Nuprl Lemma : assert-init-seg-nat-seq
∀f,g:finite-nat-seq(). (↑init-seg-nat-seq(f;g) ⇐⇒ ∃h:finite-nat-seq(). (g = f**h ∈ finite-nat-seq()))
Proof
Definitions occuring in Statement :
init-seg-nat-seq: init-seg-nat-seq(f;g),
append-finite-nat-seq: f**g,
finite-nat-seq: finite-nat-seq(),
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
init-seg-nat-seq: init-seg-nat-seq(f;g),
finite-nat-seq: finite-nat-seq(),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
mk-finite-nat-seq: f^(n),
append-finite-nat-seq: f**g,
less_than: a < b,
true: True,
squash: ↓T,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
ble_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
finite-nat-seq_wf,
assert-ble,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
set_subtype_base,
le_wf,
int_subtype_base,
assert-equal-upto-finite-nat-seq,
subtype_rel_function,
int_seg_wf,
nat_wf,
int_seg_subtype,
istype-false,
subtype_rel_self,
istype-assert,
equal-upto-finite-nat-seq_wf,
append-finite-nat-seq_wf,
mk-finite-nat-seq_wf,
add-member-int_seg2,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
lt_int_wf,
assert_of_lt_int,
istype-top,
int_seg_properties,
lelt_wf,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract-add-cancel,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
rename,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
Error :universeIsType,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
independent_pairFormation,
Error :equalityIstype,
applyEquality,
intEquality,
closedConclusion,
baseApply,
baseClosed,
sqequalBase,
addEquality,
applyLambdaEquality,
Error :productIsType,
Error :dependent_pairEquality_alt,
Error :functionIsType,
Error :functionExtensionality_alt,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageMemberEquality,
imageElimination,
Error :equalityIsType4,
universeEquality,
functionEquality
Latex:
\mforall{}f,g:finite-nat-seq(). (\muparrow{}init-seg-nat-seq(f;g) \mLeftarrow{}{}\mRightarrow{} \mexists{}h:finite-nat-seq(). (g = f**h))
Date html generated:
2019_06_20-PM-03_03_25
Last ObjectModification:
2018_11_23-PM-03_14_44
Theory : continuity
Home
Index