Nuprl Lemma : is-list-if-has-value-rec-decomp
∀[t:Base]. (if ispair(t) then t ~ <fst(t), snd(t)> else t ~ Ax fi ) supposing ((t)↓ and is-list-if-has-value-rec(t))
Proof
Definitions occuring in Statement :
is-list-if-has-value-rec: is-list-if-has-value-rec(t),
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
ispair: if z is a pair then a otherwise b,
pair: <a, b>,
base: Base,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
is-list-if-has-value-rec: is-list-if-has-value-rec(t),
is-list-if-has-value-fun: is-list-if-has-value-fun(t;n),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
or: P ∨ Q,
has-value: (a)↓,
pi1: fst(t),
pi2: snd(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
assert: ↑b,
exposed-it: exposed-it,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
exists: ∃x:A. B[x],
bnot: ¬bb
Lemmas referenced :
false_wf,
le_wf,
primrec1_lemma,
has-value-implies-dec-ispair-2,
top_wf,
ispair-bool-if-has-value,
equal_wf,
has-value_wf_base,
is-list-if-has-value-rec_wf,
bool_wf,
base_wf,
has-value-implies-dec-isaxiom-2,
btrue_wf,
eqtt_to_assert,
subtype_base_sq,
subtype_rel_self,
isaxiom-implies,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
hypothesis,
extract_by_obid,
hypothesisEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_functionElimination,
unionElimination,
because_Cache,
sqequalAxiom,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
baseClosed,
equalityTransitivity,
equalitySymmetry,
instantiate,
equalityElimination,
productElimination,
cumulativity,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[t:Base]
(if ispair(t) then t \msim{} <fst(t), snd(t)> else t \msim{} Ax fi ) supposing
((t)\mdownarrow{} and
is-list-if-has-value-rec(t))
Date html generated:
2018_05_21-PM-10_19_30
Last ObjectModification:
2017_07_26-PM-06_37_02
Theory : eval!all
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