Nuprl Lemma : fun-path-cons
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L:T List]. ∀[x,y,z:T].
uiff(z=f*(x) via [y / L];{(z = y ∈ T)
∧ ((y = (f hd(L)) ∈ T) ∧ (¬(y = hd(L) ∈ T))) ∧ hd(L)=f*(x) via L supposing 0 < ||L||
∧ x = y ∈ T supposing ¬0 < ||L||})
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
hd: hd(l),
length: ||as||,
cons: [a / b],
list: T List,
less_than: a < b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
cons: [a / b],
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
fun-path: y=f*(x) via L,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtract: n - m,
last: last(L),
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
le: A ≤ B,
nat: ℕ,
nat_plus: ℕ+,
cand: A c∧ B,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
sq_type: SQType(T),
rev_implies: P ⇐ Q
Lemmas referenced :
list-cases,
product_subtype_list,
equal_wf,
hd_wf,
decidable__le,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
select_wf,
int_seg_properties,
subtract_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_wf,
less_than_wf,
not_wf,
fun-path_wf,
cons_wf,
length_of_cons_lemma,
add-is-int-iff,
false_wf,
list_wf,
member-less_than,
length_of_nil_lemma,
reduce_hd_cons_lemma,
stuck-spread,
base_wf,
all_wf,
nil_wf,
equal-wf-T-base,
equal-wf-base,
equal-wf-base-T,
lelt_wf,
select-cons-tl,
add-subtract-cancel,
non_neg_length,
select-cons-hd,
add_nat_plus,
le_wf,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
length_wf_nat,
member_wf,
add-member-int_seg2,
le_weakening2,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
and_wf,
true_wf,
select_cons_tl,
iff_weakening_equal,
subtract-is-int-iff,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
general_arith_equation1
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
cumulativity,
because_Cache,
independent_isectElimination,
natural_numberEquality,
imageElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
addEquality,
setElimination,
rename,
functionExtensionality,
applyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
isectEquality,
functionEquality,
universeEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
lambdaFormation,
independent_functionElimination,
imageMemberEquality,
minusEquality,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[L:T List]. \mforall{}[x,y,z:T].
uiff(z=f*(x) via [y / L];\{(z = y)
\mwedge{} ((y = (f hd(L))) \mwedge{} (\mneg{}(y = hd(L)))) \mwedge{} hd(L)=f*(x) via L supposing 0 < ||L||
\mwedge{} x = y supposing \mneg{}0 < ||L||\})
Date html generated:
2018_05_21-PM-07_43_14
Last ObjectModification:
2017_07_26-PM-05_21_12
Theory : general
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