Nuprl Lemma : coerce-fetch'-commutes
∀[Val,S:Type]. ∀[n:ℕ]. ∀[AType:array{i:l}(Val;n)]. ∀[prog:Val ⟶ Val ⟶ (A-map S)].
∀j,k:ℕn.
((A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k))
(λval@k.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j))
(λval@j.(prog val@k val@j)))))
= (A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j))
(λval@j.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k))
(λval@k.(prog val@k val@j)))))
∈ (A-map S))
Proof
Definitions occuring in Statement :
A-coerce: A-coerce(AModel),
A-fetch': A-fetch'(AModel),
A-bind: A-bind(AModel),
A-map: A-map,
array-model: array-model(AType),
array: array{i:l}(Val;n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
array: array{i:l}(Val;n),
array-model: array-model(AType),
A-fetch': A-fetch'(AModel),
A-coerce: A-coerce(AModel),
A-bind: A-bind(AModel),
A-map: A-map,
pi2: snd(t),
pi1: fst(t),
idx: idx(AType),
array-monad: array-monad(AType),
M-bind: M-bind(Mnd),
M-map: M-map(mnd),
let: let,
Arr: Arr(AType),
mk_monad: mk_monad(M;return;bind),
nat: ℕ,
top: Top
Lemmas referenced :
int_seg_wf,
istype-universe,
A-map_wf,
array_wf,
nat_wf,
istype-void,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
hypothesis,
inhabitedIsType,
hypothesisEquality,
universeIsType,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
functionIsType,
applyEquality,
isect_memberEquality_alt,
because_Cache,
universeEquality,
functionExtensionality,
voidElimination,
functionExtensionality_alt
Latex:
\mforall{}[Val,S:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[AType:array\{i:l\}(Val;n)]. \mforall{}[prog:Val {}\mrightarrow{} Val {}\mrightarrow{} (A-map S)].
\mforall{}j,k:\mBbbN{}n.
((A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k))
(\mlambda{}val@k.(A-bind(array-model(AType))
(A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j))
(\mlambda{}val@j.(prog val@k val@j)))))
= (A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j))
(\mlambda{}val@j.(A-bind(array-model(AType))
(A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k))
(\mlambda{}val@k.(prog val@k val@j))))))
Date html generated:
2019_10_15-AM-10_59_31
Last ObjectModification:
2018_10_11-PM-09_54_09
Theory : monads
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