Nuprl Lemma : case-term_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
(u ∨ v) ∈ {Gamma, (phi ∨ psi) ⊢ _:A} supposing Gamma, (phi ∧ psi) ⊢ u=v:A
Proof
Definitions occuring in Statement :
case-term: (u ∨ v),
same-cubical-term: X ⊢ u=v:A,
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cubical-term: {X ⊢ _:A},
all: ∀x:A. B[x],
case-term: (u ∨ v),
context-subset: Gamma, phi,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: 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,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
same-cubical-term: X ⊢ u=v:A
Lemmas referenced :
context-subset-subtype-or,
context-subset-subtype-or2,
context-subset-subtype-and,
I_cube_wf,
context-subset_wf,
face-or_wf,
names-hom_wf,
fset_wf,
nat_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
same-cubical-term_wf,
face-and_wf,
cubical-type-cumulativity2,
subset-cubical-term2,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
cubical-term_wf,
cubical_set_cumulativity-i-j,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
face-or-eq-1,
fl-eq_wf,
cubical-term-at_wf,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
subset-cubical-type,
face-term-implies-or1,
face-term-implies-or2,
cube_set_restriction_pair_lemma,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
iff_weakening_equal,
squash_wf,
istype-universe,
face-term-at-restriction-eq-1,
istype-true,
cubical-term-at-morph,
face-and-at,
lattice-meet-idempotent,
bdd-distributive-lattice-subtype-lattice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesisEquality,
applyEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
universeIsType,
inhabitedIsType,
functionIsType,
equalityIstype,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
productEquality,
cumulativity,
isectEquality,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
independent_pairFormation,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, (phi \mvee{} psi) \mvdash{} \_\}]. \mforall{}[u:\{Gamma, phi \mvdash{} \_:A\}].
\mforall{}[v:\{Gamma, psi \mvdash{} \_:A\}].
(u \mvee{} v) \mmember{} \{Gamma, (phi \mvee{} psi) \mvdash{} \_:A\} supposing Gamma, (phi \mwedge{} psi) \mvdash{} u=v:A
Date html generated:
2020_05_20-PM-03_10_16
Last ObjectModification:
2020_04_07-PM-00_51_30
Theory : cubical!type!theory
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