Nuprl Lemma : 0-comp-cc-fst-comp-m
∀[H:j⊢]. ([0(𝕀)] o p o m = m ∈ H.𝕀.𝕀, ((q=0))p j⟶ H.𝕀)
Proof
Definitions occuring in Statement :
csm-m: m,
context-subset: Gamma, phi,
face-zero: (i=0),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
csm-comp: G o F,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
cube-context-adjoin: X.A,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-m: m,
csm-comp: G o F,
compose: f o g,
cc-adjoin-cube: (v;u),
csm-id: 1(X),
csm-adjoin: (s;u),
pi1: fst(t),
csm-ap: (s)x,
uimplies: b supposing a,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
cubical-type-at: A(a),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
face-zero: (i=0),
csm-ap-term: (t)s,
cubical-term-at: u(a),
pi2: snd(t),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
DeMorgan-algebra: DeMorganAlgebra,
uiff: uiff(P;Q),
implies: P ⇒ Q
Lemmas referenced :
cubical_set_wf,
csm-equal,
context-subset_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
face-zero_wf,
cc-snd_wf,
csm-context-subset-subtype2,
csm-m_wf,
I_cube_pair_redex_lemma,
I_cube_wf,
fset_wf,
nat_wf,
dM0_wf,
subtype_rel_self,
cubical-type-at_wf,
istype-cubical-type-at,
dM-to-FL-eq-1,
dm-neg_wf,
names_wf,
names-deq_wf,
lattice-point_wf,
free-DeMorgan-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,
subtype_rel-equal,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
interval-type-at,
dma-neg-eq-1-implies-meet-eq-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
instantiate,
introduction,
extract_by_obid,
hypothesis,
equalitySymmetry,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :memTop,
equalityTransitivity,
because_Cache,
applyEquality,
functionExtensionality,
dependent_functionElimination,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
independent_functionElimination
Latex:
\mforall{}[H:j\mvdash{}]. ([0(\mBbbI{})] o p o m = m)
Date html generated:
2020_05_20-PM-04_42_23
Last ObjectModification:
2020_04_13-PM-08_47_12
Theory : cubical!type!theory
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