Nuprl Lemma : case-type-same1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[psi:{Gamma ⊢ _:𝔽}]. ∀[B:{Gamma, psi ⊢ _}].
Gamma, phi ⊢ (if phi then A else B) = A
Proof
Definitions occuring in Statement :
case-type: (if phi then A else B),
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-type: 𝔽,
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,
uimplies: b supposing a,
case-type: (if phi then A else B),
context-subset: Gamma, phi,
all: ∀x:A. B[x],
case-cube: case-cube(phi;A;B;I;rho),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
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],
eq_atom: x =a y,
bfalse: ff,
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),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
respects-equality: respects-equality(S;T),
squash: ↓T,
cubical-type: {X ⊢ _}
Lemmas referenced :
cubical-type-equal,
cubical-type_wf,
context-subset_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
I_cube_wf,
fset_wf,
nat_wf,
fl-eq_wf,
cubical-term-at_wf,
lattice-1_wf,
face_lattice_wf,
eqtt_to_assert,
assert-fl-eq,
subtype_rel_self,
lattice-point_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,
cubical-type-at_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
names-hom_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
iff_weakening_equal,
istype-true,
subset-cubical-term,
context-subset-is-subset,
cube-set-restriction_wf,
face-type-at,
respects-equality_weakening,
squash_wf,
istype-universe,
face-term-at-restriction-eq-1,
cubical-type-ap-morph_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
istype-cubical-type-at,
cubical_type_ap_morph_pair_lemma,
cubical_type_at_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
dependent_pairEquality_alt,
lambdaEquality_alt,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
equalityTransitivity,
applyEquality,
because_Cache,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
dependent_set_memberEquality_alt,
equalityIstype,
productEquality,
cumulativity,
isectEquality,
dependent_pairFormation_alt,
promote_hyp,
independent_functionElimination,
voidElimination,
independent_pairFormation,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
functionIsType,
functionExtensionality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
\mforall{}[B:\{Gamma, psi \mvdash{} \_\}].
Gamma, phi \mvdash{} (if phi then A else B) = A
Date html generated:
2020_05_20-PM-03_08_33
Last ObjectModification:
2020_04_06-PM-00_52_14
Theory : cubical!type!theory
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