Nuprl Lemma : fix_wf1
∀[F:{F:ℕ ⟶ Type| Top ⊆r (F 0)} ]. ∀[G:⋂n:ℕ. ((F n) ⟶ (F (n + 1)))]. (fix(G) ∈ ⋂n:ℕ. (F n))
Proof
Definitions occuring in Statement :
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
subtype_rel_self,
subtract-add-cancel,
false_wf,
top_wf,
subtype_rel_wf,
le_wf,
int_term_value_add_lemma,
itermAdd_wf,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
isect_memberEquality,
setElimination,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
isectEquality,
functionEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[F:\{F:\mBbbN{} {}\mrightarrow{} Type| Top \msubseteq{}r (F 0)\} ]. \mforall{}[G:\mcap{}n:\mBbbN{}. ((F n) {}\mrightarrow{} (F (n + 1)))]. (fix(G) \mmember{} \mcap{}n:\mBbbN{}. (F n))
Date html generated:
2016_05_15-PM-04_12_51
Last ObjectModification:
2016_01_16-AM-11_07_00
Theory : general
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