Nuprl Lemma : case-type_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}].
Gamma, (phi ∨ psi) ⊢ (if phi then A else B) supposing Gamma, (phi ∧ psi) ⊢ A = B
Proof
Definitions occuring in Statement :
case-type: (if phi then A else B),
same-cubical-type: Gamma ⊢ A = B,
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,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
case-type: (if phi then A else B),
context-subset: Gamma, phi,
face-or: (a ∨ b),
cubical-term-at: u(a),
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],
so_apply: x[s],
case-cube: case-cube(phi;A;B;I;rho),
istype: istype(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
respects-equality: respects-equality(S;T),
face-and: (a ∧ b),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
same-cubical-type: Gamma ⊢ A = B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cube-set-restriction: f(s),
pi2: snd(t)
Lemmas referenced :
fset_wf,
nat_wf,
I_cube_wf,
context-subset_wf,
face-or_wf,
nh-id_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
names-hom_wf,
nh-comp_wf,
cube-set-restriction-comp,
same-cubical-type_wf,
face-and_wf,
subset-cubical-type,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
cubical-type_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
case-cube_wf,
I_cube_pair_redex_lemma,
face_lattice-1-join-irreducible,
cubical-term-at_wf,
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,
lattice-meet_wf,
lattice-join_wf,
face-type-at,
fl-eq_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
face-or-eq-1,
respects-equality_weakening,
face-term-at-restriction-eq-1,
cube_set_restriction_pair_lemma,
cubical-type-ap-morph_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
face-type-ap-morph,
fl-morph-1,
cubical-term-at-morph,
istype-cubical-type-at,
cubical-type-at_wf_face-type,
lattice-meet-eq-1,
bdd-distributive-lattice-subtype-bdd-lattice,
context-subset-restriction,
subtype_rel_wf,
cubical-type-at_wf,
cubical-type-ap-morph-id,
fl-morph_wf,
cubical-type-ap-morph-comp-general,
subset-I_cube,
context-subset-is-subset,
subtype_rel_weakening,
ext-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
productElimination,
thin,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
hypothesisEquality,
applyEquality,
equalityIstype,
because_Cache,
independent_isectElimination,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_pairEquality_alt,
Error :memTop,
setElimination,
rename,
productEquality,
cumulativity,
isectEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
sqequalIntensionalEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[B:\{Gamma, psi \mvdash{} \_\}].
Gamma, (phi \mvee{} psi) \mvdash{} (if phi then A else B) supposing Gamma, (phi \mwedge{} psi) \mvdash{} A = B
Date html generated:
2020_05_20-PM-03_08_03
Last ObjectModification:
2020_04_19-PM-01_55_23
Theory : cubical!type!theory
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