Nuprl Lemma : case-real3-req2
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:Top]. (case-real3(a;b;f) = b) supposing ∀n:ℕ+. (¬↑(f n))
Proof
Definitions occuring in Statement :
case-real3: case-real3(a;b;f),
req: x = y,
real: ℝ,
nat_plus: ℕ+,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
case-real3: case-real3(a;b;f),
real: ℝ,
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
case-real3-seq: case-real3-seq(a;b;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
real-regular,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
regular-int-seq_wf,
accelerate-req,
req_wf,
accelerate_wf,
subtype_rel_sets_simple,
nat_plus_wf,
istype-top,
real_wf,
istype-assert,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
equalitySymmetry,
dependent_set_memberEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
because_Cache,
productElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
functionEquality,
intEquality,
functionIsType,
lambdaFormation_alt,
functionExtensionality,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
setElimination,
rename
Latex:
\mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:\mBbbR{}]. \mforall{}[a:Top]. (case-real3(a;b;f) = b) supposing \mforall{}n:\mBbbN{}\msupplus{}. (\mneg{}\muparrow{}(f n))
Date html generated:
2019_10_29-AM-09_38_02
Last ObjectModification:
2019_06_14-PM-03_28_18
Theory : reals
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