Nuprl Lemma : map-to-context-subset-disjoint
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[H:j⊢].
∀[sigma:H j⟶ Gamma, (phi ∧ psi)]. ∀[I:fset(ℕ)]. (¬H(I)) supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽))
Proof
Definitions occuring in Statement :
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-and: (a ∧ b),
face-0: 0(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cube_set_map: A ⟶ B,
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
context-subset: Gamma, phi,
type-cat: TypeCat,
cube-cat: CubeCat,
op-cat: op-cat(C),
spreadn: spread4,
all: ∀x:A. B[x],
I_cube: A(I),
prop: ℙ,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
cubical-term-at: u(a),
face-0: 0(𝔽),
lattice-0: 0,
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,
empty-fset: {},
nil: [],
it: ⋅
Lemmas referenced :
cat_arrow_triple_lemma,
ob_pair_lemma,
cat_ob_pair_lemma,
I_cube_wf,
fset_wf,
nat_wf,
cube_set_map_wf,
context-subset_wf,
face-and_wf,
face-term-implies_wf,
face-0_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
void-list-equality,
nil_wf,
face-lattice-0-not-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
setElimination,
rename,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
hypothesisEquality,
independent_functionElimination,
because_Cache,
voidElimination,
universeIsType,
isectElimination,
lambdaEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
voidEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[H:j\mvdash{}].
\mforall{}[sigma:H j{}\mrightarrow{} Gamma, (phi \mwedge{} psi)]. \mforall{}[I:fset(\mBbbN{})]. (\mneg{}H(I)) supposing Gamma \mvdash{} ((phi \mwedge{} psi) {}\mRightarrow{} 0(\mBbbF{}))
Date html generated:
2020_05_20-PM-03_07_13
Last ObjectModification:
2020_04_04-PM-05_23_46
Theory : cubical!type!theory
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