Nuprl Lemma : presw-pres-c1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 ⊢ Compositon(A)].
G ⊢ (comp (cA)p+ [((phi)p ∨ (q=1)) ⊢→ (presw(G;phi;f;t;t0;cT))p+] (pres-a0(G;f;t0))p)[0(𝕀)]=pres-c1(G;phi;f;t;t0;cA):
(A)[1(𝕀)]
Proof
Definitions occuring in Statement :
presw: presw(G;phi;f;t;t0;cT),
pres-c1: pres-c1(G;phi;f;t;t0;cA),
pres-a0: pres-a0(G;f;t0),
comp_term: comp cA [phi ⊢→ u] a0,
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
partial-term-0: u[0],
same-cubical-term: X ⊢ u=v:A,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-one: (i=1),
face-or: (a ∨ b),
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm+: tau+,
csm-id-adjoin: [u],
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
same-cubical-term: X ⊢ u=v:A,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
uimplies: b supposing a,
composition-structure: Gamma ⊢ Compositon(A),
all: ∀x:A. B[x],
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
csm-ap-term: (t)s,
pi1: fst(t),
pi2: snd(t),
csm-id: 1(X),
interval-0: 0(𝕀),
csm-id-adjoin: [u],
interval-1: 1(𝕀),
cubical-type: {X ⊢ _},
pres-a0: pres-a0(G;f;t0),
pres-c1: pres-c1(G;phi;f;t;t0;cA),
csm-ap: (s)x,
compose: f o g,
csm-adjoin: (s;u),
csm-comp-structure: (cA)tau,
csm-comp: G o F,
csm+: tau+,
true: True,
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),
face-type: 𝔽,
cubical-type-at: A(a),
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
cubical-term-at: u(a),
face-or: (a ∨ b),
face-one: (i=1),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presw: presw(G;phi;f;t;t0;cT),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-type: Gamma ⊢ A = B,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
cube-context-adjoin: X.A,
context-subset: Gamma, phi,
cc-adjoin-cube: (v;u),
partial-term-0: u[0],
face-term-implies: Gamma ⊢ (phi ⇒ psi)
Lemmas referenced :
csm+_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-interval-type,
context-subset-term-subtype,
cube-context-adjoin_wf,
cubical-fun_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cubical-app_wf_fun,
thin-context-subset,
cubical-fun-subset,
subset-cubical-term,
context-subset_wf,
face-or_wf,
face-one_wf,
cc-snd_wf,
sub_cubical_set-cumulativity1,
sub_cubical_set_functionality,
context-subset-is-subset,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
composition-structure_wf,
csm-id-adjoin_wf,
interval-0_wf,
partial-term-0_wf,
constrained-cubical-term-eqcd,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
interval-1_wf,
presw_wf,
composition-function-cumulativity,
pres-a0-constraint,
cubical-type-cumulativity2,
subtype_rel_self,
csm-comp-structure_wf,
csm-comp_term,
csm+_wf_interval,
csm-comp-structure-composition-function,
comp_term_wf,
csm-face-or,
cubical-term-equal,
nat_wf,
fset_wf,
I_cube_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-join-0,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
lattice-point_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,
cubical-term-at_wf,
cubical-type-at_wf_face-type,
csm_id_adjoin_fst_term_lemma,
squash_wf,
true_wf,
istype-universe,
dM-to-FL-dM0,
iff_weakening_equal,
pres-v_wf,
csm-cubical-app,
csm-id-adjoin_wf-interval-1,
cubical-term-eqcd,
csm-cubical-fun,
face-term-implies-same,
csm-ap-id-term,
face-term-implies_wf,
csm-ap-term-wf-subset,
composition-function_wf,
thin-context-subset-adjoin,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term_wf,
cube_set_map_wf,
cube_set_map_cumulativity-i-j,
csm-comp-assoc,
csm-comp_wf,
csm-id-comp,
csm+-id,
csm-id_wf,
csm+-comp-csm+-interval,
cc-fst-csm-adjoin,
context-adjoin-subset4,
I_cube_pair_redex_lemma,
cc-adjoin-cube_wf,
pres-a0_wf,
csm-comp-term,
csm-context-subset-subtype2,
subset-cubical-type,
lattice-1_wf,
csm-ap-id-type,
sub_cubical_set_self,
subset-cubical-term2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
Error :memTop,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
setElimination,
rename,
dependent_functionElimination,
lambdaEquality_alt,
cumulativity,
universeEquality,
productElimination,
functionExtensionality,
natural_numberEquality,
isectEquality,
productEquality,
imageElimination,
inhabitedIsType,
imageMemberEquality,
baseClosed,
independent_functionElimination,
equalityIstype,
lambdaFormation_alt,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[f:\{G.\mBbbI{} \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[t:\{G.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[t0:\{G \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} t[0]]\}]. \mforall{}[cT:G.\mBbbI{} +\mvdash{} Compositon(T)]. \mforall{}[cA:G.\mBbbI{} \mvdash{} Compositon(A)].
G \mvdash{} (comp (cA)p+ [((phi)p \mvee{} (q=1)) \mvdash{}\mrightarrow{} (presw(G;phi;f;t;t0;cT))p+]
(pres-a0(G;f;t0))p)[0(\mBbbI{})]=pres-c1(G;phi;f;t;t0;cA):(A)[1(\mBbbI{})]
Date html generated:
2020_05_20-PM-05_28_50
Last ObjectModification:
2020_05_02-PM-03_33_55
Theory : cubical!type!theory
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