Nuprl Lemma : adjacent-cons
∀[T:Type]
∀x,y,u:T. ∀L:T List. (adjacent(T;[u / L];x;y) ⇐⇒ 0 < ||L|| ∧ (((x = u ∈ T) ∧ (y = hd(L) ∈ T)) ∨ adjacent(T;L;x;y)))
Proof
Definitions occuring in Statement :
adjacent: adjacent(T;L;x;y),
hd: hd(l),
length: ||as||,
cons: [a / b],
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
adjacent: adjacent(T;L;x;y),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
false: False,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
le: A ≤ B,
so_apply: x[s],
rev_implies: P ⇐ Q,
sq_type: SQType(T),
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
less_than': less_than'(a;b)
Lemmas referenced :
length_of_cons_lemma,
int_seg_properties,
decidable__lt,
length_wf,
subtract-is-int-iff,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
false_wf,
exists_wf,
int_seg_wf,
subtract_wf,
equal_wf,
select_wf,
cons_wf,
decidable__le,
non_neg_length,
less_than_wf,
or_wf,
hd_wf,
list_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
select0,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
select-cons-tl,
general_arith_equation1,
add-member-int_seg2,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
productElimination,
isectElimination,
because_Cache,
hypothesisEquality,
setElimination,
rename,
cumulativity,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
addEquality,
productEquality,
universeEquality,
instantiate,
independent_functionElimination,
inlFormation,
inrFormation,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}x,y,u:T. \mforall{}L:T List.
(adjacent(T;[u / L];x;y) \mLeftarrow{}{}\mRightarrow{} 0 < ||L|| \mwedge{} (((x = u) \mwedge{} (y = hd(L))) \mvee{} adjacent(T;L;x;y)))
Date html generated:
2018_05_21-PM-06_32_44
Last ObjectModification:
2017_07_26-PM-04_51_47
Theory : general
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