Nuprl Lemma : accum_split_iseg2
∀[T,A:Type].
∀x:A. ∀g:(T List × A) ⟶ A. ∀f:(T List × A) ⟶ 𝔹. ∀L1,L2:T List.
(L1 ≤ L2
⇒ (∀LL1,LL2:(T List × A) List. ∀X,Y:T List. ∀z1,z2:A.
(((LL1 = LL2 ∈ ((T List × A) List)) ∧ X ≤ Y ∧ (z1 = z2 ∈ A))
∨ (∃Z:T List
∃ZZ:(T List × A) List. (((LL1 @ [<Z, z1> / ZZ]) = LL2 ∈ ((T List × A) List)) ∧ X ≤ Z))) supposing
((accum_split(g;x;f;L2) = <LL2, Y, z2> ∈ ((T List × A) List × T List × A)) and
(accum_split(g;x;f;L1) = <LL1, X, z1> ∈ ((T List × A) List × T List × A)))))
Proof
Definitions occuring in Statement :
accum_split: accum_split(g;x;f;L),
iseg: l1 ≤ l2,
append: as @ bs,
cons: [a / b],
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
spreadn: spread3,
or: P ∨ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
accum_split_iseg,
accum_split_wf,
list_wf,
iseg_wf,
bool_wf,
istype-universe,
length_wf,
append_wf,
cons_wf,
length-append,
or_wf,
equal_wf,
exists_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
rename,
equalityIstype,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality_alt,
setElimination,
sqequalRule,
independent_pairEquality,
productIsType,
universeIsType,
productEquality,
functionIsType,
instantiate,
universeEquality,
unionIsType,
applyLambdaEquality,
because_Cache,
Error :memTop,
hyp_replacement,
functionEquality,
productElimination,
unionEquality,
spreadEquality
Latex:
\mforall{}[T,A:Type].
\mforall{}x:A. \mforall{}g:(T List \mtimes{} A) {}\mrightarrow{} A. \mforall{}f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}. \mforall{}L1,L2:T List.
(L1 \mleq{} L2
{}\mRightarrow{} (\mforall{}LL1,LL2:(T List \mtimes{} A) List. \mforall{}X,Y:T List. \mforall{}z1,z2:A.
(((LL1 = LL2) \mwedge{} X \mleq{} Y \mwedge{} (z1 = z2))
\mvee{} (\mexists{}Z:T List
\mexists{}ZZ:(T List \mtimes{} A) List. (((LL1 @ [<Z, z1> / ZZ]) = LL2) \mwedge{} X \mleq{} Z))) supposing
((accum\_split(g;x;f;L2) = <LL2, Y, z2>) and
(accum\_split(g;x;f;L1) = <LL1, X, z1>))))
Date html generated:
2020_05_20-AM-08_12_37
Last ObjectModification:
2020_01_17-PM-05_34_37
Theory : general
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