Nuprl Lemma : subtype_pos_polymorphism_test
((⋂T:Type. (ℙ ⟶ ℤ ⟶ (T × T + T))) ⊆r (ℙ ⟶ ℤ ⟶ (Void × Void + Void)))
∧ ((ℙ ⟶ ℤ ⟶ (Void × Void + Void)) ⊆r (⋂T:Type. (ℙ ⟶ ℤ ⟶ (T × T + T))))
Proof
Definitions occuring in Statement :
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
union: left + right,
int: ℤ,
void: Void,
universe: Type
Definitions unfolded in proof :
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_rel_dep_function,
subtype_rel_union,
subtype_rel_product
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaEquality,
isectElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis,
isectEquality,
universeEquality,
functionEquality,
cumulativity,
intEquality,
unionEquality,
productEquality,
hypothesisEquality,
thin,
isect_memberEquality,
cut,
applyEquality,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
sqequalRule,
because_Cache,
independent_isectElimination,
lambdaFormation,
voidElimination
Latex:
((\mcap{}T:Type. (\mBbbP{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} (T \mtimes{} T + T))) \msubseteq{}r (\mBbbP{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} (Void \mtimes{} Void + Void)))
\mwedge{} ((\mBbbP{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} (Void \mtimes{} Void + Void)) \msubseteq{}r (\mcap{}T:Type. (\mBbbP{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} (T \mtimes{} T + T))))
Date html generated:
2016_05_15-PM-07_49_13
Last ObjectModification:
2015_12_27-AM-11_07_20
Theory : general
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