Nuprl Lemma : ts-refinement-composition
∀ts1,ts2,ts3:transition-system{i:l}. ∀g:ts-type(ts3) ⟶ ts-type(ts2). ∀f:ts-type(ts2) ⟶ ts-type(ts1).
(ts-refinement(ts1;ts2;f) ⇒ ts-refinement(ts2;ts3;g) ⇒ ts-refinement(ts1;ts3;f o g))
Proof
Definitions occuring in Statement :
ts-refinement: ts-refinement(ts1;ts2;f),
ts-type: ts-type(ts),
transition-system: transition-system{i:l},
compose: f o g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ts-refinement: ts-refinement(ts1;ts2;f),
ts-reachable: ts-reachable(ts),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
guard: {T},
compose: f o g,
cand: A c∧ B,
infix_ap: x f y,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rel-preserving-star-reachable,
ts-type_wf,
ts-init_wf,
ts-rel_wf,
ts-reachable_wf,
subtype_rel_wf,
ts-final_wf,
ts-refinement_wf,
transition-system_wf,
rel_star_transitivity,
ts-init_wf_reachable,
rel_star_wf,
rel_rel_star,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
productElimination,
because_Cache,
sqequalRule,
applyEquality,
setElimination,
rename,
lambdaEquality,
setEquality,
universeEquality,
cumulativity,
functionExtensionality,
functionEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productEquality
Latex:
\mforall{}ts1,ts2,ts3:transition-system\{i:l\}. \mforall{}g:ts-type(ts3) {}\mrightarrow{} ts-type(ts2).
\mforall{}f:ts-type(ts2) {}\mrightarrow{} ts-type(ts1).
(ts-refinement(ts1;ts2;f) {}\mRightarrow{} ts-refinement(ts2;ts3;g) {}\mRightarrow{} ts-refinement(ts1;ts3;f o g))
Date html generated:
2018_05_21-PM-08_01_10
Last ObjectModification:
2017_07_26-PM-05_38_00
Theory : general
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