Nuprl Lemma : same-binding-not-bound
∀[vs,ws:varname() List].
∀v,w:varname().
((↑same-binding(vs;ws;v;w)) ⇒ (¬(v ∈ vs)) ⇒ {(¬(w ∈ ws)) ∧ (w = v ∈ varname()) ∧ (||vs|| = ||ws|| ∈ ℤ)})
Proof
Definitions occuring in Statement :
same-binding: same-binding(vs;ws;v;w),
varname: varname(),
l_member: (x ∈ l),
length: ||as||,
list: T List,
assert: ↑b,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
same-binding: same-binding(vs;ws;v;w),
nil: [],
it: ⋅,
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
rev_implies: P ⇐ Q,
band: p ∧b q
Lemmas referenced :
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,
varname_wf,
list-cases,
length_of_nil_lemma,
l_member_wf,
squash_wf,
true_wf,
list_wf,
istype-universe,
nil_wf,
subtype_rel_self,
iff_weakening_equal,
istype-void,
iff_weakening_uiff,
assert_wf,
eq_var_wf,
equal_wf,
assert-eq_var,
istype-assert,
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,
length_of_cons_lemma,
istype-nat,
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,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
bnot_wf,
bool_cases,
band_wf,
btrue_wf,
same-binding_wf,
bfalse_wf,
not_wf,
iff_transitivity,
assert_of_bnot,
assert_of_band,
cons_member,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
productElimination,
independent_pairEquality,
functionIsTypeImplies,
inhabitedIsType,
axiomEquality,
isectIsTypeImplies,
unionElimination,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
functionIsType,
equalityIstype,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
equalityElimination,
cumulativity,
productEquality,
productIsType,
inlFormation_alt,
hyp_replacement,
inrFormation_alt
Latex:
\mforall{}[vs,ws:varname() List].
\mforall{}v,w:varname().
((\muparrow{}same-binding(vs;ws;v;w)) {}\mRightarrow{} (\mneg{}(v \mmember{} vs)) {}\mRightarrow{} \{(\mneg{}(w \mmember{} ws)) \mwedge{} (w = v) \mwedge{} (||vs|| = ||ws||)\})
Date html generated:
2020_05_19-PM-09_53_06
Last ObjectModification:
2020_03_09-PM-04_07_59
Theory : terms
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