Nuprl Lemma : seq-append-bar
∀k:ℕ. ∀s:ℕk ⟶ ℕ. ∀x:ℕ. ∀Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.
((∀f:ℕ ⟶ ℕ. ∃n:ℕ. ∀m:{n...}. Q[m + k;seq-append(k;m;s;f)])
⇒ (∀f:ℕ ⟶ ℕ. ∃n:ℕ. ∀m:{n...}. Q[m + k;seq-append(k + 1;m;s.x@k;f)]))
Proof
Definitions occuring in Statement :
seq-add: s.x@n,
seq-append: seq-append(n;m;s1;s2),
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
false: False,
not: ¬A,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
int_upper: {i...},
guard: {T},
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
so_apply: x[s],
seq-append: seq-append(n;m;s1;s2),
seq-add: s.x@n,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
lelt: i ≤ j < k,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_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_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_wf,
int_upper_wf,
all_wf,
int_upper_properties,
itermAdd_wf,
int_term_value_add_lemma,
seq-append_wf,
upper_subtype_nat,
sq_stable__le,
seq-add_wf,
int_seg_wf,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
int_seg_subtype,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
zero-add,
add-zero,
add-commutes,
le-add-cancel,
exists_wf,
decidable__equal_int,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eq_int_wf,
assert_of_eq_int,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
int_eqEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
productElimination,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
multiplyEquality,
functionEquality,
cumulativity,
universeEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
lessCases,
isect_memberFormation,
axiomSqEquality,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
int_eqReduceFalseSq
Latex:
\mforall{}k:\mBbbN{}. \mforall{}s:\mBbbN{}k {}\mrightarrow{} \mBbbN{}. \mforall{}x:\mBbbN{}. \mforall{}Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}.
((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}m:\{n...\}. Q[m + k;seq-append(k;m;s;f)])
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}m:\{n...\}. Q[m + k;seq-append(k + 1;m;s.x@k;f)]))
Date html generated:
2019_06_20-PM-02_54_52
Last ObjectModification:
2018_08_20-PM-09_36_09
Theory : continuity
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