Nuprl Lemma : case-type-comp-partition
∀[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 ⊢ Compositon((if phi then A else B))
supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽)) ∧ Gamma ⊢ (1(𝔽) ⇒ (phi ∨ psi))
Proof
Definitions occuring in Statement :
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
composition-structure: Gamma ⊢ Compositon(A),
case-type: (if phi then A else B),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-and: (a ∧ b),
face-1: 1(𝔽),
face-0: 0(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
same-cubical-type: Gamma ⊢ A = B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
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
Lemmas referenced :
case-type-comp-disjoint,
composition-structure-subset,
context-subset_wf,
face-or_wf,
case-type_wf,
face-term-implies_wf,
face-and_wf,
face-0_wf,
face-1_wf,
composition-structure_wf,
cubical-type_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
face-1-implies-subset,
same-cubical-type-0,
subtype-context-subset-0,
context-subset-subtype,
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
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
sqequalRule,
axiomEquality,
productIsType,
universeIsType,
instantiate,
inhabitedIsType,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
equalityIstype,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
setElimination,
rename
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 \mvdash{} Compositon((if phi then A else B))
supposing Gamma \mvdash{} ((phi \mwedge{} psi) {}\mRightarrow{} 0(\mBbbF{})) \mwedge{} Gamma \mvdash{} (1(\mBbbF{}) {}\mRightarrow{} (phi \mvee{} psi))
Date html generated:
2020_05_20-PM-05_19_29
Last ObjectModification:
2020_04_18-PM-07_58_52
Theory : cubical!type!theory
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