Nuprl Lemma : fun-path-append1
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L:T List]. ∀[x,y,z:T].
(z=f*(x) via L @ [x]) supposing ((¬(y = x ∈ T)) and (y = (f x) ∈ T) and z=f*(y) via L)
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
fun-path: y=f*(x) via L,
and: P ∧ Q,
not: ¬A,
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],
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
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],
true: True,
last: last(L),
cand: A c∧ B,
sq_type: SQType(T),
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
list_induction,
uall_wf,
isect_wf,
fun-path_wf,
equal_wf,
not_wf,
append_wf,
cons_wf,
nil_wf,
list_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
member-less_than,
length_wf,
select_wf,
length-append,
length_of_cons_lemma,
length_of_nil_lemma,
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,
add-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
int_seg_wf,
stuck-spread,
base_wf,
add-subtract-cancel,
fun-path-cons,
list-cases,
product_subtype_list,
reduce_hd_cons_lemma,
less_than_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
nat_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
productElimination,
independent_pairEquality,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
natural_numberEquality,
addEquality,
setElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
universeEquality,
hypothesis_subsumption,
imageMemberEquality,
instantiate,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[L:T List]. \mforall{}[x,y,z:T].
(z=f*(x) via L @ [x]) supposing ((\mneg{}(y = x)) and (y = (f x)) and z=f*(y) via L)
Date html generated:
2018_05_21-PM-07_43_57
Last ObjectModification:
2017_07_26-PM-05_21_40
Theory : general
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