Nuprl Lemma : mk-dp_wf
∀[self:DualPlanePrimitives]. ∀[sqs:∀x,y:Vec. SqStable((x # y))]. ∀[st:∀x,y:Vec. Stable{(x ⊥ y)}].
∀[sor:∀x:Vec. ∀y:{y:Vec| (x # y)} . ∀z:Vec. ((z # x) ∨ (z # y))]. ∀[crs:∀a:Vec. ∀b:{b:Vec| (a # b)} .
(∃c:Vec [((a ⊥ c) ∧ (b ⊥ c))])].
∀[nt:∀a:Vec. (∃c:Vec [((a ⊥ c) ∧ (a # c))])].
(mk-dp(self;sqs;st;sor;crs;nt) ∈ DualPlaneStructure)
Proof
Definitions occuring in Statement :
mk-dp: mk-dp(p;sqs;st;sor;crs;nt),
dual-plane-structure: DualPlaneStructure,
dp-perp: (x ⊥ y),
dp-sep: (x # y),
dp-vec: Vec,
dual-plane-primitives: DualPlanePrimitives,
sq_stable: SqStable(P),
stable: Stable{P},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mk-dp: mk-dp(p;sqs;st;sor;crs;nt),
dual-plane-structure: DualPlaneStructure,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
or: P ∨ Q,
record+: record+,
record-update: r[x := v],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
record-select: r.x,
dp-sep: (x # y),
dp-vec: Vec,
top: Top,
eq_atom: x =a y,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
dp-perp: (x ⊥ y),
dual-plane-primitives: DualPlanePrimitives,
record: record(x.T[x])
Lemmas referenced :
dp-vec_wf,
sq_exists_wf,
dp-perp_wf,
dp-sep_wf,
stable_wf,
sq_stable_wf,
dual-plane-primitives_wf,
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
atom_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
istype-void,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-atom,
subtype_rel_self,
subtype_rel_universe1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
productEquality,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
setIsType,
setElimination,
rename,
unionIsType,
dependentIntersection_memberEquality,
functionExtensionality_alt,
tokenEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
atomEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
instantiate,
cumulativity,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
equalityIsType4,
equalityIsType1,
functionExtensionality,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
universeEquality,
functionEquality
Latex:
\mforall{}[self:DualPlanePrimitives]. \mforall{}[sqs:\mforall{}x,y:Vec. SqStable((x \# y))]. \mforall{}[st:\mforall{}x,y:Vec. Stable\{(x \mbot{} y)\}].
\mforall{}[sor:\mforall{}x:Vec. \mforall{}y:\{y:Vec| (x \# y)\} . \mforall{}z:Vec. ((z \# x) \mvee{} (z \# y))]. \mforall{}[crs:\mforall{}a:Vec. \mforall{}b:\{b:Vec| (a \# b)\}\000C .
(\mexists{}c:Vec [((a \mbot{} c)
\mwedge{} (b \mbot{} c))])].
\mforall{}[nt:\mforall{}a:Vec. (\mexists{}c:Vec [((a \mbot{} c) \mwedge{} (a \# c))])].
(mk-dp(self;sqs;st;sor;crs;nt) \mmember{} DualPlaneStructure)
Date html generated:
2019_10_16-AM-11_29_32
Last ObjectModification:
2018_10_10-PM-02_14_49
Theory : matrices
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