Nuprl Lemma : reduce_wf
∀[A,B:Type]. ∀[f:A ⟶ B ⟶ B]. ∀[k:B]. ∀[as:A List]. (reduce(f;k;as) ∈ B)
Proof
Definitions occuring in Statement :
reduce: reduce(f;k;as),
list: T List,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
true: True,
less_than': less_than'(a;b),
not: ¬A,
le: A ≤ B,
and: P ∧ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
top: Top,
or: P ∨ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x]
Lemmas referenced :
list_wf,
reduce_cons_lemma,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
add-swap,
minus-minus,
less-iff-le,
not-ge-2,
subtract_wf,
equal_wf,
le_wf,
add-commutes,
minus-one-mul-top,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
le_antisymmetry_iff,
sq_stable__le,
spread_cons_lemma,
product_subtype_list,
reduce_nil_lemma,
list-cases,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties
Rules used in proof :
Error :universeIsType,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
Error :functionIsType,
Error :inhabitedIsType,
functionEquality,
universeEquality,
Error :isect_memberFormation_alt,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionExtensionality,
instantiate,
intEquality,
minusEquality,
independent_pairFormation,
dependent_set_memberEquality,
addEquality,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
productElimination,
hypothesis_subsumption,
promote_hyp,
voidEquality,
unionElimination,
applyEquality,
cumulativity,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[k:B]. \mforall{}[as:A List]. (reduce(f;k;as) \mmember{} B)
Date html generated:
2019_06_20-PM-00_38_51
Last ObjectModification:
2018_09_26-PM-02_05_41
Theory : list_0
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