Nuprl Lemma : reduce2_shift
∀[A,T:Type]. ∀[L:T List]. ∀[k:A]. ∀[i:ℕ]. ∀[f:T ⟶ {i..i + ||L||-} ⟶ A ⟶ A].
(reduce2(f;k;i;L) = reduce2(λx,i,l. (f x (i - 1) l);k;i + 1;L) ∈ A)
Proof
Definitions occuring in Statement :
reduce2: reduce2(f;k;i;as),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
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,
so_lambda: λ2x.t[x],
nat: ℕ,
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),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
so_apply: x[s],
le: A ≤ B,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
nat_wf,
int_seg_wf,
length_wf,
equal_wf,
reduce2_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
add-is-int-iff,
intformless_wf,
int_formula_prop_less_lemma,
false_wf,
lelt_wf,
list_wf,
length_of_nil_lemma,
reduce2_nil_lemma,
length_of_cons_lemma,
reduce2_cons_lemma,
squash_wf,
true_wf,
non_neg_length,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
add-subtract-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
setElimination,
rename,
because_Cache,
addEquality,
functionExtensionality,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
axiomEquality,
lambdaFormation,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[L:T List]. \mforall{}[k:A]. \mforall{}[i:\mBbbN{}]. \mforall{}[f:T {}\mrightarrow{} \{i..i + ||L||\msupminus{}\} {}\mrightarrow{} A {}\mrightarrow{} A].
(reduce2(f;k;i;L) = reduce2(\mlambda{}x,i,l. (f x (i - 1) l);k;i + 1;L))
Date html generated:
2017_10_01-AM-08_35_03
Last ObjectModification:
2017_07_26-PM-04_25_39
Theory : list!
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