Nuprl Lemma : append-finite-nat-seq-1
∀n,a,b:finite-nat-seq(). ∀x:ℕ.
((↑init-seg-nat-seq(n**λi.x^(1);a**b)) ⇒ ((↑init-seg-nat-seq(a;n)) ∨ (↑init-seg-nat-seq(n**λi.x^(1);a))))
Proof
Definitions occuring in Statement :
init-seg-nat-seq: init-seg-nat-seq(f;g),
append-finite-nat-seq: f**g,
mk-finite-nat-seq: f^(n),
finite-nat-seq: finite-nat-seq(),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
finite-nat-seq: finite-nat-seq(),
decidable: Dec(P),
or: P ∨ Q,
mk-finite-nat-seq: f^(n),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append-finite-nat-seq: f**g,
pi1: fst(t),
pi2: snd(t),
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
top: Top,
true: True,
squash: ↓T,
lelt: i ≤ j < k,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
assert_wf,
init-seg-nat-seq_wf,
append-finite-nat-seq_wf,
mk-finite-nat-seq_wf,
false_wf,
le_wf,
int_seg_wf,
nat_wf,
finite-nat-seq_wf,
decidable__le,
assert-init-seg-nat-seq2,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
subtype_rel_function,
int_seg_subtype,
add-is-int-iff,
set_subtype_base,
int_subtype_base,
subtype_rel_self,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
lambdaEquality,
productElimination,
rename,
dependent_functionElimination,
addEquality,
setElimination,
unionElimination,
inrFormation,
functionExtensionality,
applyEquality,
because_Cache,
independent_functionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lessCases,
isect_memberFormation,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
approximateComputation,
int_eqEquality,
intEquality,
hyp_replacement,
applyLambdaEquality,
functionEquality,
baseApply,
closedConclusion,
inlFormation
Latex:
\mforall{}n,a,b:finite-nat-seq(). \mforall{}x:\mBbbN{}.
((\muparrow{}init-seg-nat-seq(n**\mlambda{}i.x\^{}(1);a**b))
{}\mRightarrow{} ((\muparrow{}init-seg-nat-seq(a;n)) \mvee{} (\muparrow{}init-seg-nat-seq(n**\mlambda{}i.x\^{}(1);a))))
Date html generated:
2019_06_20-PM-03_03_42
Last ObjectModification:
2018_08_20-PM-09_41_20
Theory : continuity
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