Nuprl Lemma : fun-path-member-connected
∀[T:Type]. ∀f:T ⟶ T. ∀L:T List. ∀x,y:T. ∀a:T. ((a ∈ L) ⇒ {x is f*(a) ∧ a is f*(y)}) supposing x=f*(y) via L
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
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],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
guard: {T},
so_apply: x[s],
so_apply: x[s1;s2;s3],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
fun-path: y=f*(x) via L,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b
Lemmas referenced :
fun-path-induction,
l_member_wf,
all_wf,
fun-connected_wf,
list_wf,
cons_member,
nil_wf,
member_singleton,
fun-connected_weakening_eq,
cons_wf,
equal_wf,
strict-fun-connected-step,
not_wf,
squash_wf,
true_wf,
iff_weakening_equal,
fun-connected_transitivity,
fun-connected_weakening,
member-less_than,
length_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,
fun-path_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
productEquality,
hypothesis,
functionExtensionality,
applyEquality,
independent_functionElimination,
because_Cache,
productElimination,
inlFormation,
independent_pairFormation,
independent_isectElimination,
equalitySymmetry,
axiomEquality,
rename,
voidElimination,
unionElimination,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
addEquality,
setElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T. \mforall{}L:T List. \mforall{}x,y:T.
\mforall{}a:T. ((a \mmember{} L) {}\mRightarrow{} \{x is f*(a) \mwedge{} a is f*(y)\}) supposing x=f*(y) via L
Date html generated:
2018_05_21-PM-07_46_10
Last ObjectModification:
2017_07_26-PM-05_23_41
Theory : general
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