Nuprl Lemma : map-fst-zip
∀[as,bs:Top List]. map(λp.(fst(p));zip(as;bs)) ~ as supposing ||as|| = ||bs|| ∈ ℤ
Proof
Definitions occuring in Statement :
zip: zip(as;bs),
length: ||as||,
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
lambda: λx.A[x],
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
decidable: Dec(P),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
pi1: fst(t),
uiff: uiff(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
top_wf,
list-cases,
length_of_nil_lemma,
zip_nil_lemma,
map_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
length_of_cons_lemma,
length_wf,
istype-nat,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
istype-void,
list_wf,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
zip_cons_nil_lemma,
le_weakening2,
non_neg_length,
decidable__lt,
zip_cons_cons_lemma,
map_cons_lemma,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomSqEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
because_Cache,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
instantiate,
equalityTransitivity,
applyLambdaEquality,
addEquality,
applyEquality,
dependent_set_memberEquality_alt,
imageElimination,
baseApply,
closedConclusion,
intEquality,
pointwiseFunctionality
Latex:
\mforall{}[as,bs:Top List]. map(\mlambda{}p.(fst(p));zip(as;bs)) \msim{} as supposing ||as|| = ||bs||
Date html generated:
2020_05_19-PM-09_50_03
Last ObjectModification:
2020_02_27-PM-04_02_40
Theory : list_1
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