Nuprl Lemma : tag-by-subtype-tag-case
∀[T,S:Type]. ∀z,x:Atom. (z×T ⊆r x:S ⇐⇒ (¬↑z =a x) ∨ (T ⊆r S))
Proof
Definitions occuring in Statement :
tag-by: z×T,
tag-case: z:T,
assert: ↑b,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
not: ¬A,
true: True,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
tag-by: z×T,
tag-case: z:T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
top: Top,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
istype-void,
subtype_rel_wf,
tag-by_wf,
tag-case_wf,
subtype_rel_product,
equal-wf-base,
atom_subtype_base,
ifthenelse_wf,
top_wf,
istype-atom,
istype-universe,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
inrFormation_alt,
lambdaEquality_alt,
universeIsType,
because_Cache,
independent_functionElimination,
natural_numberEquality,
voidElimination,
axiomEquality,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
inlFormation_alt,
setEquality,
atomEquality,
applyEquality,
setIsType,
universeEquality,
setElimination,
rename,
isect_memberEquality_alt,
unionIsType,
functionIsType,
dependent_set_memberEquality_alt,
productIsType,
applyLambdaEquality,
sqequalBase,
independent_pairEquality
Latex:
\mforall{}[T,S:Type]. \mforall{}z,x:Atom. (z\mtimes{}T \msubseteq{}r x:S \mLeftarrow{}{}\mRightarrow{} (\mneg{}\muparrow{}z =a x) \mvee{} (T \msubseteq{}r S))
Date html generated:
2019_10_15-AM-11_29_46
Last ObjectModification:
2019_06_25-PM-01_21_26
Theory : general
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