Nuprl Lemma : apply_larger_list
∀[B:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[q:ℕm + 1]. ∀[A:ℕn ⟶ Type]. ∀[lst:k:{q..n-} ⟶ (A k)]. ∀[r:ℕm]. ∀[a:A r].
∀[f:funtype(n - m;λx.(A (x + m));B)].
((apply_gen(n;λx.if (x =z r) then a else lst x fi ) m f) = (apply_gen(n;lst) m f) ∈ B)
Proof
Definitions occuring in Statement :
apply_gen: apply_gen(n;lst),
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
uiff: uiff(P;Q),
less_than: a < b,
funtype: funtype(n;A;T),
apply_gen: apply_gen(n;lst),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
squash: ↓T,
true: True
Lemmas referenced :
int_seg_properties,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base-T,
int_subtype_base,
subtype_base_sq,
ge_wf,
less_than_wf,
funtype_wf,
int_seg_wf,
add-member-int_seg2,
lelt_wf,
decidable__lt,
add-member-int_seg1,
nat_wf,
primrec0_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
subtype_rel-equal,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
primrec-unroll,
equal-wf-base,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
mul-distributes-right,
zero-add,
itermMultiply_wf,
int_term_value_mul_lemma,
primrec_wf,
zero-mul,
add-zero,
bool_cases
Rules used in proof :
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_pairFormation,
productElimination,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
intWeakElimination,
lambdaFormation,
axiomEquality,
functionExtensionality,
functionEquality,
universeEquality,
isect_memberFormation,
equalityElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
multiplyEquality,
minusEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1]. \mforall{}[q:\mBbbN{}m + 1]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[lst:k:\{q..n\msupminus{}\} {}\mrightarrow{} (A k)]. \mforall{}[r:\mBbbN{}m].
\mforall{}[a:A r]. \mforall{}[f:funtype(n - m;\mlambda{}x.(A (x + m));B)].
((apply\_gen(n;\mlambda{}x.if (x =\msubz{} r) then a else lst x fi ) m f) = (apply\_gen(n;lst) m f))
Date html generated:
2017_10_01-AM-09_03_48
Last ObjectModification:
2017_07_26-PM-04_44_35
Theory : bags
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