Nuprl Lemma : list_accum_invariant2
∀[T,A:Type].
∀f:A ⟶ T ⟶ A
∀[P:A ⟶ ℙ]
∀L:T List. ∀a:A.
(P[a]
⇒ (∀a:A. ∀x:T. ((x ∈ L) ⇒ P[a] ⇒ P[f[a;x]]))
⇒ P[accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a)])
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list_accum: list_accum,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
list_accum_wf,
list_wf,
list_accum_nil_lemma,
nil_wf,
list_accum_cons_lemma,
cons_member,
equal_wf,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
applyEquality,
hypothesis,
universeEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
productElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[T,A:Type].
\mforall{}f:A {}\mrightarrow{} T {}\mrightarrow{} A
\mforall{}[P:A {}\mrightarrow{} \mBbbP{}]
\mforall{}L:T List. \mforall{}a:A.
(P[a]
{}\mRightarrow{} (\mforall{}a:A. \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} P[a] {}\mRightarrow{} P[f[a;x]]))
{}\mRightarrow{} P[accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a)])
Date html generated:
2016_05_14-PM-01_40_44
Last ObjectModification:
2015_12_26-PM-05_29_54
Theory : list_1
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