Nuprl Lemma : compatible-composition_wf
∀[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) ∈ ℙ{[i | j'']} supposing Gamma, (phi ∧ psi) ⊢ A = B
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),
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
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],
prop: ℙ,
member: t ∈ T
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),
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
same-cubical-type: Gamma ⊢ A = B,
btrue: tt,
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
face-lattice: face-lattice(T;eq),
face_lattice: face_lattice(I),
record-select: r.x,
lattice-point: Point(l),
face-presheaf: 𝔽,
functor-ob: ob(F),
I_cube: A(I),
constant-cubical-type: (X),
face-type: 𝔽,
pi1: fst(t),
cubical-type-at: A(a),
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
context-subset: Gamma, phi,
implies: P ⇒ Q,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
composition-function: composition-function{j:l,i:l}(Gamma;A),
composition-structure: Gamma ⊢ Compositon(A),
guard: {T},
true: True,
squash: ↓T,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
cubical_set_wf,
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
context-subset_wf,
face-and_wf,
face-type_wf,
cubical-term-eqcd,
csm-ap-type_wf,
subset-cubical-type,
face-term-implies-subset,
face-term-and-implies1,
context-subset-adjoin-subtype,
csm-id-adjoin_wf,
interval-0_wf,
csm-ap-term_wf,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
same-cubical-type_wf,
face-term-and-implies2,
composition-structure_wf,
cubical-type_wf,
istype-cubical-term,
sub_cubical_set_self,
cube_set_map_subtype3,
context-subset-map,
csm-face-type,
context-subset-is-subset,
subset-cubical-term,
context-subset-subtype-and2,
context-subset-subtype-and,
csm-same-cubical-type,
face-1-implies-subset,
face-1_wf,
csm-face-term-implies,
nat_wf,
fset_wf,
I_cube_wf,
lattice-1_wf,
subtype_rel_self,
cubical-term-at_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
face_lattice_wf,
lattice-point_wf,
I_cube_pair_redex_lemma,
csm-face-1,
csm-id-adjoin_wf-interval-1,
cubical_set_cumulativity-i-j,
equal_functionality_wrt_subtype_rel2,
sub_cubical_set_functionality,
true_wf,
squash_wf,
thin-context-subset-adjoin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
functionEquality,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
Error :memTop,
inhabitedIsType,
isectEquality,
productEquality,
equalityIstype,
rename,
setElimination,
dependent_functionElimination,
lambdaFormation_alt,
hyp_replacement,
independent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination
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) \mmember{} \mBbbP{}\{[i | j'']\}
supposing Gamma, (phi \mwedge{} psi) \mvdash{} A = B
Date html generated:
2020_05_20-PM-05_15_05
Last ObjectModification:
2020_05_02-PM-01_24_12
Theory : cubical!type!theory
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