Nuprl Lemma : equiv-path2-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)].
((equiv-path2(G;A;B;cA;cB;f))[1(𝕀)] = cB ∈ G +⊢ Compositon(B))
Proof
Definitions occuring in Statement :
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
cubical-equiv: Equiv(T;A),
interval-1: 1(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
subtype_rel: A ⊆r B,
csm-comp-structure: (cA)tau,
csm-comp: G o F,
compose: f o g,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
pi2: snd(t),
prop: ℙ,
pi1: fst(t),
cubical-type: {X ⊢ _},
face-term-implies: Gamma ⊢ (phi ⇒ psi),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type-at: A(a),
face-type: 𝔽,
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
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,
same-cubical-type: Gamma ⊢ A = B
Lemmas referenced :
cc-snd_wf,
interval-type_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
cc-fst_wf_interval,
case-type_wf,
face-zero_wf,
face-one_wf,
thin-context-subset,
same-cubical-type-zero-and-one,
face-0_wf,
csm-glue-comp-agrees,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-1,
csm-comp-structure_wf2,
face-or_wf,
cubical-equiv-by-cases_wf,
composition-structure_wf,
istype-cubical-term,
cubical-equiv_wf,
cubical-type_wf,
cubical_set_wf,
subset-cubical-type,
context-subset_wf,
context-subset-is-subset,
case-type-comp-disjoint,
csm-comp-structure_wf,
csm-context-subset-subtype2,
face-term-implies_wf,
face-zero-and-one,
iff_weakening_equal,
face-term-implies-same,
csm-face-or,
csm-face-zero,
csm-face-one,
face-one-interval-1,
interval-1_wf,
face-term-implies-or2,
squash_wf,
true_wf,
face-type_wf,
subtype_rel_self,
csm-case-type-comp,
case-type-comp-false-true,
csm-id_wf,
composition-structure-subset,
face-zero-interval-1,
lattice-point_wf,
face_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,
cubical-term-at_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-universe,
subtype_rel_wf,
csm-comp-structure-id,
csm-ap-id-type,
sub_cubical_set_self,
csm-case-type,
case-type-same2,
face-1_wf,
context-1-subset,
same-cubical-type-0,
face-and_wf,
face-term-implies-subset,
face-term-and-implies1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
instantiate,
because_Cache,
independent_isectElimination,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
Error :memTop,
universeEquality,
setElimination,
rename,
productEquality,
cumulativity,
isectEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(A;B)\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)].
\mforall{}[cB:G +\mvdash{} Compositon(B)].
((equiv-path2(G;A;B;cA;cB;f))[1(\mBbbI{})] = cB)
Date html generated:
2020_05_20-PM-07_29_03
Last ObjectModification:
2020_04_28-PM-04_51_49
Theory : cubical!type!theory
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