Nuprl Lemma : fseg_extend
∀[T:Type]
∀l1:T List. ∀v:T. ∀l2:T List.
(fseg(T;l1;l2) ⇒ fseg(T;[v / l1];l2) supposing ||l1|| < ||l2|| c∧ (l2[||l2|| - ||l1|| + 1] = v ∈ T))
Proof
Definitions occuring in Statement :
fseg: fseg(T;L1;L2),
select: L[n],
length: ||as||,
cons: [a / b],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fseg: fseg(T;L1;L2),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
cand: A c∧ B,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
cons: [a / b],
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
ge: i ≥ j ,
le: A ≤ B,
true: True,
nat_plus: ℕ+,
less_than': less_than'(a;b),
guard: {T},
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
last: last(L)
Lemmas referenced :
member-less_than,
length_wf,
length_wf_nat,
equal_wf,
nat_wf,
list-cases,
null_nil_lemma,
less_than_wf,
list_ind_nil_lemma,
satisfiable-full-omega-tt,
intformless_wf,
itermVar_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
product_subtype_list,
null_cons_lemma,
false_wf,
last_lemma,
exists_wf,
list_wf,
append_wf,
cons_wf,
select_wf,
subtract_wf,
decidable__le,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
non_neg_length,
decidable__lt,
append_assoc,
squash_wf,
true_wf,
list_ind_cons_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
last_wf,
le_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
select_append_front,
length-append,
lelt_wf,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
axiomEquality,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
hypothesis_subsumption,
because_Cache,
productEquality,
addEquality,
independent_pairFormation,
universeEquality,
applyEquality,
equalityTransitivity,
imageMemberEquality,
baseClosed,
independent_functionElimination,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]
\mforall{}l1:T List. \mforall{}v:T. \mforall{}l2:T List.
(fseg(T;l1;l2)
{}\mRightarrow{} fseg(T;[v / l1];l2) supposing ||l1|| < ||l2|| c\mwedge{} (l2[||l2|| - ||l1|| + 1] = v))
Date html generated:
2018_05_21-PM-06_30_18
Last ObjectModification:
2017_07_26-PM-04_50_38
Theory : general
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