Nuprl Lemma : map_is_nil
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[l:A List]. uiff(map(f;l) = [] ∈ (B List);l = [] ∈ (A List))
Proof
Definitions occuring in Statement :
map: map(f;as),
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
uiff_wf,
equal-wf-T-base,
list_wf,
map_wf,
map_nil_lemma,
nil_wf,
equal-wf-base,
map_cons_lemma,
null_nil_lemma,
btrue_wf,
and_wf,
equal_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
baseClosed,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
lambdaFormation,
rename,
productElimination,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
setElimination,
applyEquality,
independent_pairEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[l:A List]. uiff(map(f;l) = [];l = [])
Date html generated:
2019_06_20-PM-00_39_16
Last ObjectModification:
2018_08_07-PM-02_14_00
Theory : list_0
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