Nuprl Lemma : ppcc-test6
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[as,L:A List]. ∀[n:ℤ].
||L @ as|| = (n + n) ∈ ℤ ⇐⇒ ||map(f;as)|| = n ∈ ℤ supposing ||L|| = ||as|| ∈ ℤ
Proof
Definitions occuring in Statement :
length: ||as||,
map: map(f;as),
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A
Lemmas referenced :
map-length,
equal-wf-T-base,
length_wf,
append_wf,
int_subtype_base,
subtype_base_sq,
length_append,
subtype_rel_list,
top_wf,
map_wf,
equal_wf,
list_wf,
non_neg_length,
length-append,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
intEquality,
cumulativity,
hypothesisEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
addEquality,
lambdaEquality,
functionExtensionality,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
universeEquality,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[as,L:A List]. \mforall{}[n:\mBbbZ{}].
||L @ as|| = (n + n) \mLeftarrow{}{}\mRightarrow{} ||map(f;as)|| = n supposing ||L|| = ||as||
Date html generated:
2018_05_21-PM-09_04_18
Last ObjectModification:
2017_07_26-PM-06_27_14
Theory : general
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