Nuprl Lemma : free-vars-all-vars
∀[opr:Type]. ∀t:term(opr). ∀x:varname(). ((x ∈ free-vars(t)) ⇒ (x ∈ all-vars(t)))
Proof
Definitions occuring in Statement :
all-vars: all-vars(t),
free-vars: free-vars(t),
term: term(opr),
varname: varname(),
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
false: False,
so_apply: x[s],
all-vars: all-vars(t),
varterm: varterm(v),
cons: [a / b],
free-vars: free-vars(t),
free-vars-aux: free-vars-aux(bnds;t),
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
bfalse: ff,
bound-term: bound-term(opr),
pi2: snd(t),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
or: P ∨ Q,
pi1: fst(t)
Lemmas referenced :
term-induction,
varname_wf,
l_member_wf,
free-vars_wf,
subtype_rel_list,
not_wf,
equal-wf-T-base,
nullvar_wf,
istype-void,
all-vars_wf,
term_wf,
varterm_wf,
mkterm_wf,
bound-term_wf,
list_wf,
istype-universe,
member-all-vars-mkterm,
member-free-vars-mkterm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
applyEquality,
setEquality,
baseClosed,
independent_isectElimination,
setElimination,
rename,
setIsType,
universeIsType,
because_Cache,
functionIsType,
equalityIstype,
independent_functionElimination,
lambdaFormation_alt,
inhabitedIsType,
productElimination,
instantiate,
universeEquality,
dependent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
inrFormation_alt,
productIsType,
unionIsType
Latex:
\mforall{}[opr:Type]. \mforall{}t:term(opr). \mforall{}x:varname(). ((x \mmember{} free-vars(t)) {}\mRightarrow{} (x \mmember{} all-vars(t)))
Date html generated:
2020_05_19-PM-09_56_30
Last ObjectModification:
2020_03_09-PM-04_09_23
Theory : terms
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