Nuprl Lemma : find-ge-val_wf
∀[T:Type]
∀[test:T ⟶ 𝔹]. ∀[n:ℤ]. ∀[f:{n...} ⟶ T].
find-ge-val(test;f;n) ∈ v:T × {n':ℤ| (n ≤ n') ∧ (v = (f n') ∈ T) ∧ test v = tt}
supposing ∃m:{n...}. ∀k:{m...}. test (f k) = tt
supposing value-type(T)
Proof
Definitions occuring in Statement :
find-ge-val: find-ge-val(test;f;n),
int_upper: {i...},
value-type: value-type(T),
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
find-ge-val: find-ge-val(test;f;n),
int_upper: {i...},
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
nat_plus: ℕ+,
spreadn: spread3,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
find-xover-val_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
istype-less_than,
intformand_wf,
int_formula_prop_and_lemma,
bool_wf,
btrue_wf,
istype-int_upper,
upper_subtype_upper,
sq_stable__le,
value-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
Error :dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
spreadEquality,
Error :dependent_pairEquality_alt,
Error :setIsType,
Error :productIsType,
Error :equalityIstype,
applyEquality,
setElimination,
rename,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
Error :functionIsType,
imageMemberEquality,
baseClosed,
imageElimination,
Error :isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}[test:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbZ{}]. \mforall{}[f:\{n...\} {}\mrightarrow{} T].
find-ge-val(test;f;n) \mmember{} v:T \mtimes{} \{n':\mBbbZ{}| (n \mleq{} n') \mwedge{} (v = (f n')) \mwedge{} test v = tt\}
supposing \mexists{}m:\{n...\}. \mforall{}k:\{m...\}. test (f k) = tt
supposing value-type(T)
Date html generated:
2019_06_20-PM-01_16_37
Last ObjectModification:
2019_01_09-PM-04_13_25
Theory : int_2
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