Nuprl Lemma : case-cube_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[I:fset(ℕ)].
∀[rho:Gamma, (phi ∨ psi)(I)].
(case-cube(phi;A;B;I;rho) ∈ Type)
Proof
Definitions occuring in Statement :
case-cube: case-cube(phi;A;B;I;rho),
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
face-or: (a ∨ b),
cubical-term-at: u(a),
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
case-cube: case-cube(phi;A;B;I;rho),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
or: P ∨ Q
Lemmas referenced :
I_cube_pair_redex_lemma,
face_lattice-1-join-irreducible,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
I_cube_wf,
context-subset_wf,
face-or_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical-term_wf,
cubical_set_wf,
fl-eq_wf,
lattice-1_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
assert_wf,
eqtt_to_assert,
assert-fl-eq,
cubical-type-at_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
setElimination,
rename,
sqequalRule,
hypothesisEquality,
isectElimination,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
universeIsType,
independent_isectElimination,
productElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
baseClosed,
dependent_set_memberEquality_alt,
equalityIstype,
independent_pairFormation,
functionIsType,
voidElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[B:\{Gamma, psi \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})].
\mforall{}[rho:Gamma, (phi \mvee{} psi)(I)].
(case-cube(phi;A;B;I;rho) \mmember{} Type)
Date html generated:
2020_05_20-PM-03_07_26
Last ObjectModification:
2020_04_06-PM-00_51_27
Theory : cubical!type!theory
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