Nuprl Lemma : isl-prior
∀[T:Type]
∀f:ℕ ⟶ (T + Top). ∀n:ℕ.
let m,x = outl(prior(n;f))
in ((f m) = (inl x) ∈ (T + Top)) ∧ (∀k:{m + 1..n-}. (¬↑isl(f k)))
supposing ↑isl(prior(n;f))
Proof
Definitions occuring in Statement :
prior: prior(n;f),
int_seg: {i..j-},
nat: ℕ,
outl: outl(x),
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
inl: inl x,
union: left + right,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uimplies: b supposing a,
true: True,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
bfalse: ff
Lemmas referenced :
prior-cases,
prior_wf,
nat_wf,
int_seg_wf,
unit_wf2,
true_wf,
equal_wf,
top_wf,
int_seg_subtype_nat,
false_wf,
all_wf,
not_wf,
assert_wf,
isl_wf,
int_seg_properties,
nat_properties,
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
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
cumulativity,
functionExtensionality,
applyEquality,
unionEquality,
productEquality,
natural_numberEquality,
setElimination,
rename,
unionElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_pairFormation,
inlEquality,
because_Cache,
addEquality,
lambdaEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:\mBbbN{} {}\mrightarrow{} (T + Top). \mforall{}n:\mBbbN{}.
let m,x = outl(prior(n;f))
in ((f m) = (inl x)) \mwedge{} (\mforall{}k:\{m + 1..n\msupminus{}\}. (\mneg{}\muparrow{}isl(f k)))
supposing \muparrow{}isl(prior(n;f))
Date html generated:
2017_10_01-AM-09_12_10
Last ObjectModification:
2017_07_26-PM-04_47_56
Theory : general
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