Nuprl Lemma : has-value-equality-fix
∀[T,E,S:Type]. ∀[G:T ⟶ partial(E)]. ∀[g:T ⟶ T]. ((G fix(g))↓ ∈ ℙ) supposing value-type(E) ∧ (⊥ ∈ T)
Proof
Definitions occuring in Statement :
partial: partial(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 :
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
pi2: snd(t),
pi1: fst(t),
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
has-value: (a)↓,
nat: ℕ,
rev_implies: P ⇐ Q
Lemmas referenced :
pair-eta,
subtype_rel_product,
partial_wf,
top_wf,
istype-top,
istype-void,
has-value_wf_base,
member_wf,
istype-universe,
value-type_wf,
has-value-extensionality,
fun_exp0_lemma,
equal_wf,
squash_wf,
true_wf,
fun_exp_wf,
nat_wf,
le_weakening2,
le_wf,
subtype_rel_self,
iff_weakening_equal,
int_subtype_base,
istype-int,
less_than_wf,
primrec-wf2,
equal-wf-base,
has-value_wf-partial,
is-exception_wf,
fixpoint-upper-bound,
has-value-monotonic,
set_subtype_base
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairEquality,
hypothesisEquality,
Error :inhabitedIsType,
hypothesis,
Error :lambdaFormation_alt,
thin,
pointwiseFunctionality,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionEquality,
Error :lambdaEquality_alt,
because_Cache,
Error :functionIsType,
Error :universeIsType,
independent_isectElimination,
Error :isect_memberEquality_alt,
voidElimination,
baseApply,
closedConclusion,
baseClosed,
applyLambdaEquality,
independent_pairFormation,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
Error :equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
Error :productIsType,
Error :equalityIsType4,
Error :isect_memberFormation_alt,
axiomEquality,
compactness,
rename,
setElimination,
Error :dependent_set_memberEquality_alt,
Error :setIsType,
axiomSqleEquality,
hyp_replacement,
sqleRule,
divergentSqle,
sqleReflexivity,
intEquality
Latex:
\mforall{}[T,E,S:Type].
\mforall{}[G:T {}\mrightarrow{} partial(E)]. \mforall{}[g:T {}\mrightarrow{} T]. ((G fix(g))\mdownarrow{} \mmember{} \mBbbP{}) supposing value-type(E) \mwedge{} (\mbot{} \mmember{} T)
Date html generated:
2019_06_20-PM-00_34_02
Last ObjectModification:
2018_10_07-PM-09_32_01
Theory : partial_1
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