preAddPathRCKM⇔ (2.01)
∃RefinedClinicalKnowledgeModel0;rckmPath! :decisionPathRCKM• [def.preAddPathRCKM] (2.02)
AddPathRCKM
⇔
∃RefinedClinicalKnowledgeModel0;rckmPath! :decisionPathRCKM• [def.AddPathRCKM] (2.03)
RCKM6=∅⇒head(domdppm?) =rootRCKM∧ (2.04)
∀pos:N|pos∈domrefinements?•pos>1∧
pos≤(#(domdppm?) +#(randppm?))∧ (2.05)
ran(domrckmPath!)⊂randecisionPathConditionRCKM∧ (2.06)
ran(ranrckmPath!)⊂ranConclusionRCKM∧ (2.07) (ran(ranrckmPath!)∩randecisionPathConditionRCKM)⊂
randecisionPathConditionRCKM∧ (2.08)
0≤decPathRCKMAccuracy(rckmPath!)≤100∧ (2.09)
head(domrckmPath!)/∈ranConclusionRCKM∩randecisionPathConditionRCKM∧ (2.10)
∃dp:decisionPathRCKM|dp∈RCKM•
domrckmPath!=domdp\last(domdp)⇒last(domdp) =ranrckmPath!∧ (2.11)
domrckmPath!=∃pckm:decisionPathCKM|pckm∈CKM•
dom(ppm?)∪dompckm∧ (2.12)
ranrckmPaht!=randppm?∧ (2.13)
∀r:RefinedTreatmentPlan|r∈refinements?•
rckmPath!=a/h{tp:TreatmentPlan•(1 . . domr,tp)}domrckmPath!, ranr,
{tp:TreatmentPlan•(domr+1 . . #(domrckmPath!),tp)}domrckmPath!i ∧ (2.14)
decisionPathRCKM0=decisionPathRCKM∪ {domrckmPath!7→ranrckmPath!} ∧ (2.15)
decisionPathConditionRCKM0=decisionPathConditionRCKM∪domrckmPath!∧ (2.16)
refinedTPlan0=refinedTPlan∪refinements?∧ (2.17)
refinementsDecPath0=refinementsDecPath∪ {refinements?7→dppm?} ∧ (2.18)
ConclusionRCKM0=ConclusionRCKM∪ranrckmPath!∧ (2.19)
decPathRCKMAccuracy0=decPathRCKMAccuracy∪
{rckmPath!7→decPathRCKMAccuracy(rckmPath!)} ∧ (2.20)
accuracyRCKM0=accuracyRCKM×#RCKM+decPathRCKMAccuracy
0(rckmPath!)
#RCKM+1 ∧ (2.21)
#RCKM0=#RCKM+1∧ (2.22)
evidences0=evidences∪decPathEvidences(dppm?)∧ (2.23)
decPathRCKMEvidences0=decPathRCKMEvidences∪
{rckmPath!7→decPathEvidences(dppm?)} ∧ (2.24)
RCKM0=RCKM⊕ {domrckmPath!7→ranrckmPath!} ∧ (2.25)
refinedCKM0=refinedCKM⊕ {rckmPath!7→CKM} ∧ (2.26)
rootRCKM0=rootRCKM=head(domdppm? (2.27))
⇔
∃rckmPath! :decisionPathRCKM; [def.RefinedClinicalKnowledgeModel0] (2.28)
decisionPathConditionRCKM0:FConditionKMs; (2.29)
ConclusionRCKM0:FTreatmentPlan; (2.30)
decisionPathRCKM0:ConditionKMs7→TreatmentPlan; (2.31)
decPathRCKMAccuracy0:decisionPathRCKM0 →accuracy; (2.32)
evidences0:FEvidences; (2.33)
decPathRCKMEvidences0:decisionPathRCKM0 7→Evidences; (2.34)
refinedTPlan0:FRefinedTreatmentPlan; (2.35)
RCKM0:FdecisionPathRCKM; (2.36)
refinedCKM0:decisionPathRCKM0 →CKM; (2.37)
refinementsDecPath0:RefinedTreatmentPlandecisionPath; (2.38)
rootRCKM0: seqCondition; (2.39)
accuracyRCKM0:F Z; (2.40)
refinedCKMsAccuracy0:RCKM0 →accuracy• (2.41) (1)..decisionPathConditionRCKM0=domdecisionPathRCKM0 ∧ (2.42)
(2)..ConclusionRCKM0=randecisionPathRCKM0 ∧ (2.43) (3)..(ranConclusionRCKM0 ∩randecisionPathConditionRCKM0)⊂randecisionPathConditionRCKM0 ∧ (2.44) (4)..head(decisionPathConditionRCKM0)∈/ranConclusionRCKM0 ∩randecisionPathConditionRCKM0 ∧ (2.45)
(5)..evidences0=randecPathRCKMEvidences0 ∧ (2.46)
(6)..refinedTPlan0=domrefinementsDecPath0 ∧ (2.47)
Supplementary Appendix B. Simplification of primed statements using logical proofs 814
This section describes the detailed steps used to prove the primed statements in Proof2(line2.42 to 815
2.54). The primed statements are evolved using fundamental laws of set theory and deduction rules to 816
obtain the simplified form. All proofs (Proof5-15) are straightforward and instructions are provided for 817
each logical statement. 818
We introduce the necessary definitions (if required) before each proof in order to clarify the logical steps 819
in the corresponding and subsequent proofs. Proof3provides the simplification of the first prime statement 820
in PProof2(line2.42), which is concluded to the simplified statement of the R-CKM model ((Axiom3: line 821
11). In addition to the one-point rule (Definition2), the following basic definitions (Definitions4,5) are used 822
to deduce the final conclusion. 823
Proof4simplifies the primed statement in Proof2(line2.43) to the refined statement of the R-CKM 824
model (Axiom 3: line 12). Using the one-point rule (line 4.02), set subtraction, and ran properties 825
(line4.03-4.05), the proof is easily concluded. Theranproperty for the union is defined as follows. 826
Continued.. 1from Proof2
(7)..0≤accuracyRCKM0≤100∧ (2.48)
(8)..RCKM0=domrefinedCKM0∧ (2.49)
(9)..∀dp:decisionPathRCKM0|dp∈RCKM0•
head(domdp)∈/ranConclusionRCKM0∩randecisionPathConditionRCKM0∧ (2.50)
(10)..∃dp:decisionPathRCKM0o
9dp1:decisionPathRCKM
0|dp,dp1∈RCKM0•
last(domdp) =randp1⇔domdp1=domdp\last(domdp)∧ (2.51) (11)..accuracyRCKM0= (
letpathsAcc=={pathsAcc:Z|RCKM0 6=∅∧
(∀dp:decisionPathRCKM0|dp∈RCKM0•pathsAcc=
decPathRCKMAccuracy0(dp) +pathsAcc)})/#RCKM0∧ (2.52)
(12)..∀p
rckm:decisionPathRCKM0|prckm∈RCKM0•
∃ppm:decisionPath,pckm:decisionPathCKM|
ppm∈PM∧pckm∈CKM•domprckm=domppm∪dompckm∧ (2.53)
(13)..RCKM6=∅⇒rootRCKM0=rootRCKM∧ ( 2.54)
RCKM6=∅⇒head(domdppm?) =rootRCKM∧ (2.55) ∀pos:N|pos∈domrefinements?•pos>1∧
pos≤(#(domdppm?) +#(randppm?))∧ (2.56)
ran(domrckmPath!)⊂randecisionPathConditionRCKM∧ (2.57)
ran(ranrckmPath!)⊂ranConclusionRCKM∧ (2.58) (ran(ranrckmPath!)∩randecisionPathConditionRCKM)⊂
randecisionPathConditionRCKM∧ (2.59)
0≤decPathRCKMAccuracy(rckmPath!)≤100∧ (2.60)
head(domrckmPath!)∈/ranConclusionRCKM∩randecisionPathConditionRCKM∧ (2.61) ∃dp:decisionPathRCKM|dp∈RCKM•
domrckmPath!=domdp\last(domdp)⇒last(domdp) =ranrckmPath!∧ (2.62)
domrckmPath!=∃pckm:decisionPathCKM|pckm∈CKM•
dom(ppm?)∪dompckm∧ (2.63)
ranrckmPaht!=randppm?∧ (2.64)
∀r:RefinedTreatmentPlan|r∈refinements?•
rckmPath!=a/h{tp:TreatmentPlan•(1 . . domr,tp)}domrckmPath!, ranr,
{tp:TreatmentPlan•(domr+1 . . #(domrckmPath!),tp)}domrckmPath!i ∧(2.65)
decisionPathRCKM0=decisionPathRCKM∪ {domrckmPath!7→ranrckmPath!} ∧ (2.66)
decisionPathConditionRCKM0=decisionPathConditionRCKM∪
domrckmPath!∧ (2.67)
refinedTPlan0=refinedTPlan∪refinements?∧ (2.68) refinementsDecPath0=refinementsDecPath∪ {refinements?7→dppm?} ∧ (2.69) ConclusionRCKM0=ConclusionRCKM∪ranrckmPath!∧ (2.70) decPathRCKMAccuracy0=decPathRCKMAccuracy∪
{rckmPath!7→decPathRCKMAccuracy(rckmPath!)} ∧ (2.71)
accuracyRCKM0=accuracyRCKM×#RCKM+decPathRCKMAccuracy0(rckmPath!)
#RCKM+1 ∧ (2.72)
#RCKM0=#RCKM+1∧ (2.73)
evidences0=evidences∪decPathEvidences(dppm?)∧ (2.74) decPathRCKMEvidences0=decPathRCKMEvidences∪
{rckmPath!7→decPathEvidences(dppm?)} ∧ (2.75)
RCKM0=RCKM⊕ {domrckmPath!7→ranrckmPath!} ∧ (2.76) refinedCKM0=refinedCKM⊕ {rckmPath!7→CKM} ∧ (2.77) rootRCKM0=rootRCKM=head(domdppm?) (2.78)
Definition4:For any two functions f and g, the dom property for the union is defined as follows;
dom(f∪g)⇔domf∪domg
Definition 4: domoverunion
Definition5:For any two sets a and b, the set subtraction is formally defined as follows; a\b={x∈a|x/∈b}
Definition 5: Set subtraction
Definition6:For any two functions f and g, the ran property for the union is defined as follows;
ran(f∪g)⇔ranf∪rang
Proof5simplifies the primed statement in Proof2(line2.44) using the one-point rule (line5.02), the 827
definition of range (line5.03 using Definition6), and other laws and principles of set theory, which are 828
described in the following definitions. 829
Definition7:For any two sets a and b, the following property holds; a∪b=a⇔b⊂a
Definition 7: Union Properties
Definition8:Set intersection is distributive over A set union. For sets r, s, and t, the set intersection distribution over a union set can be defined as follows;
r∩(s∪t) = (r∩s)∪(r∩t)
Definition 8: Set intersection distribution law over union
Definition9:For sets a, b, and c, the following definition holds; a∪b⊂c⇒(a⊂c∧b⊂c)
Definition 9: Set union and proper subset
Using the one-point rule (line 6.01) and definitions of basic set theory (lines 6.02 -6.04), Proof 6
830
concludes the primed statement in Proof2(line2.45) into the R-CKM model (Axiom3: line 14). 831
Proof7concludes the primed statement in Proof2(line2.46) into the R-CKM model (Axiom3: line 15) 832
using the one-point rule (line7.02) and definitions of basic set theory (lines7.03 -7.05). 833
Using the one-point rule (line 8.02) and definitions of basic set theory (lines 8.03 -8.05), Proof 8
834
concludes the primed statement in Proof2(line2.47) into the R-CKM model (Axiom3: line 16). 835
Proof9concludes the primed statement in Proof2(line2.48) into the R-CKM model (Schema3: line 8). 836
This proof is straightforward and its conclusion is reached by using the one-point rule (line9.02,9.09) and 837
solving the inequalities with fundamental mathematics. The proof is logically decomposed into two parts 838
(lines9.03-9.07 and lines9.08-9.11). Each part is proven separately and the final statement is concluded 839
(line9.12). 840
The remaining proofs (Proof 10-Proof 15) use the same pattern of logical proofs to simplify the 841
remaining primed statements of Proof2(line2.49-line 2.54). Each step in the proofs is provided with 842
instructive definitions, and necessary definitions are included where explanation is required. 843
Definition10:The union (∪) of two functions is not always a function. However,⊕is the same as a union but ensures that combinations of the two functions are also a function. For two functions f and g,⊕is defined as follows:
f⊕g= (domg−Cf)∪g
For functions f and g, the dom property for⊕is defined as follows;
dom(f⊕g)⇔ domf⊕domg
Definition 10: domproperty over⊕
Definition11:Modus ponens, or implication elimination, is a simple argument form and rule inference in logic. For predicates p and q, the modus ponens can be formally represented as follows;
p⇒q,q`p