Nuprl Lemma : has-value-equality-fix-bar
∀[T,E,S:Type]. ∀[G:T ⟶ bar(E)]. ∀[g:T ⟶ T]. ((G fix(g))↓ ∈ ℙ) supposing value-type(E) ∧ (⊥ ∈ T)
Proof
Definitions occuring in Statement :
bar: bar(T),
value-type: value-type(T),
has-value: (a)↓,
bottom: ⊥,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
fix: fix(F),
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
pi2: snd(t),
pi1: fst(t),
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
has-value: (a)↓,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
true: True,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
bar_wf,
value-type_wf,
pair-eta,
subtype_rel_product,
top_wf,
istype-top,
istype-void,
has-value_wf_base,
pi1_wf_top,
member_wf,
has-value-extensionality,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
fun_exp0_lemma,
subtract-1-ge-0,
equal_wf,
fun_exp_wf,
squash_wf,
true_wf,
nat_wf,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
decidable__le,
le_wf,
iff_weakening_equal,
has-value_wf-bar,
is-exception_wf,
fixpoint-upper-bound,
has-value-monotonic,
set_subtype_base,
int_subtype_base
Rules used in proof :
functionIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
universeIsType,
sqequalRule,
productIsType,
equalityIsType4,
baseClosed,
because_Cache,
universeEquality,
isect_memberFormation_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
independent_pairEquality,
lambdaFormation_alt,
pointwiseFunctionality,
productElimination,
applyEquality,
functionEquality,
lambdaEquality_alt,
closedConclusion,
independent_isectElimination,
voidElimination,
baseApply,
applyLambdaEquality,
independent_pairFormation,
instantiate,
imageElimination,
compactness,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
functionIsTypeImplies,
unionElimination,
dependent_set_memberEquality_alt,
imageMemberEquality,
axiomSqleEquality,
equalityIsType1,
hyp_replacement,
sqleRule,
divergentSqle,
sqleReflexivity,
intEquality
Latex:
\mforall{}[T,E,S:Type]. \mforall{}[G:T {}\mrightarrow{} bar(E)]. \mforall{}[g:T {}\mrightarrow{} T]. ((G fix(g))\mdownarrow{} \mmember{} \mBbbP{}) supposing value-type(E) \mwedge{} (\mbot{} \mmember{} T)
Date html generated:
2019_10_16-AM-11_37_45
Last ObjectModification:
2018_10_11-PM-11_40_03
Theory : bar!type
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