Nuprl Lemma : compatible-composition-disjoint
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽))
Proof
Definitions occuring in Statement :
compatible-composition: compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB),
composition-structure: Gamma ⊢ Compositon(A),
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},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
compatible-composition: compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB),
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
not: ¬A,
implies: P ⇒ Q,
false: False,
cube-context-adjoin: X.A,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
so_apply: x[s]
Lemmas referenced :
map-to-context-subset-disjoint,
cube-context-adjoin_wf,
interval-type_wf,
istype-cubical-term,
face-type_wf,
cube_set_map_wf,
context-subset_wf,
face-and_wf,
face-term-implies_wf,
face-0_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-type_wf,
context-subset-subtype-and,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
csm-ap-term_wf,
empty-context-eq-lemma,
I_cube_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
interval-type-at-is-point,
lattice-0_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
bounded-lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-point_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
instantiate,
hypothesis,
independent_isectElimination,
universeIsType,
inhabitedIsType,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :memTop,
dependent_functionElimination,
rename,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality
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)].
compatible-composition\{j:l, i:l\}(Gamma; phi; psi; A; B; cA; cB)
supposing Gamma \mvdash{} ((phi \mwedge{} psi) {}\mRightarrow{} 0(\mBbbF{}))
Date html generated:
2020_05_20-PM-05_15_20
Last ObjectModification:
2020_04_18-PM-06_18_00
Theory : cubical!type!theory
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