Nuprl Lemma : same-cubical-type-by-cases
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A,B:{Gamma, (phi ∨ psi) ⊢ _}].
(Gamma, (phi ∨ psi) ⊢ A = B) supposing (Gamma, phi ⊢ A = B and Gamma, psi ⊢ A = B)
Proof
Definitions occuring in Statement :
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
same-cubical-type: Gamma ⊢ A = B,
all: ∀x:A. B[x],
context-subset: Gamma, phi,
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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-type-equal3,
context-subset_wf,
face-or_wf,
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,
fset_wf,
nat_wf,
istype-cubical-type-at,
names-hom_wf,
same-cubical-type_wf,
context-subset-subtype-or,
context-subset-subtype-or2,
cubical-type_wf,
cubical-term_wf,
cubical_set_wf,
cubical-type-at_wf,
squash_wf,
true_wf,
cube_set_restriction_pair_lemma,
cube-set-restriction_wf,
istype-universe,
face-term-at-restriction-eq-1,
lattice-1_wf,
iff_weakening_equal,
cubical-type-ap-morph_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
sqequalRule,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
universeIsType,
productElimination,
independent_functionElimination,
unionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
universeEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,B:\{Gamma, (phi \mvee{} psi) \mvdash{} \_\}].
(Gamma, (phi \mvee{} psi) \mvdash{} A = B) supposing (Gamma, phi \mvdash{} A = B and Gamma, psi \mvdash{} A = B)
Date html generated:
2020_05_20-PM-03_01_31
Last ObjectModification:
2020_04_04-PM-05_16_35
Theory : cubical!type!theory
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