Nuprl Lemma : case-type-comp_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) ∈ Gamma, (phi ∨ psi) ⊢ Compositon((if phi then A else B))) supposing
(compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) and
Gamma, (phi ∧ psi) ⊢ A = B)
Proof
Definitions occuring in Statement :
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
compatible-composition: compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB),
composition-structure: Gamma ⊢ Compositon(A),
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],
uimplies: b supposing a,
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
composition-function: composition-function{j:l,i:l}(Gamma;A),
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
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,
same-cubical-type: Gamma ⊢ A = B,
cand: A c∧ B,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-0: 0(𝕀),
csm-ap-term: (t)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
squash: ↓T,
interval-type: 𝕀,
csm+: tau+,
csm-comp: G o F,
cc-snd: q,
cc-fst: p,
compose: f o g,
true: True,
interval-1: 1(𝕀),
case-type: (if phi then A else B),
cubical-term-at: u(a),
pi2: snd(t),
case-term: (u ∨ v),
case-cube: case-cube(phi;A;B;I;rho),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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,
face-1: 1(𝔽),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
respects-equality: respects-equality(S;T)
Lemmas referenced :
case-type_wf,
case-type-comp_wf1,
constrained-cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
context-subset_wf,
face-or_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
thin-context-subset-adjoin,
istype-cubical-term,
face-type_wf,
cube_set_map_wf,
uniform-comp-function_wf,
compatible-composition_wf,
same-cubical-type_wf,
face-and_wf,
subset-cubical-type,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
subtype_rel_self,
cubical-term-eqcd,
csm-id-adjoin_wf-interval-1,
csm-case-term,
csm-case-type,
csm-face-or,
csm-face-term-implies,
face-1_wf,
subset-cubical-term,
context-subset-is-subset,
I_cube_pair_redex_lemma,
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,
cubical-term-at_wf,
lattice-1_wf,
csm-face-1,
face-forall_wf,
csm-face-type,
csm-same-cubical-type,
context-subset-subtype-and,
context-subset-subtype-and2,
csm-face-and,
context-subset-map,
cube_set_map_subtype3,
sub_cubical_set_self,
sub_cubical_set_functionality,
context-iterated-subset1,
case-type-same1,
sub_cubical_set_transitivity,
context-iterated-subset2,
cc-fst_wf_interval,
context-adjoin-subset2,
face-forall-implies,
csm-id-adjoin_wf,
interval-0_wf,
face-forall-implies-0,
sub_cubical_set_functionality2,
thin-context-subset,
context-subset-swap,
squash_wf,
true_wf,
csm-face-forall,
csm-comp_wf,
csm+_wf_interval,
face-forall-map,
context-subset-term-subtype,
case-type-same2,
cubical_type_ap_morph_pair_lemma,
cubical_type_at_pair_lemma,
csm+_wf,
subtype_rel-equal,
cubical-term_wf,
csm-interval-type,
istype-universe,
iff_weakening_equal,
face-type-at,
fl-eq_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,
face-forall-implies-1,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
face-term-implies_wf,
face-forall-or,
implies-face-forall-holds,
face-or-eq-1,
csm-comp-term,
subset-cubical-term2,
subtype-respects-equality,
cubical-type-at_wf,
interval-1_wf,
face-and-eq-1,
csm-context-subset-subtype2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
inhabitedIsType,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
rename,
promote_hyp,
functionEquality,
lambdaEquality_alt,
cumulativity,
universeEquality,
setElimination,
Error :memTop,
productEquality,
isectEquality,
independent_pairFormation,
hyp_replacement,
productElimination,
applyLambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
voidElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[B:\{Gamma, psi \mvdash{} \_\}].
\mforall{}[cA:Gamma, phi \mvdash{} Compositon(A)]. \mforall{}[cB:Gamma, psi \mvdash{} Compositon(B)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB)
\mmember{} Gamma, (phi \mvee{} psi) \mvdash{} Compositon((if phi then A else B))) supposing
(compatible-composition\{j:l, i:l\}(Gamma; phi; psi; A; B; cA; cB) and
Gamma, (phi \mwedge{} psi) \mvdash{} A = B)
Date html generated:
2020_05_20-PM-05_18_25
Last ObjectModification:
2020_04_18-PM-07_36_16
Theory : cubical!type!theory
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