Nuprl Lemma : reject_cons_tl
∀[T:Type]. ∀[a:T]. ∀[as:T List]. ∀[i:ℤ].
([a / as]\[i] = [a / as\[i - 1]] ∈ (T List)) supposing ((i ≤ ||as||) and 0 < i)
Proof
Definitions occuring in Statement :
length: ||as||,
reject: as\[i],
cons: [a / b],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
reject: as\[i],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
top: Top,
le: A ≤ B,
false: False,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
le_wf,
length_wf,
less_than_wf,
list_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
reduce_tl_cons_lemma,
less_than_transitivity1,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
list_ind_cons_lemma,
cons_wf,
reject_wf,
subtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
because_Cache,
intEquality,
universeEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[a:T]. \mforall{}[as:T List]. \mforall{}[i:\mBbbZ{}].
([a / as]\mbackslash{}[i] = [a / as\mbackslash{}[i - 1]]) supposing ((i \mleq{} ||as||) and 0 < i)
Date html generated:
2019_06_20-PM-00_40_17
Last ObjectModification:
2018_09_26-PM-02_47_36
Theory : list_0
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