Nuprl Lemma : pres-invariant
∀[G,H: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)].
(pres f [phi ⊢→ t] t0
= pres f [phi ⊢→ t] t0
∈ {G ⊢ _:(Path_(A)[1(𝕀)] pres-c1(G;phi;f;t;t0;cA) pres-c2(G;phi;f;t;t0;cT))})
supposing H = G ∈ CubicalSet{j}
Proof
Definitions occuring in Statement :
pres: pres f [phi ⊢→ t] t0,
pres-c2: pres-c2(G;phi;f;t;t0;cT),
pres-c1: pres-c1(G;phi;f;t;t0;cA),
composition-structure: Gamma ⊢ Compositon(A),
path-type: (Path_A a b),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm-id-adjoin: [u],
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
prop: ℙ,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
true: True,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
csm-ap: (s)x,
csm-adjoin: (s;u),
csm-id: 1(X),
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
interval-1: 1(𝕀),
cubical-type: {X ⊢ _}
Lemmas referenced :
context-subset-term-subtype,
cube-context-adjoin_wf,
interval-type_wf,
cubical-fun_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
cubical-app_wf_fun,
thin-context-subset,
cubical-fun-subset,
pres_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
subtype_rel_self,
composition-structure_wf,
equal_wf,
csm-ap-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
partial-term-0_wf,
constrained-cubical-term-eqcd,
istype-cubical-term,
context-subset_wf,
cubical-type_wf,
cubical_set_wf,
cubical-term-eqcd,
path-type_wf,
csm-id-adjoin_wf-interval-1,
composition-function-cumulativity,
pres-c1_wf,
pres-c2_wf,
cubical-term_wf,
squash_wf,
true_wf,
iff_weakening_equal,
istype-universe,
cube_set_map_wf,
subtype_rel-equal,
interval-1_wf,
cubical-type-cumulativity,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term_wf,
composition-function_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesisEquality,
applyEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
instantiate,
hypothesis,
sqequalRule,
Error :memTop,
because_Cache,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productIsType,
equalityIstype,
inhabitedIsType,
hyp_replacement,
applyLambdaEquality,
universeIsType,
independent_isectElimination,
dependent_functionElimination,
rename,
setElimination,
lambdaEquality_alt,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
universeEquality
Latex:
\mforall{}[G,H: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)].
(pres f [phi \mvdash{}\mrightarrow{} t] t0 = pres f [phi \mvdash{}\mrightarrow{} t] t0)
supposing H = G
Date html generated:
2020_05_20-PM-05_32_52
Last ObjectModification:
2020_05_02-PM-03_55_40
Theory : cubical!type!theory
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