Nuprl Lemma : iseg-transition-lemma
∀[T:Type]. ∀[P:(T List) ⟶ ℙ].
∀L:T List. ∀x:T.
((∃L1:T List. (L1 ≤ L @ [x] ∧ P[L1])) ∧ (¬(∃L1:T List. (L1 ≤ L ∧ P[L1])))
⇐⇒ P[L @ [x]] ∧ (¬(∃L1:T List. (L1 ≤ L ∧ P[L1]))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
cand: A c∧ B,
or: P ∨ Q,
squash: ↓T,
true: True,
less_than: a < b,
less_than': less_than'(a;b),
cons: [a / b],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
exists_wf,
list_wf,
iseg_wf,
append_wf,
cons_wf,
nil_wf,
not_wf,
iseg_weakening,
iseg_append_iff,
equal_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
cons_iseg,
iseg_nil,
null_nil_lemma,
null_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
universeEquality,
because_Cache,
dependent_pairFormation,
dependent_functionElimination,
functionEquality,
unionElimination,
addLevel,
hyp_replacement,
equalitySymmetry,
levelHypothesis,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
\mforall{}L:T List. \mforall{}x:T.
((\mexists{}L1:T List. (L1 \mleq{} L @ [x] \mwedge{} P[L1])) \mwedge{} (\mneg{}(\mexists{}L1:T List. (L1 \mleq{} L \mwedge{} P[L1])))
\mLeftarrow{}{}\mRightarrow{} P[L @ [x]] \mwedge{} (\mneg{}(\mexists{}L1:T List. (L1 \mleq{} L \mwedge{} P[L1]))))
Date html generated:
2016_10_25-AM-10_54_48
Last ObjectModification:
2016_07_12-AM-07_02_05
Theory : general
Home
Index