Nuprl Lemma : ml-reduce-sq
∀[A,B:Type].
(∀[f:A ⟶ B ⟶ B]. ∀[l:A List]. ∀[b:B]. (ml-reduce(f;b;l) ~ reduce(f;b;l))) supposing
((valueall-type(A) ∧ A) and
valueall-type(B))
Proof
Definitions occuring in Statement :
ml-reduce: ml-reduce(f;b;l),
reduce: reduce(f;k;as),
list: T List,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
ml-reduce: ml-reduce(f;b;l),
top: Top,
ml_apply: f(x),
spreadcons: spreadcons,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
has-value: (a)↓,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
uiff: uiff(P;Q),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
sq_type: SQType(T),
less_than: a < b,
bfalse: ff
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
reduce_nil_lemma,
function-value-type,
valueall-type-value-type,
value-type_wf,
function-valueall-type,
valueall-type-has-valueall,
evalall-reduce,
null_nil_lemma,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
reduce_cons_lemma,
list_wf,
list-valueall-type,
cons_wf,
null_cons_lemma,
valueall-type_wf,
reduce_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
cumulativity,
productElimination,
applyEquality,
because_Cache,
unionElimination,
voidEquality,
callbyvalueReduce,
sqleReflexivity,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
functionEquality,
independent_pairFormation,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
imageElimination,
addEquality,
dependent_set_memberEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
instantiate,
productEquality,
universeEquality,
functionExtensionality
Latex:
\mforall{}[A,B:Type].
(\mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[l:A List]. \mforall{}[b:B]. (ml-reduce(f;b;l) \msim{} reduce(f;b;l))) supposing
((valueall-type(A) \mwedge{} A) and
valueall-type(B))
Date html generated:
2017_09_29-PM-05_51_00
Last ObjectModification:
2017_05_10-PM-06_50_40
Theory : ML
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