Nuprl Lemma : same-cubical-term-by-cases
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[a,b:{Gamma, (phi ∨ psi) ⊢ _:A}].
  (Gamma, (phi ∨ psi) ⊢ a=b:A) supposing (Gamma, phi ⊢ a=b:A and Gamma, psi ⊢ a=b:A)
Proof
Definitions occuring in Statement : 
same-cubical-term: X ⊢ u=v:A
, 
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]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
same-cubical-term: X ⊢ u=v:A
, 
subtype_rel: A ⊆r B
, 
cubical-term-at: u(a)
, 
context-subset: Gamma, phi
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
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
, 
so_apply: x[s]
Lemmas referenced : 
subset-cubical-term, 
context-subset_wf, 
face-or_wf, 
face-term-implies-subset, 
face-term-implies-or1, 
face-term-implies-or2, 
same-cubical-term_wf, 
subset-cubical-type, 
cubical-term_wf, 
cubical-type-cumulativity2, 
cubical_set_cumulativity-i-j, 
cubical-type_wf, 
face-type_wf, 
cubical_set_wf, 
I_cube_wf, 
fset_wf, 
nat_wf, 
cubical-term-equal, 
I_cube_pair_redex_lemma, 
face-or-eq-1, 
cubical-term-at_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
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
independent_isectElimination, 
because_Cache, 
universeIsType, 
applyEquality, 
sqequalRule, 
instantiate, 
functionExtensionality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_functionElimination, 
Error :memTop, 
setElimination, 
rename, 
productElimination, 
independent_functionElimination, 
unionElimination, 
dependent_set_memberEquality_alt, 
equalityIstype, 
inhabitedIsType, 
lambdaEquality_alt, 
productEquality, 
cumulativity, 
isectEquality, 
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}].  \mforall{}[phi,psi:\{Gamma  \mvdash{}  \_:\mBbbF{}\}].  \mforall{}[A:\{Gamma,  (phi  \mvee{}  psi)  \mvdash{}  \_\}].
\mforall{}[a,b:\{Gamma,  (phi  \mvee{}  psi)  \mvdash{}  \_:A\}].
    (Gamma,  (phi  \mvee{}  psi)  \mvdash{}  a=b:A)  supposing  (Gamma,  phi  \mvdash{}  a=b:A  and  Gamma,  psi  \mvdash{}  a=b:A)
Date html generated:
2020_05_20-PM-03_01_44
Last ObjectModification:
2020_04_06-AM-10_32_41
Theory : cubical!type!theory
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