theory caseCam=caseParaComparator:
lemma posAssAndSet:
shows "
ALL M N i. len = M - N -->
set (PosAssertOfLine (map match [N ..<M])
((λ i. (map (λ j. (if (i=j) then ONE else DontCare)) [N ..<M])) ( i))) ⊆
{chaos} Un {Is1 (match (i::nat))}" (is "?P len")
proof(induct_tac len)
show "?P 0"
proof(simp del:cktSat_def)qed
fix n
assume b0:"?P n"
show "?P (Suc n)"
proof((rule allI)+,(rule impI)+)
fix M N i
assume b1:"Suc n = M - N"
have b3:"n=M - (Suc N)" by(cut_tac b1,arith)
have b4:"EX M'. M=Suc M'& N <=M'"
by(cut_tac b1,arith)
then obtain M' where b5:"M=Suc M'& N <=M'"
by blast
then have b6:"[N..<M]=[ N..<M']@[M']"
apply -
apply auto
done
have b7:"[N..<M]=N#[Suc N ..<M]"
proof(rule upt_conv_Cons,cut_tac b5,simp)qed
show "set (PosAssertOfLine (map match [N..<M])
(map (λj. if i = j then ONE else DontCare) [N..<M]))
⊆ {chaos} ∪ {Is1 (match i)}"
apply(cut_tac b0 b7,case_tac "i = N",simp add:Let_def,
drule_tac x="M" in spec,
drule_tac x="Suc N" in spec,
cut_tac b3,simp,simp add:Let_def,
drule_tac x="M" in spec,
drule_tac x="Suc N" in spec,
cut_tac b3,simp)
done
qed
qed
lemma posAssAndSet:
shows "
set (PosAssertOfLine (map match [N ..<M])
((λ i. (map (λ j. (if (i=j) then ONE else DontCare)) [N ..<M])) ( i))) ⊆
{chaos} Un {Is1 (match (i::nat))}"
proof( cut_tac posAssAndSet[where len="M - N" and match="match"] ,
drule_tac x="M" in spec ,
drule_tac x="N" in spec,
drule_tac x="i"in spec, simp )qed
lemma camAux1:
assumes a1:"nl:netlists" and
a2:"isClosed nl" and
a3:"orLine=(λ i. (map (λ j. (if (i=j) then ONE else DontCare)) [0..<M]))" and
a5:"matches=map match [0..<M]"
shows "cktSat nl ((Is1 (match i)) \<leadsto>
andLists (PosAssertOfLine matches (orLine i)))" (is "?P i")
proof(rule_tac B="andLists [Is1 (match i)]" in steTrans)
show "cktSat nl (Is1 (match i) \<leadsto> andLists [Is1 (match i)])" thm andListsAllI1
proof(rule andListsAllI)
show "∀a. a ∈ set [Is1 (match i)] --> cktSat nl (Is1 (match i) \<leadsto> a)"
proof(force intro:steRef)qed
qed
next
show "cktSat nl (andLists [Is1 (match i)] \<leadsto> andLists (PosAssertOfLine matches (orLine i)))"
proof(rule andListSubsetEq)
have "set (PosAssertOfLine (map match [0 ..<M])
((λ i. (map (λ j. (if (i=j) then ONE else DontCare)) [0 ..<M])) ( i))) ⊆
{chaos} Un {Is1 (match (i::nat))}"
proof(rule posAssAndSet)qed
then show "set (PosAssertOfLine matches (orLine i)) ∩ - {chaos} ⊆ set [Is1 (match i)]"
apply -
apply(cut_tac a3 a5,simp,force)
done
qed
qed
lemma camCase:
assumes a0:"css=map (λi. (map (λj. c i j) [0..<N])) [0..<M]" and
a0a:"xorGs={g. EX j i. j<N &i<M &g=Gate (c i j) [(Tag j),(storedTag i j) ] xorTab}" and
a1:"xorTab=[[ONE,ONE],[ZERO,ZERO]]" and a2:"andTab=[andLine]" and
a3:"isClosed nl" and
a6:"andLine=map (λ n. ONE) [0..<N]" and
a7:"andGs={g. EX j i .j<N &i<M & g=Gate (match i) (map (λj. c i j) [0..<N]) andTab}" and
a8:" (xorGs Un andGs Un {orG} )⊆ nl " and
a9:"nl:netlists" and
a10:"orLine=(λ i. (map (λ j. (if (i=j) then ONE else DontCare)) [0..<M]))" and
a11:"orTab=map orLine [0..<M]" and
a12:"matches= map match [0..<M]" and
a13:"orG=Gate hit matches orTab" and
a14:"antOfTag = map (λj. Isb (Tag j) (bvOfTag j)) [0..<N]" and
a15:"antOfTs = map (λi. (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N])) [0..<M]" and
a16:"GpOfUnHitI=(λ i. (EX j. j<N & (bvOfTag j)~=( bvOfTs i j)))" and
a17:"GpOfHitI=(λ i. (ALL j. j<N --> (bvOfTag j)=( bvOfTs i j)))" and
a18:"ant=andLists (antOfTag @ (concat antOfTs))" and
a19:"cons0=GpOfUnHit ->T (Is0 hit)" and
a20:"cons1=GpOfHit ->T (Is1 hit)" and
a21:"GpOfUnHit=(ALL i. i<M --> (GpOfUnHitI i))" and
a22:"GpOfHit=(EX i. i<M & (GpOfHitI i))" and
a23:"0<M" and
a24:"0<N"
shows "cktSat nl (ant \<leadsto> (cons0 andT cons1) )"
proof(rule steConjI)
let ?asLs="map (Is0) matches"
show " cktSat nl (ant \<leadsto> cons0)"
proof(cut_tac a19 a19,simp del:cktSat_def,rule steImpI1)
assume b0:"GpOfUnHit"
show "cktSat nl (ant \<leadsto> Is0 hit)" thm orTabPropF
proof(rule_tac ?B="(andLists ?asLs)" in steTrans)
show " cktSat nl (andLists (map Is0 matches) \<leadsto> Is0 hit)"
proof(rule_tac ?asLists=" ?asLs" in orTabPropF)
show "Gate hit matches orTab ∈ nl"
by(cut_tac a13 a8, simp)
show "∀l. l ∈ set orTab --> length l = length matches"
proof(cut_tac a12 a11 a10,simp,force)qed
show "∀line. line mem orTab --> (∃as. as mem map Is0 matches ∧ isNegAssertOfLine as matches line)"
proof(rule allI,rule impI)
fix line
assume c1:"line mem orTab"
show " ∃as. as mem map Is0 matches ∧ isNegAssertOfLine as matches line"
proof -
from c1 have "EX i. i<M & line =(orLine i)"
proof(cut_tac a10 a11 a12 c1, simp add:mem_iff,force)qed
then obtain i where d1:"i<M & line =(orLine i)"
by blast
show " ∃as. as mem map Is0 matches ∧ isNegAssertOfLine as matches line" thm exI
proof(rule_tac x="Is0 (match i)" in exI)
show "Is0 (match i) mem map Is0 matches ∧
isNegAssertOfLine (Is0 (match i)) matches line"
proof(cut_tac a10 a11 a12 d1,simp add:mem_iff set_zip,force)qed
qed
qed
qed
qed
show "cktSat nl (ant \<leadsto> andLists (map Is0 matches))"
proof(rule andListsAllI)
show a1:"∀a. a ∈ set (map Is0 matches) --> cktSat nl (ant \<leadsto> a)"
proof(rule allI,rule impI)
fix a
assume b1:" a ∈ set (map Is0 matches)"
from b1 have "EX i. i<M & a=Is0 (match i)"
proof(cut_tac a10 a11 a12 b1,simp add:mem_iff,force)qed
then obtain i where d1:"i<M &a=Is0 (match i)"
by blast
let ?antOfAs="antOfTag"
let ?antOfBs=" (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N])"
let ?ant="andLists (?antOfAs @?antOfBs)"
show "cktSat nl (ant \<leadsto> a)"
proof(cut_tac d1,simp del:cktSat_def,
rule_tac ?B="?ant" in steTrans)
show " cktSat nl (ant \<leadsto> andLists
(antOfTag @ map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]))"
proof(cut_tac a18 a15,simp del:cktSat_def,rule andListCat )
show "cktSat nl ((andLists antOfTag) \<leadsto> andLists antOfTag)"
by(rule steRef)
show "cktSat nl
(andLists
(concat (map (λi. map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) [0..<M])) \<leadsto>
andLists (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]))"
proof(rule andListSubsetEq)
show " set (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N])∩ - {chaos}
⊆ set (concat (map (λi. map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) [0..<M]))"
proof(cut_tac d1,simp del:cktSat_def,force)qed
qed
qed
let ?xorGs="{g. EX j . j<N &i<M &g=Gate (c i j) [(Tag j),(storedTag i j)] xorTab}"
let ?c="c i"
let ?n="N"
let ?cs="map (c i) [0..<N]"
let ?andG="Gate (match i) ?cs andTab"
let ?b="storedTag i"
let ?a="Tag"
let ?bvOfAs="bvOfTag "
let ?bvOfBs="bvOfTs i"
let ?antOfAs =" map (λj. Isb (Tag j) (bvOfTag j)) [0..<N]"
let ?antOfBs ="map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]"
let ?Gp0.0 = "(∃j<N. bvOfTag j ≠bvOfTs i j )"
let ?Gp1.0 =" (∀j<N. bvOfTag j= bvOfTs i j)"
let ?cons0.0 = "(∃j<N. bvOfTag j ≠ (bvOfTs i j)) ->T Is0 (match i)"
let ?cons1.0 = "(∀j<N. bvOfTag j = (bvOfTs i j)) ->T Is1 (match i)"
show "cktSat nl (andLists (antOfTag @ map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) \<leadsto> Is0 (match i))"
proof(rule_tac ?xorTab ="xorTab" and ?andLine="andLine" and
?andTab="andTab" and ?xorGLs="?xorGs" and ?c="?c" and
?n="N" and ?cs="?cs" and ?andG="?andG" and ?a="?a" and
?b="?b" and ?bvOfAs="?bvOfAs" and ?bvOfBs="?bvOfBs" and
?antOfAs="?antOfAs" and ?antOfBs ="?antOfBs" and
?Gp0.0="?Gp0.0" and ?Gp1.0 ="?Gp1.0" and?cons0.0="?cons0.0" and
?cons1.0 = "?cons1.0"
in caseParaComparatorNeq,(assumption)+,cut_tac d1,simp,
cut_tac a6,simp,(force)+, cut_tac d1 a8 a0a a7 a24 a23,simp,force,(assumption)+,(simp+),
(assumption)+,simp,
cut_tac a14,simp
)
show "∃j<N. bvOfTag j ≠ bvOfTs i j"
proof(cut_tac d1 b0 a16 a21,simp)qed
qed
qed
qed
qed
qed
qed
show "cktSat nl (ant \<leadsto> cons1)"
proof(cut_tac a20 a20,simp del:cktSat_def,rule steImpI1)
assume b0:"GpOfHit"
from b0 obtain i where b1:"i<M &GpOfHitI i"
proof(cut_tac a22,simp,auto)qed
show "cktSat nl (ant \<leadsto> Is1 hit)"
proof(rule_tac ?B="andLists (PosAssertOfLine matches (orLine i))" in steTrans)
show " cktSat nl (ant \<leadsto> andLists (PosAssertOfLine matches (orLine i)))"
proof(rule_tac ?B="Is1 (match i)" in steTrans)
show "cktSat nl (Is1 (match i) \<leadsto>andLists ( PosAssertOfLine matches (orLine i)))"
proof(rule camAux1) qed
next
let ?xorGs="{g. EX j . j<N &i<M &g=Gate (c i j) [(Tag j),(storedTag i j)] xorTab}"
let ?c="c i"
let ?n="N"
let ?cs="map (c i) [0..<N]"
let ?andG="Gate (match i) ?cs andTab"
let ?b="storedTag i"
let ?a="Tag"
let ?bvOfAs="bvOfTag "
let ?bvOfBs="bvOfTs i"
let ?antOfAs =" map (λj. Isb (Tag j) (bvOfTag j)) [0..<N]"
let ?antOfBs ="map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]"
let ?Gp0.0 = "(∃j<N. bvOfTag j ≠bvOfTs i j )"
let ?Gp1.0 =" (∀j<N. bvOfTag j= bvOfTs i j)"
let ?cons0.0 = "(∃j<N. bvOfTag j ≠ (bvOfTs i j)) ->T Is0 (match i)"
let ?cons1.0 = "(∀j<N. bvOfTag j = (bvOfTs i j)) ->T Is1 (match i)"
let ?ant="andLists (?antOfAs@?antOfBs)"
have c1:"cktSat nl (?ant \<leadsto> Is1 (match i))" thm caseParaComparatorEq
proof(rule_tac ?xorTab ="xorTab" and ?andLine="andLine" and
?andTab="andTab" and ?xorGLs="?xorGs" and ?c="?c" and
?n="N" and ?cs="?cs" and ?andG="?andG" and ?a="?a" and
?b="?b" and ?bvOfAs="?bvOfAs" and ?bvOfBs="?bvOfBs" and
?antOfAs="?antOfAs" and ?antOfBs ="?antOfBs" and
?Gp0.0="?Gp0.0" and ?Gp1.0 ="?Gp1.0" and?cons0.0="?cons0.0" and
?cons1.0 = "?cons1.0"
in caseParaComparatorEq,(assumption)+,
cut_tac b1,simp,
cut_tac a6,simp,(force)+
, cut_tac b1 a8 a0a a7 a24 a23,simp,
force,
(assumption)+,(simp+),
(assumption)+,simp+
)
show " ∀j<N. bvOfTag j = bvOfTs i j"
proof(cut_tac b0 b1 a17,force)qed
qed
have c2:"cktSat nl (ant \<leadsto> ?ant)"
proof(cut_tac a18 a15,simp del:cktSat_def,rule andListCat )
show "cktSat nl ((andLists antOfTag) \<leadsto> andLists (map (λj. Isb (Tag j) (bvOfTag j)) [0..<N]))"
by(cut_tac a14, simp del:cktSat_def, rule steRef)
show "cktSat nl
(andLists
(concat (map (λi. map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) [0..<M])) \<leadsto>
andLists (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]))"
proof(rule andListSubsetEq)
show " set (map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N])∩ - {chaos}
⊆ set (concat (map (λi. map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) [0..<M]))"
proof(cut_tac b1,simp del:cktSat_def,force)qed
qed
qed
with c1 show " cktSat nl (ant \<leadsto> Is1 (match i))"
proof(rule_tac steTrans)qed
qed
next
show "cktSat nl (andLists (PosAssertOfLine matches (orLine i)) \<leadsto> Is1 hit)"
proof(rule orTabPropT)
show "Gate hit matches orTab ∈ nl"
by(cut_tac a13 a8, simp)
show "∀l. l ∈ set orTab --> length l = length matches"
proof(cut_tac a12 a11 a10,simp,force)qed
show "orLine i mem orTab"
proof(cut_tac a10 a11 b1,simp add:mem_iff,force)qed
qed
qed
qed
qed
end
lemma posAssAndSet:
∀M N i. len = M - N --> set (PosAssertOfLine (map match [N..<M]) (map (λj. if i = j then ONE else DontCare) [N..<M])) ⊆ {chaos} ∪ {Is1 (match i)}
lemma posAssAndSet:
set (PosAssertOfLine (map match [N..<M]) (map (λj. if i = j then ONE else DontCare) [N..<M])) ⊆ {chaos} ∪ {Is1 (match i)}
lemma camAux1:
[|nl ∈ netlists; isClosed nl; orLine = (λi. map (λj. if i = j then ONE else DontCare) [0..<M]); matches = map match [0..<M]|] ==> cktSat nl (Is1 (match i) \<leadsto> andLists (PosAssertOfLine matches (orLine i)))
lemma camCase:
[|css = map (λi. map (c i) [0..<N]) [0..<M]; xorGs = {g. ∃j i. j < N ∧ i < M ∧ g = Gate (c i j) [Tag j, storedTag i j] xorTab}; xorTab = [[ONE, ONE], [ZERO, ZERO]]; andTab = [andLine]; isClosed nl; andLine = map (λn. ONE) [0..<N]; andGs = {g. ∃j i. j < N ∧ i < M ∧ g = Gate (match i) (map (c i) [0..<N]) andTab}; xorGs ∪ andGs ∪ {orG} ⊆ nl; nl ∈ netlists; orLine = (λi. map (λj. if i = j then ONE else DontCare) [0..<M]); orTab = map orLine [0..<M]; matches = map match [0..<M]; orG = Gate hit matches orTab; antOfTag = map (λj. Isb (Tag j) (bvOfTag j)) [0..<N]; antOfTs = map (λi. map (λj. Isb (storedTag i j) (bvOfTs i j)) [0..<N]) [0..<M]; GpOfUnHitI = (λi. ∃j<N. bvOfTag j ≠ bvOfTs i j); GpOfHitI = (λi. ∀j<N. bvOfTag j = bvOfTs i j); ant = andLists (antOfTag @ concat antOfTs); cons0.0 = GpOfUnHit ->T Is0 hit; cons1.0 = GpOfHit ->T Is1 hit; GpOfUnHit = (∀i<M. GpOfUnHitI i); GpOfHit = (∃i<M. GpOfHitI i); 0 < M; 0 < N|] ==> cktSat nl (ant \<leadsto> cons0.0 andT cons1.0)