Nuprl Lemma : sq_stable__ex_nat_plus
∀[P:ℕ+ ⟶ ℙ]. ((∀m:ℕ+. Dec(P[m])) ⇒ SqStable(∃m:ℕ+. P[m]))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
int_upper: {i...},
nat_plus: ℕ+,
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True,
istype: istype(T),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__ex_int_upper,
subtype_rel_dep_function,
nat_plus_wf,
int_upper_wf,
istype-int_upper,
subtype_rel_sets_simple,
le_wf,
istype-int,
less_than_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
istype-void,
zero-add,
le-add-cancel,
istype-le,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
decidable__le,
nat_plus_properties,
subtype_rel_self,
squash_wf,
decidable_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
natural_numberEquality,
Error :isect_memberFormation_alt,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
sqequalRule,
Error :lambdaEquality_alt,
universeEquality,
Error :universeIsType,
independent_isectElimination,
intEquality,
Error :lambdaFormation_alt,
productElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
imageElimination,
because_Cache,
imageMemberEquality,
baseClosed,
productEquality,
Error :functionIsType
Latex:
\mforall{}[P:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}m:\mBbbN{}\msupplus{}. Dec(P[m])) {}\mRightarrow{} SqStable(\mexists{}m:\mBbbN{}\msupplus{}. P[m]))
Date html generated:
2019_06_20-PM-01_17_05
Last ObjectModification:
2019_01_09-PM-03_01_59
Theory : int_2
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