Nuprl Lemma : respects-equality-list
∀[A,B:Type]. respects-equality(A List;B List) supposing respects-equality(A;B)
Proof
Definitions occuring in Statement :
list: T List,
uimplies: b supposing a,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
respects-equality: respects-equality(S;T),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
subtract: n - m,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__respects-equality,
list_wf,
respects-equality_wf,
istype-universe,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
subtract-1-ge-0,
istype-nat,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
istype-base,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
istype-void,
le_weakening2,
length_wf,
istype-sqequal,
sq_stable__le,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
subtract_wf,
minus-zero,
subtype_base_sq,
add-swap,
nil_wf,
tl_wf,
equal_wf,
squash_wf,
true_wf,
length_tl,
decidable__le,
istype-false,
not-ge-2,
less-iff-le,
le-add-cancel2,
subtype_rel_self,
iff_weakening_equal,
hd_wf,
le_weakening,
reduce_tl_nil_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
cons_wf,
equal_functionality_wrt_subtype_rel2,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
voidElimination,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :equalityIstype,
Error :setIsType,
applyEquality,
intEquality,
closedConclusion,
because_Cache,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
Error :equalityIsType1,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
applyLambdaEquality,
addEquality,
minusEquality,
cumulativity,
Error :dependent_set_memberEquality_alt,
baseApply,
independent_pairFormation,
setEquality
Latex:
\mforall{}[A,B:Type]. respects-equality(A List;B List) supposing respects-equality(A;B)
Date html generated:
2019_06_20-PM-00_40_13
Last ObjectModification:
2018_11_23-PM-02_17_25
Theory : list_0
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