Nuprl Lemma : iseg_extend
∀[T:Type]. ∀l1:T List. ∀v:T. ∀l2:T List. (l1 ≤ l2 ⇒ l1 @ [v] ≤ l2 supposing ||l1|| < ||l2|| c∧ (l2[||l1||] = v ∈ T))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
select: L[n],
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
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,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
iseg: 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,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
subtract: n - m,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
nat: ℕ,
less_than': less_than'(a;b),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b]
Lemmas referenced :
member-less_than,
length_wf,
tl_wf,
equal_wf,
list_wf,
append_wf,
cons_wf,
nil_wf,
less_than_wf,
select_wf,
non_neg_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
exists_wf,
append_assoc,
length-append,
decidable__lt,
add-is-int-iff,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
false_wf,
length_wf_nat,
nat_wf,
squash_wf,
true_wf,
select_append_back,
lelt_wf,
iff_weakening_equal,
minus-one-mul,
add-mul-special,
zero-mul,
subtype_base_sq,
int_subtype_base,
and_wf,
list_induction,
length_of_nil_lemma,
list_ind_cons_lemma,
stuck-spread,
base_wf,
list_ind_nil_lemma,
reduce_tl_nil_lemma,
length_of_cons_lemma,
reduce_tl_cons_lemma
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_pairFormation,
productEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
pointwiseFunctionality,
equalityTransitivity,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
dependent_set_memberEquality,
addEquality,
setElimination,
applyEquality,
imageMemberEquality,
independent_functionElimination,
instantiate,
functionEquality
Latex:
\mforall{}[T:Type]
\mforall{}l1:T List. \mforall{}v:T. \mforall{}l2:T List.
(l1 \mleq{} l2 {}\mRightarrow{} l1 @ [v] \mleq{} l2 supposing ||l1|| < ||l2|| c\mwedge{} (l2[||l1||] = v))
Date html generated:
2017_04_17-AM-07_31_05
Last ObjectModification:
2017_02_27-PM-04_08_58
Theory : list_1
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