Nuprl Lemma : fun-path-member
∀[T:Type]. ∀f:T ⟶ T. ∀x,y:T. ∀L:T List. {(x ∈ L) ∧ (y ∈ L)} supposing x=f*(y) via L
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
fun-path: y=f*(x) via L,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtract: n - m,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff
Lemmas referenced :
member-less_than,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
subtract_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
int_seg_wf,
hd_member,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
l_member_wf,
last_member,
fun-path_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
axiomEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
addEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}x,y:T. \mforall{}L:T List. \{(x \mmember{} L) \mwedge{} (y \mmember{} L)\} supposing x=f*(y) via L
Date html generated:
2018_05_21-PM-07_43_01
Last ObjectModification:
2017_07_26-PM-05_21_01
Theory : general
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