Nuprl Lemma : p-first-cons
∀[A,B:Type]. ∀[L:(A ⟶ (B + Top)) List]. ∀[f:A ⟶ (B + Top)]. (p-first([f / L]) = [f?p-first(L)] ∈ (A ⟶ (B + Top)))
Proof
Definitions occuring in Statement :
p-conditional: [f?g],
p-first: p-first(L),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
top_wf,
list_wf,
equal_wf,
squash_wf,
true_wf,
p-first-append,
cons_wf,
nil_wf,
p-conditional_wf,
p-first_wf,
iff_weakening_equal,
p-first-singleton,
p-conditional-to-p-first
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
functionEquality,
cumulativity,
hypothesisEquality,
unionEquality,
isectElimination,
axiomEquality,
because_Cache,
universeEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[L:(A {}\mrightarrow{} (B + Top)) List]. \mforall{}[f:A {}\mrightarrow{} (B + Top)]. (p-first([f / L]) = [f?p-first(L)])
Date html generated:
2018_05_21-PM-06_44_28
Last ObjectModification:
2017_07_26-PM-04_55_07
Theory : general
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