM-09_56_13
Last ObjectModification:
2020_03_09-PM-04_09_15
Theory : terms
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