Nuprl Lemma : case-type-comp_wf1
∀[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)
∈ composition-function{j:l,i:l}(Gamma, (phi ∨ psi);(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),
composition-function: composition-function{j:l,i:l}(Gamma;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,
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
composition-function: composition-function{j:l,i:l}(Gamma;A),
subtype_rel: A ⊆r B,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x],
implies: P ⇒ Q,
context-subset: Gamma, phi,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
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,
csm-id-adjoin: [u],
csm-id: 1(X),
same-cubical-type: Gamma ⊢ A = B,
cand: A c∧ B,
interval-0: 0(𝕀),
csm-ap-term: (t)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
composition-structure: Gamma ⊢ Compositon(A),
guard: {T},
squash: ↓T,
interval-1: 1(𝕀),
compatible-composition: compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB),
true: True,
same-cubical-term: X ⊢ u=v:A,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-term: {X ⊢ _:A},
respects-equality: respects-equality(S;T),
case-type: (if phi then A else B),
cubical-term-at: u(a)
Lemmas referenced :
case-type_wf,
csm-face-or,
csm-face-term-implies,
context-subset_wf,
face-or_wf,
face-1_wf,
subset-cubical-term,
context-subset-is-subset,
face-type_wf,
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,
subtype_rel_self,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-face-1,
face-forall_wf,
csm-ap-term_wf,
csm-face-type,
csm-same-cubical-type,
face-and_wf,
context-subset-subtype-and,
context-subset-subtype-and2,
csm-face-and,
context-subset-map,
cube_set_map_subtype3,
sub_cubical_set_self,
csm-ap-type_wf,
subset-cubical-type,
sub_cubical_set_functionality,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical-term-eqcd,
cube_set_map_wf,
cubical_set_wf,
compatible-composition_wf,
same-cubical-type_wf,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
composition-structure_wf,
cubical-type_wf,
istype-cubical-term,
csm-case-type,
context-iterated-subset1,
case-type-same1,
sub_cubical_set_transitivity,
context-iterated-subset2,
cc-fst_wf_interval,
context-adjoin-subset2,
face-forall-implies,
context-subset-swap,
sub_cubical_set_functionality2,
csm-id-adjoin_wf,
interval-0_wf,
face-forall-implies-0,
thin-context-subset,
context-subset-term-subtype,
face-forall-map,
case-type-same2,
interval-1_wf,
cube_set_map_subtype2,
face-term-and-implies,
face-forall-implies-1,
case-term-wf4,
squash_wf,
true_wf,
face-forall-and,
context-adjoin-subset,
istype-universe,
cubical-term_wf,
composition-in-subset,
same-cubical-type-by-cases,
face-term-or-implies,
face-term-implies-or,
face-term-implies_wf,
csm-id-adjoin-subset,
iff_weakening_equal,
csm-id-adjoin_wf-interval-1,
face-1-implies-subset,
sub_cubical_set_wf,
face-forall-or,
implies-face-forall-holds,
context-iterated-subset0,
subset-cubical-term2,
respects-equality-set,
cubical-type-at_wf,
all_wf,
names-hom_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
subset-I_cube,
subtype-respects-equality,
subtype_rel_dep_function,
respects-equality-context-subset-term,
same-cubical-term-by-cases,
case-term_wf2,
cubical_type_ap_morph_pair_lemma,
case-term-equal-right,
context-iterated-subset,
case-term-equal-left
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,
functionExtensionality,
sqequalRule,
rename,
Error :memTop,
applyEquality,
lambdaFormation_alt,
dependent_functionElimination,
setElimination,
equalityIstype,
universeIsType,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
universeEquality,
hyp_replacement,
independent_functionElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productElimination,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
inrFormation_alt,
inlFormation_alt,
functionEquality,
functionIsType
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{} composition-function\{j:l,i:l\}(Gamma, (phi \mvee{} psi);(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_17_00
Last ObjectModification:
2020_04_18-PM-06_13_32
Theory : cubical!type!theory
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