Nuprl Lemma : pi1-if-ispair-append-nil
∀[l:Base]. (fst((l @ [])) ~ fst(l)) supposing ((↑ispair(l @ [])) and (l @ [])↓)
Proof
Definitions occuring in Statement :
append: as @ bs,
nil: [],
has-value: (a)↓,
assert: ↑b,
bfalse: ff,
btrue: tt,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
ispair: if z is a pair then a otherwise b,
base: Base,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
append: as @ bs,
list_ind: list_ind,
has-value: (a)↓,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi1: fst(t),
top: Top,
nil: [],
it: ⋅,
bfalse: ff,
false: False,
not: ¬A,
prop: ℙ
Lemmas referenced :
has-value-implies-dec-ispair-2,
pi1_cons_lemma,
top_wf,
has-value-implies-dec-isaxiom-2,
bottom_diverge,
assert_wf,
has-value_wf_base,
is-exception_wf,
btrue_wf,
bfalse_wf,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
callbyvalueCallbyvalue,
hypothesis,
callbyvalueReduce,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
unionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
axiomSqEquality,
Error :universeIsType,
isectElimination,
ispairCases,
divergentSqle,
baseClosed,
baseApply,
closedConclusion,
Error :inhabitedIsType
Latex:
\mforall{}[l:Base]. (fst((l @ [])) \msim{} fst(l)) supposing ((\muparrow{}ispair(l @ [])) and (l @ [])\mdownarrow{})
Date html generated:
2019_06_20-PM-00_39_24
Last ObjectModification:
2018_09_26-PM-02_09_54
Theory : list_0
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