Nuprl Lemma : same-cubical-type-zero-and-one
∀[G:j⊢]. ∀[A,B:{G, 0(𝔽) ⊢ _}]. ∀[i:{G ⊢ _:𝕀}]. G, ((i=0) ∧ (i=1)) ⊢ A = B
Proof
Definitions occuring in Statement :
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-and: (a ∧ b),
face-0: 0(𝔽),
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
same-cubical-type: Gamma ⊢ A = B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
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],
and: P ∧ Q,
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
same-cubical-type-0,
context-subset-subtype,
face-0_wf,
face-and_wf,
face-zero_wf,
face-one_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
face-type_wf,
face-zero-and-one,
subtype_rel_self,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
iff_weakening_equal,
lattice-1_wf,
interval-type_wf,
context-subset_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaFormation_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality,
cumulativity,
isectEquality,
productElimination,
independent_functionElimination,
equalityIstype,
setElimination,
rename,
inhabitedIsType,
axiomEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G, 0(\mBbbF{}) \mvdash{} \_\}]. \mforall{}[i:\{G \mvdash{} \_:\mBbbI{}\}]. G, ((i=0) \mwedge{} (i=1)) \mvdash{} A = B
Date html generated:
2020_05_20-PM-03_01_03
Last ObjectModification:
2020_04_06-AM-10_32_52
Theory : cubical!type!theory
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