Nuprl Lemma : fst-transprt-sigma
∀[X:j⊢]. ∀[A:{X.𝕀 ⊢ _}]. ∀[B:{X.𝕀.A ⊢ _}]. ∀[cA:X.𝕀 +⊢ Compositon(A)]. ∀[cB:X.𝕀.A +⊢ Compositon(B)].
∀[pr:{X ⊢ _:(Σ A B)[0(𝕀)]}].
(transprt(X;sigma_comp(cA;cB);pr).1 = transprt(X;cA;pr.1) ∈ {X ⊢ _:(A)[1(𝕀)]})
Proof
Definitions occuring in Statement :
sigma_comp: sigma_comp(cA;cB),
transprt: transprt(G;cA;a0),
composition-structure: Gamma ⊢ Compositon(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fst: p.1,
cubical-sigma: Σ A B,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-fst: p.1,
sigma_comp: sigma_comp(cA;cB),
transprt: transprt(G;cA;a0),
cubical-pair: cubical-pair(u;v),
comp_term: comp cA [phi ⊢→ u] a0,
let: let,
pi1: fst(t),
fill_term: fill cA [phi ⊢→ u] a0,
csm-ap-term: (t)s,
csm-comp-structure: (cA)tau,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
face-0: 0(𝔽),
interval-type: 𝕀,
csm-id: 1(X),
csm-comp: G o F,
csm-adjoin: (s;u),
compose: f o g,
constant-cubical-type: (X),
member: t ∈ T,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
all: ∀x:A. B[x],
cubical-type: {X ⊢ _},
cc-snd: q,
interval-0: 0(𝕀),
csm-ap-type: (AF)s,
cc-fst: p,
respects-equality: respects-equality(S;T),
implies: P ⇒ Q,
prop: ℙ,
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),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
composition-structure_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-id-type,
csm-comp-structure_wf,
csm-id_wf,
fill_term_1,
face-0_wf,
csm-face-0,
empty-context-subset-lemma3,
composition-structure-cumulativity,
subtype_rel-equal,
csm-ap-type_wf,
istype-cubical-term,
cubical_set_cumulativity-i-j,
cubical-sigma_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
csm-cubical-sigma,
cubical-term-eqcd,
member_wf,
cubical-fst_wf,
csm-adjoin_wf,
csm-comp_wf,
cc-fst_wf,
cc-snd_wf,
context-subset_wf,
thin-context-subset,
respects-equality-context-subset-term,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-id-adjoin_wf-interval-1,
equals-transprt,
subtype_rel_self,
transprt_wf,
csm-comp-structure-id,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeIsType,
Error :memTop,
independent_isectElimination,
dependent_set_memberEquality_alt,
dependent_functionElimination,
applyLambdaEquality,
cumulativity,
universeEquality,
setElimination,
rename,
productElimination,
equalityIstype,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X.\mBbbI{} \mvdash{} \_\}]. \mforall{}[B:\{X.\mBbbI{}.A \mvdash{} \_\}]. \mforall{}[cA:X.\mBbbI{} +\mvdash{} Compositon(A)]. \mforall{}[cB:X.\mBbbI{}.A +\mvdash{} Compositon(B)].
\mforall{}[pr:\{X \mvdash{} \_:(\mSigma{} A B)[0(\mBbbI{})]\}].
(transprt(X;sigma\_comp(cA;cB);pr).1 = transprt(X;cA;pr.1))
Date html generated:
2020_05_20-PM-05_00_40
Last ObjectModification:
2020_04_18-PM-00_21_35
Theory : cubical!type!theory
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