Nuprl Lemma : csm-m-comp-0
∀[H:j⊢]. ([0(𝕀)] o p = m o [0(𝕀)] ∈ H.𝕀 ij⟶ H.𝕀)
Proof
Definitions occuring in Statement :
csm-m: m,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
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,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
cube-context-adjoin: X.A,
csm-m: m,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-comp: G o F,
cc-fst: p,
compose: f o g,
pi1: fst(t),
csm-adjoin: (s;u),
csm-ap: (s)x,
csm-id: 1(X),
cc-adjoin-cube: (v;u),
uimplies: b supposing a,
interval-presheaf: 𝕀,
guard: {T},
and: P ∧ Q,
squash: ↓T,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dM0: 0
Lemmas referenced :
cubical_set_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-comp_wf,
cc-fst_wf,
csm-id-adjoin_wf-interval-0,
csm-m_wf,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
istype-cubical-type-at,
I_cube_wf,
fset_wf,
nat_wf,
csm-equal2,
interval-type-at,
lattice_properties,
dM_wf,
bdd-distributive-lattice-subtype-lattice,
DeMorgan-algebra-subtype,
subtype_rel_transitivity,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
lattice_wf,
equal_wf,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM0_wf,
iff_weakening_equal,
lattice-meet-0,
bdd-distributive-lattice-subtype-bdd-lattice,
bdd-lattice_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
instantiate,
introduction,
extract_by_obid,
hypothesis,
thin,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
productElimination,
dependent_pairEquality_alt,
independent_isectElimination,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
productEquality,
cumulativity,
isectEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_functionElimination
Latex:
\mforall{}[H:j\mvdash{}]. ([0(\mBbbI{})] o p = m o [0(\mBbbI{})])
Date html generated:
2020_05_20-PM-04_42_38
Last ObjectModification:
2020_04_13-PM-00_59_52
Theory : cubical!type!theory
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