Nuprl Lemma : before-map
∀[T,T':Type].
∀f:T ⟶ T'. ∀L:T List. ∀x',y':T'.
(x' before y' ∈ map(f;L) ⇐⇒ ∃x,y:T. (x before y ∈ L ∧ ((f x) = x' ∈ T') ∧ ((f y) = y' ∈ T')))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
top: Top,
implies: P ⇒ Q,
so_apply: x[s],
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
false: False,
or: P ∨ Q,
cand: A c∧ B
Lemmas referenced :
istype-universe,
map_cons_lemma,
istype-void,
map_nil_lemma,
list_wf,
equal_wf,
map_wf,
l_before_wf,
iff_wf,
list_induction,
nil_wf,
nil_before,
cons_wf,
cons_before,
l_member_wf,
member_map
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
equalityIstype,
productIsType,
because_Cache,
functionIsType,
rename,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
applyEquality,
productEquality,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
dependent_pairFormation_alt,
productElimination,
independent_pairFormation,
unionIsType,
unionElimination,
equalityTransitivity,
equalitySymmetry,
inlFormation_alt,
inrFormation_alt,
setElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[T,T':Type].
\mforall{}f:T {}\mrightarrow{} T'. \mforall{}L:T List. \mforall{}x',y':T'.
(x' before y' \mmember{} map(f;L) \mLeftarrow{}{}\mRightarrow{} \mexists{}x,y:T. (x before y \mmember{} L \mwedge{} ((f x) = x') \mwedge{} ((f y) = y')))
Date html generated:
2019_10_15-AM-10_21_33
Last ObjectModification:
2019_08_05-PM-02_05_29
Theory : list_1
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