theory_file
stringlengths 5
95
| lemma_name
stringlengths 5
250
| lemma_command
stringlengths 17
21.3k
⌀ | lemma_object
stringlengths 5
62.8k
| template
stringlengths 7
63.6k
| symbols
listlengths 0
89
| defs
listlengths 0
63
|
|---|---|---|---|---|---|---|
Stream-Fusion/StreamFusion
|
StreamFusion.zipWithS_cong
|
lemma zipWithS_cong:
"f = f' \<Longrightarrow> a \<approx> a' \<Longrightarrow> b \<approx> b' \<Longrightarrow>
zipWithS\<cdot>f\<cdot>a\<cdot>b \<approx> zipWithS\<cdot>f\<cdot>a'\<cdot>b'"
|
?f = ?f' \<Longrightarrow> ?a \<approx> ?a' \<Longrightarrow> ?b \<approx> ?b' \<Longrightarrow> zipWithS\<cdot>?f\<cdot>?a\<cdot>?b \<approx> zipWithS\<cdot>?f\<cdot>?a'\<cdot>?b'
|
\<lbrakk>x_1 = x_2; ?H1 x_3 x_4; ?H2 x_5 x_6\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 (?H6 ?H7 x_1) x_3) x_5) (?H8 (?H9 (?H10 ?H11 x_1) x_4) x_6)
|
[
"StreamFusion.Either.is_Left",
"StreamFusion.mapS",
"Groups.zero_class.zero",
"StreamFusion.zipWithS",
"Stream.bisimilar",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Both.Both_case"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.foldrS_cong
|
lemma foldrS_cong:
fixes a :: "('a, 's) Stream"
fixes b :: "('a, 't) Stream"
shows "f = g \<Longrightarrow> z = w \<Longrightarrow> a \<approx> b \<Longrightarrow> foldrS\<cdot>f\<cdot>z\<cdot>a = foldrS\<cdot>g\<cdot>w\<cdot>b"
|
?f = ?g \<Longrightarrow> ?z = ?w \<Longrightarrow> ?a \<approx> ?b \<Longrightarrow> foldrS\<cdot>?f\<cdot>?z\<cdot>?a = foldrS\<cdot>?g\<cdot>?w\<cdot>?b
|
\<lbrakk>x_1 = x_2; x_3 = x_4; ?H1 x_5 x_6\<rbrakk> \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_1) x_3) x_5 = ?H6 (?H7 (?H8 ?H9 x_2) x_4) x_6
|
[
"Cfun.cfun.Rep_cfun",
"Stream.bisimilar",
"StreamFusion.foldrS",
"StreamFusion.Both_bisim"
] |
[
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToStep_simps'(1)
|
lemma enumFromToStep_simps' [simp]:
"x \<le> y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>(up\<cdot>x)) =
Yield\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>(up\<cdot>(x+1)))"
"\<not> x \<le> y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>(up\<cdot>x)) = Done"
|
?x \<le> ?y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>?y)\<cdot>(up\<cdot>(up\<cdot>?x)) = Yield\<cdot>(up\<cdot>?x)\<cdot>(up\<cdot>(up\<cdot>(?x + 1)))
|
x_1 \<le> x_2 \<Longrightarrow> ?H1 (?H2 ?H3 (?H4 ?H5 x_2)) (?H6 ?H7 (?H4 ?H5 x_1)) = ?H1 (?H2 ?H8 (?H4 ?H5 x_1)) (?H6 ?H7 (?H4 ?H5 (?H9 x_1 ?H10)))
|
[
"Groups.plus_class.plus",
"Stream.Step.Yield",
"Groups.one_class.one",
"StreamFusion.enumFromToStep",
"Up.up",
"Cfun.cfun.Rep_cfun"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"class one =\n fixes one :: 'a (\"1\")",
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_appendS
|
lemma unstream_appendS:
"a \<noteq> \<bottom> \<Longrightarrow> b \<noteq> \<bottom> \<Longrightarrow>
unstream\<cdot>(appendS\<cdot>a\<cdot>b) = appendL\<cdot>(unstream\<cdot>a)\<cdot>(unstream\<cdot>b)"
|
?a \<noteq> \<bottom> \<Longrightarrow> ?b \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(appendS\<cdot>?a\<cdot>?b) = appendL\<cdot>(unstream\<cdot>?a)\<cdot>(unstream\<cdot>?b)
|
\<lbrakk>x_1 \<noteq> ?H1; x_2 \<noteq> ?H2\<rbrakk> \<Longrightarrow> ?H3 ?H4 (?H5 (?H6 ?H7 x_1) x_2) = ?H8 (?H9 ?H10 (?H11 ?H12 x_1)) (?H13 ?H14 x_2)
|
[
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom",
"StreamFusion.Maybe.match_Just",
"LazyList.appendL",
"StreamFusion.appendS",
"Stream.unstream"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_foldrS
|
lemma unfold_foldrS:
assumes "s \<noteq> \<bottom>" shows "foldrS\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s) = foldrL\<cdot>f\<cdot>z\<cdot>(unfold\<cdot>h\<cdot>s)"
|
?s \<noteq> \<bottom> \<Longrightarrow> foldrS\<cdot>?f\<cdot>?z\<cdot>(Stream\<cdot>?h\<cdot>?s) = foldrL\<cdot>?f\<cdot>?z\<cdot>(unfold\<cdot>?h\<cdot>?s)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_2) x_3) (?H6 (?H7 ?H8 x_4) x_1) = ?H9 (?H10 (?H11 ?H12 x_2) x_3) (?H13 (?H14 ?H15 x_4) x_1)
|
[
"Stream.Stream.Stream",
"Stream.unfold",
"StreamFusion.foldrS",
"LazyList.foldrL",
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_mapStep
|
lemma unfold_mapStep:
fixes f :: "'a \<rightarrow> 'b" and h :: "'s \<rightarrow> ('a, 's) Step"
assumes "s \<noteq> \<bottom>"
shows "unfold\<cdot>(mapStep\<cdot>f\<cdot>h)\<cdot>s = mapL\<cdot>f\<cdot>(unfold\<cdot>h\<cdot>s)"
|
?s \<noteq> \<bottom> \<Longrightarrow> unfold\<cdot>(mapStep\<cdot>?f\<cdot>?h)\<cdot>?s = mapL\<cdot>?f\<cdot>(unfold\<cdot>?h\<cdot>?s)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 ?H4 (?H5 (?H6 ?H7 x_2) x_3)) x_1 = ?H8 (?H9 ?H10 x_2) (?H11 (?H12 ?H13 x_3) x_1)
|
[
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"StreamFusion.mapStep",
"StreamFusion.Both_take",
"LazyList.mapL",
"Stream.unfold"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_filterStep
|
lemma unfold_filterStep:
fixes p :: "'a \<rightarrow> tr" and h :: "'s \<rightarrow> ('a, 's) Step"
assumes "s \<noteq> \<bottom>"
shows "unfold\<cdot>(filterStep\<cdot>p\<cdot>h)\<cdot>s = filterL\<cdot>p\<cdot>(unfold\<cdot>h\<cdot>s)"
|
?s \<noteq> \<bottom> \<Longrightarrow> unfold\<cdot>(filterStep\<cdot>?p\<cdot>?h)\<cdot>?s = filterL\<cdot>?p\<cdot>(unfold\<cdot>?h\<cdot>?s)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 ?H4 (?H5 (?H6 ?H7 x_2) x_3)) x_1 = ?H8 (?H9 ?H10 x_2) (?H2 (?H3 ?H4 x_3) x_1)
|
[
"StreamFusion.filterStep",
"Cfun.cfun.Rep_cfun",
"Stream.unfold",
"Adm.compact",
"LazyList.filterL",
"StreamFusion.Either.match_Right",
"Orderings.ord_class.min",
"Pcpo.pcpo_class.bottom"
] |
[
"definition compact :: \"'a::cpo \\<Rightarrow> bool\"\n where \"compact k = adm (\\<lambda>x. k \\<notsqsubseteq> x)\"",
"class ord =\n fixes less_eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n and less :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_foldrS
|
lemma unstream_foldrS:
"a \<noteq> \<bottom> \<Longrightarrow> foldrS\<cdot>f\<cdot>z\<cdot>a = foldrL\<cdot>f\<cdot>z\<cdot>(unstream\<cdot>a)"
|
?a \<noteq> \<bottom> \<Longrightarrow> foldrS\<cdot>?f\<cdot>?z\<cdot>?a = foldrL\<cdot>?f\<cdot>?z\<cdot>(unstream\<cdot>?a)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_2) x_3) x_1 = ?H6 (?H7 (?H8 ?H9 x_2) x_3) (?H10 ?H11 x_1)
|
[
"Domain.defl_set",
"LazyList.appendL",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Both.Both_case",
"LazyList.LList.LCons",
"Stream.unstream",
"Pcpo.pcpo_class.bottom",
"StreamFusion.foldrS",
"LazyList.foldrL"
] |
[
"definition defl_set :: \"'a::bifinite defl \\<Rightarrow> 'a set\"\nwhere \"defl_set A = {x. cast\\<cdot>A\\<cdot>x = x}\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.appendS_cong
|
lemma appendS_cong:
fixes f :: "'a \<rightarrow> 'b"
fixes a :: "('a, 's) Stream"
fixes b :: "('a, 't) Stream"
shows "a \<approx> a' \<Longrightarrow> b \<approx> b' \<Longrightarrow> appendS\<cdot>a\<cdot>b \<approx> appendS\<cdot>a'\<cdot>b'"
|
?a \<approx> ?a' \<Longrightarrow> ?b \<approx> ?b' \<Longrightarrow> appendS\<cdot>?a\<cdot>?b \<approx> appendS\<cdot>?a'\<cdot>?b'
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_3) (?H7 (?H8 ?H9 x_2) x_4)
|
[
"Stream.bisimilar",
"StreamFusion.appendS",
"StreamFusion.Both.Abs_Both",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Either_abs",
"Cfun.cfcomp_syn",
"StreamFusion.zipWithStep"
] |
[
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.filterS_cong
|
lemma filterS_cong:
fixes p :: "'a \<rightarrow> tr"
fixes a :: "('a, 's) Stream"
fixes b :: "('a, 't) Stream"
shows "p = q \<Longrightarrow> a \<approx> b \<Longrightarrow> filterS\<cdot>p\<cdot>a \<approx> filterS\<cdot>q\<cdot>b"
|
?p = ?q \<Longrightarrow> ?a \<approx> ?b \<Longrightarrow> filterS\<cdot>?p\<cdot>?a \<approx> filterS\<cdot>?q\<cdot>?b
|
\<lbrakk>x_1 = x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 (?H3 ?H4 x_1) x_3) (?H5 (?H6 ?H7 x_2) x_4)
|
[
"Stream.bisimilar",
"StreamFusion.filterS",
"Cfun.cfun.Rep_cfun",
"StreamFusion.mapS"
] |
[
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.mapS_cong
|
lemma mapS_cong:
fixes f :: "'a \<rightarrow> 'b"
fixes a :: "('a, 's) Stream"
fixes b :: "('a, 't) Stream"
shows "f = g \<Longrightarrow> a \<approx> b \<Longrightarrow> mapS\<cdot>f\<cdot>a \<approx> mapS\<cdot>g\<cdot>b"
|
?f = ?g \<Longrightarrow> ?a \<approx> ?b \<Longrightarrow> mapS\<cdot>?f\<cdot>?a \<approx> mapS\<cdot>?g\<cdot>?b
|
\<lbrakk>x_1 = x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_1) x_3) (?H6 (?H7 ?H8 x_2) x_4)
|
[
"Cfun.cfun.Rep_cfun",
"StreamFusion.mapS",
"Fixrec.run",
"Stream.bisimilar",
"Porder.below_class.below",
"StreamFusion.Either_finite",
"Product_Type.prod.fst"
] |
[
"definition\n run :: \"'a match \\<rightarrow> 'a::pcpo\" where\n \"run = (\\<Lambda> m. sscase\\<cdot>\\<bottom>\\<cdot>(fup\\<cdot>ID)\\<cdot>(Rep_match m))\"",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\"",
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToS_cong
|
lemma enumFromToS_cong:
"x = x' \<Longrightarrow> y = y' \<Longrightarrow> enumFromToS\<cdot>x\<cdot>y \<approx> enumFromToS\<cdot>x'\<cdot>y'"
|
?x = ?x' \<Longrightarrow> ?y = ?y' \<Longrightarrow> enumFromToS\<cdot>?x\<cdot>?y \<approx> enumFromToS\<cdot>?x'\<cdot>?y'
|
\<lbrakk>x_1 = x_2; x_3 = x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 (?H3 ?H4 x_1) x_3) (?H2 (?H3 ?H4 x_2) x_4)
|
[
"StreamFusion.enumFromToS",
"Cfun.cfun.Rep_cfun",
"Stream.bisimilar",
"StreamFusion.L.L"
] |
[
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.isodefl_Either
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl ?fb ?db \<Longrightarrow> isodefl (Either_map\<cdot>?fa\<cdot>?fb) (Either_defl\<cdot>?da\<cdot>?db)
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_3) (?H7 (?H8 ?H9 x_2) x_4)
|
[
"Domain.isodefl",
"StreamFusion.Switch.is_S2",
"StreamFusion.Either_map",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Either_defl",
"Fixrec.fail",
"StreamFusion.Both_rep"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\"",
"definition\n fail :: \"'a match\" where\n \"fail = Abs_match (sinl\\<cdot>ONE)\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.isodefl_Both
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl ?fb ?db \<Longrightarrow> isodefl (Both_map\<cdot>?fa\<cdot>?fb) (Both_defl\<cdot>?da\<cdot>?db)
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_3) (?H7 (?H8 ?H9 x_2) x_4)
|
[
"StreamFusion.Both_defl",
"Domain.isodefl",
"StreamFusion.appendStep",
"StreamFusion.Both_map",
"Porder.po_class.Lub",
"Cfun.cfun.Rep_cfun",
"StreamFusion.L.Abs_L"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\"",
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_mapS
|
lemma unstream_mapS:
fixes f :: "'a \<rightarrow> 'b" and a :: "('a, 's) Stream"
shows "a \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(mapS\<cdot>f\<cdot>a) = mapL\<cdot>f\<cdot>(unstream\<cdot>a)"
|
?a \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(mapS\<cdot>?f\<cdot>?a) = mapL\<cdot>?f\<cdot>(unstream\<cdot>?a)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 ?H3 (?H4 (?H5 ?H6 x_2) x_1) = ?H7 (?H8 ?H9 x_2) (?H10 ?H11 x_1)
|
[
"Pcpo.pcpo_class.bottom",
"StreamFusion.mapS",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Switch_rep",
"Map_Functions.u_map",
"LazyList.mapL",
"LazyList.appendL",
"Stream.unstream",
"StreamFusion.Switch.is_S1"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_filterS
|
lemma unstream_filterS:
"a \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(filterS\<cdot>p\<cdot>a) = filterL\<cdot>p\<cdot>(unstream\<cdot>a)"
|
?a \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(filterS\<cdot>?p\<cdot>?a) = filterL\<cdot>?p\<cdot>(unstream\<cdot>?a)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 ?H3 (?H4 (?H5 ?H6 x_2) x_1) = ?H7 (?H8 ?H9 x_2) (?H2 ?H3 x_1)
|
[
"Cfun.cfun.Rep_cfun",
"Porder.po_class.chain",
"Stream.unstream",
"Pcpo.pcpo_class.bottom",
"LazyList.filterL",
"StreamFusion.filterS"
] |
[
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.zipWithL_eq
|
lemma zipWithL_eq:
"zipWithL\<cdot>f\<cdot>xs\<cdot>ys = unstream\<cdot>(zipWithS\<cdot>f\<cdot>(stream\<cdot>xs)\<cdot>(stream\<cdot>ys))"
|
zipWithL\<cdot>?f\<cdot>?xs\<cdot>?ys = unstream\<cdot>(zipWithS\<cdot>?f\<cdot>(stream\<cdot>?xs)\<cdot>(stream\<cdot>?ys))
|
?H1 (?H2 (?H3 ?H4 x_1) x_2) x_3 = ?H5 ?H6 (?H7 (?H8 (?H9 ?H10 x_1) (?H11 ?H12 x_2)) (?H13 ?H14 x_3))
|
[
"LazyList.zipWithL",
"Cfun.ID",
"StreamFusion.zipWithS",
"Stream.stream",
"LazyList.filterL",
"Stream.unstream",
"Cfun.cfun.Rep_cfun"
] |
[
"definition ID :: \"'a \\<rightarrow> 'a\"\n where \"ID = (\\<Lambda> x. x)\"",
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToStep_simps'(2)
|
lemma enumFromToStep_simps' [simp]:
"x \<le> y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>(up\<cdot>x)) =
Yield\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>(up\<cdot>(x+1)))"
"\<not> x \<le> y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>(up\<cdot>x)) = Done"
|
\<not> ?x \<le> ?y \<Longrightarrow> enumFromToStep\<cdot>(up\<cdot>?y)\<cdot>(up\<cdot>(up\<cdot>?x)) = Done
|
\<not> x_1 \<le> x_2 \<Longrightarrow> ?H1 (?H2 ?H3 (?H4 ?H5 x_2)) (?H6 ?H7 (?H4 ?H5 x_1)) = ?H8
|
[
"Stream.Step.Done",
"Cfun.cfun.Rep_cfun",
"Porder.po_class.chain",
"Up.up",
"StreamFusion.enumFromToStep",
"Porder.po_class.Lub"
] |
[
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\"",
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.oo_LAM
|
lemma oo_LAM [simp]: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
|
cont ?g \<Longrightarrow> ?f oo (\<Lambda> x. ?g x) = (\<Lambda> x. ?f\<cdot>(?g x))
|
?H1 x_1 \<Longrightarrow> ?H2 x_2 (?H3 x_1) = ?H4 (\<lambda>y_1. ?H5 x_2 (x_1 y_1))
|
[
"Cfun.cfun.Rep_cfun",
"StreamFusion.enumFromToStep",
"Cfun.cfun.Abs_cfun",
"Cont.cont",
"Deflation.deflation",
"Cfun.cfcomp_syn",
"StreamFusion.Switch.is_S1",
"StreamFusion.Switch.Abs_Switch"
] |
[
"definition cont :: \"('a::cpo \\<Rightarrow> 'b::cpo) \\<Rightarrow> bool\"\n where \"cont f = (\\<forall>Y. chain Y \\<longrightarrow> range (\\<lambda>i. f (Y i)) <<| f (\\<Squnion>i. Y i))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_enumFromToStep
|
lemma unfold_enumFromToStep:
"unfold\<cdot>(enumFromToStep\<cdot>(up\<cdot>y))\<cdot>(up\<cdot>n) = enumFromToL\<cdot>n\<cdot>(up\<cdot>y)"
|
unfold\<cdot>(enumFromToStep\<cdot>(up\<cdot>?y))\<cdot>(up\<cdot>?n) = enumFromToL\<cdot>?n\<cdot>(up\<cdot>?y)
|
?H1 (?H2 ?H3 (?H4 ?H5 (?H6 ?H7 x_1))) (?H8 ?H9 x_2) = ?H10 (?H11 ?H12 x_2) (?H6 ?H7 x_1)
|
[
"StreamFusion.Switch.Switch_case",
"LazyList.concatMapL",
"LazyList.enumFromToL",
"StreamFusion.enumFromToStep",
"Up.up",
"Porder.po_class.Lub",
"Stream.unfold",
"Cfun.cfun.Rep_cfun"
] |
[
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\"",
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_concatMapS
|
lemma unstream_concatMapS:
"unstream\<cdot>(concatMapS\<cdot>f\<cdot>a) = concatMapL\<cdot>(unstream oo f)\<cdot>(unstream\<cdot>a)"
|
unstream\<cdot>(concatMapS\<cdot>?f\<cdot>?a) = concatMapL\<cdot>(unstream oo ?f)\<cdot>(unstream\<cdot>?a)
|
?H1 ?H2 (?H3 (?H4 ?H5 x_1) x_2) = ?H6 (?H7 ?H8 (?H9 ?H10 x_1)) (?H11 ?H12 x_2)
|
[
"Cfun.cfcomp_syn",
"LazyList.zipWithL",
"Fixrec.run",
"Stream.unstream",
"LazyList.concatMapL",
"StreamFusion.concatMapS",
"Cfun.cfun.Rep_cfun",
"Fixrec.match_up"
] |
[
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition\n run :: \"'a match \\<rightarrow> 'a::pcpo\" where\n \"run = (\\<Lambda> m. sscase\\<cdot>\\<bottom>\\<cdot>(fup\\<cdot>ID)\\<cdot>(Rep_match m))\"",
"definition\n match_up :: \"'a::cpo u \\<rightarrow> ('a \\<rightarrow> 'c match) \\<rightarrow> 'c match\"\nwhere\n \"match_up = (\\<Lambda> x k. fup\\<cdot>k\\<cdot>x)\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.concatMapL_eq
|
lemma concatMapL_eq:
"concatMapL\<cdot>f\<cdot>xs = unstream\<cdot>(concatMapS\<cdot>(stream oo f)\<cdot>(stream\<cdot>xs))"
|
concatMapL\<cdot>?f\<cdot>?xs = unstream\<cdot>(concatMapS\<cdot>(stream oo ?f)\<cdot>(stream\<cdot>?xs))
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 ?H8 (?H9 ?H10 x_1)) (?H11 ?H12 x_2))
|
[
"Stream.stream",
"Stream.unstream",
"StreamFusion.Switch.S2",
"StreamFusion.concatMapS",
"LazyList.concatMapL",
"Cfun.cfcomp_syn",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Maybe_map"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.Either_map_unfold
| null |
Either_map\<cdot>?a\<cdot>?b = Either_abs oo ssum_map\<cdot>?a\<cdot>?b oo Either_rep
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_1) x_2) ?H10)
|
[
"StreamFusion.Either_map",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Maybe.is_Nothing",
"StreamFusion.Either_rep",
"Cfun.cfcomp_syn",
"Map_Functions.ssum_map",
"StreamFusion.Either_abs",
"StreamFusion.Maybe_bisim"
] |
[
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition ssum_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<oplus> 'c \\<rightarrow> 'b \\<oplus> 'd\"\n where \"ssum_map = (\\<Lambda> f g. sscase\\<cdot>(sinl oo f)\\<cdot>(sinr oo g))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.Both_map_unfold
| null |
Both_map\<cdot>?a\<cdot>?b = Both_abs oo sprod_map\<cdot>?a\<cdot>?b oo Both_rep
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_1) x_2) ?H10)
|
[
"Cfun.cfun.Rep_cfun",
"Map_Functions.ssum_map",
"Cfun.cfcomp_syn",
"StreamFusion.Both_abs",
"Map_Functions.sprod_map",
"StreamFusion.Both.Abs_Both",
"StreamFusion.Both_rep",
"StreamFusion.Both_map",
"Stream.unfold"
] |
[
"definition ssum_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<oplus> 'c \\<rightarrow> 'b \\<oplus> 'd\"\n where \"ssum_map = (\\<Lambda> f g. sscase\\<cdot>(sinl oo f)\\<cdot>(sinr oo g))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition sprod_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<otimes> 'c \\<rightarrow> 'b \\<otimes> 'd\"\n where \"sprod_map = (\\<Lambda> f g. ssplit\\<cdot>(\\<Lambda> x y. (:f\\<cdot>x, g\\<cdot>y:)))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.appendL_eq
|
lemma appendL_eq: "appendL\<cdot>xs\<cdot>ys = unstream\<cdot>(appendS\<cdot>(stream\<cdot>xs)\<cdot>(stream\<cdot>ys))"
|
appendL\<cdot>?xs\<cdot>?ys = unstream\<cdot>(appendS\<cdot>(stream\<cdot>?xs)\<cdot>(stream\<cdot>?ys))
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 ?H8 (?H9 ?H10 x_1)) (?H9 ?H10 x_2))
|
[
"Cfun.cfun.Rep_cfun",
"StreamFusion.L_abs",
"LazyList.appendL",
"Stream.stream",
"StreamFusion.Switch.Switch_case",
"StreamFusion.Both.Both_case",
"Stream.unstream",
"StreamFusion.appendS"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.Maybe_map_unfold
| null |
Maybe_map\<cdot>?a = Maybe_abs oo ssum_map\<cdot>ID\<cdot>?a oo Maybe_rep
|
?H1 ?H2 x_1 = ?H3 ?H4 (?H5 (?H6 (?H7 ?H8 ?H9) x_1) ?H10)
|
[
"Cfun.ID",
"StreamFusion.Maybe_map",
"StreamFusion.Both_bisim",
"StreamFusion.Maybe_abs",
"StreamFusion.Maybe_rep",
"Cfun.cfcomp_syn",
"StreamFusion.Switch_bisim",
"Cfun.cfun.Rep_cfun",
"Map_Functions.ssum_map"
] |
[
"definition ID :: \"'a \\<rightarrow> 'a\"\n where \"ID = (\\<Lambda> x. x)\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition ssum_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<oplus> 'c \\<rightarrow> 'b \\<oplus> 'd\"\n where \"ssum_map = (\\<Lambda> f g. sscase\\<cdot>(sinl oo f)\\<cdot>(sinr oo g))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.foldrL_eq
|
lemma foldrL_eq:
"foldrL\<cdot>f\<cdot>z\<cdot>xs = foldrS\<cdot>f\<cdot>z\<cdot>(stream\<cdot>xs)"
|
foldrL\<cdot>?f\<cdot>?z\<cdot>?xs = foldrS\<cdot>?f\<cdot>?z\<cdot>(stream\<cdot>?xs)
|
?H1 (?H2 (?H3 ?H4 x_1) x_2) x_3 = ?H5 (?H6 (?H7 ?H8 x_1) x_2) (?H9 ?H10 x_3)
|
[
"StreamFusion.Either_abs",
"Stream.stream",
"StreamFusion.Switch.match_S2",
"StreamFusion.Maybe.Maybe_case",
"StreamFusion.foldrS",
"Tr.cifte_syn",
"Cfun.cfun.Rep_cfun",
"LazyList.foldrL"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2",
"abbreviation cifte_syn :: \"[tr, 'c, 'c] \\<Rightarrow> 'c\" (\"(If (_)/ then (_)/ else (_))\" [0, 0, 60] 60)\n where \"If b then e1 else e2 \\<equiv> tr_case\\<cdot>e1\\<cdot>e2\\<cdot>b\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.mapL_eq
|
lemma mapL_eq: "mapL\<cdot>f\<cdot>xs = unstream\<cdot>(mapS\<cdot>f\<cdot>(stream\<cdot>xs))"
|
mapL\<cdot>?f\<cdot>?xs = unstream\<cdot>(mapS\<cdot>?f\<cdot>(stream\<cdot>?xs))
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 ?H8 x_1) (?H9 ?H10 x_2))
|
[
"Stream.stream",
"Stream.unstream",
"StreamFusion.mapS",
"LazyList.mapL",
"Cfun.cfun.Rep_cfun"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.filterL_eq
|
lemma filterL_eq: "filterL\<cdot>p\<cdot>xs = unstream\<cdot>(filterS\<cdot>p\<cdot>(stream\<cdot>xs))"
|
filterL\<cdot>?p\<cdot>?xs = unstream\<cdot>(filterS\<cdot>?p\<cdot>(stream\<cdot>?xs))
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 ?H8 x_1) (?H9 ?H10 x_2))
|
[
"LazyList.filterL",
"StreamFusion.Either.Right",
"Stream.stream",
"Cfun.cfun.Rep_cfun",
"StreamFusion.L_bisim",
"StreamFusion.filterS",
"Stream.unstream"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.L_map_unfold
| null |
L_map\<cdot>?a = L_abs oo u_map\<cdot>?a oo L_rep
|
?H1 ?H2 x_1 = ?H3 ?H4 (?H5 (?H6 ?H7 x_1) ?H8)
|
[
"Cfun.cfun.Rep_cfun",
"Map_Functions.u_map",
"StreamFusion.L_rep",
"StreamFusion.L_abs",
"Cfun.cfcomp_syn",
"StreamFusion.L_map"
] |
[
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.deflation_Either_map
| null |
deflation ?a \<Longrightarrow> deflation ?b \<Longrightarrow> deflation (Either_map\<cdot>?a\<cdot>?b)
|
\<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_2)
|
[
"Representable.domain_class.defl",
"StreamFusion.Switch.S1",
"Cfun.cfun.Rep_cfun",
"Stream.stream",
"StreamFusion.Either.match_Left",
"Deflation.deflation",
"StreamFusion.Either_map"
] |
[
"class \"domain\" = predomain_syn + pcpo +\n fixes emb :: \"'a \\<rightarrow> udom\"\n fixes prj :: \"udom \\<rightarrow> 'a\"\n fixes defl :: \"'a itself \\<Rightarrow> udom defl\"\n assumes ep_pair_emb_prj: \"ep_pair emb prj\"\n assumes cast_DEFL: \"cast\\<cdot>(defl TYPE('a)) = emb oo prj\"\n assumes liftemb_eq: \"liftemb = u_map\\<cdot>emb\"\n assumes liftprj_eq: \"liftprj = u_map\\<cdot>prj\"\n assumes liftdefl_eq: \"liftdefl TYPE('a) = liftdefl_of\\<cdot>(defl TYPE('a))\"",
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.deflation_Both_map
| null |
deflation ?a \<Longrightarrow> deflation ?b \<Longrightarrow> deflation (Both_map\<cdot>?a\<cdot>?b)
|
\<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_2)
|
[
"StreamFusion.Both_map",
"Fixrec.match_up",
"Deflation.deflation",
"Cfun.cfun.Rep_cfun",
"StreamFusion.enumFromToStep"
] |
[
"definition\n match_up :: \"'a::cpo u \\<rightarrow> ('a \\<rightarrow> 'c match) \\<rightarrow> 'c match\"\nwhere\n \"match_up = (\\<Lambda> x k. fup\\<cdot>k\\<cdot>x)\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_enumFromToS
|
lemma unstream_enumFromToS:
"unstream\<cdot>(enumFromToS\<cdot>x\<cdot>y) = enumFromToL\<cdot>x\<cdot>y"
|
unstream\<cdot>(enumFromToS\<cdot>?x\<cdot>?y) = enumFromToL\<cdot>?x\<cdot>?y
|
?H1 ?H2 (?H3 (?H4 ?H5 x_1) x_2) = ?H6 (?H7 ?H8 x_1) x_2
|
[
"StreamFusion.appendStep",
"StreamFusion.Maybe.match_Nothing",
"Stream.unstream",
"LazyList.enumFromToL",
"StreamFusion.enumFromToS",
"Cfun.cfun.Rep_cfun",
"Porder.below_class.below"
] |
[
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToL_eq
|
lemma enumFromToL_eq: "enumFromToL\<cdot>x\<cdot>y = unstream\<cdot>(enumFromToS\<cdot>x\<cdot>y)"
|
enumFromToL\<cdot>?x\<cdot>?y = unstream\<cdot>(enumFromToS\<cdot>?x\<cdot>?y)
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 ?H8 x_1) x_2)
|
[
"Porder.below_class.not_below",
"LazyList.enumFromToL",
"Fixrec.succeed",
"Stream.unstream",
"StreamFusion.enumFromToS",
"Cfun.cfun.Rep_cfun"
] |
[
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"definition\n succeed :: \"'a \\<rightarrow> 'a match\" where\n \"succeed = (\\<Lambda> x. Abs_match (sinr\\<cdot>(up\\<cdot>x)))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.stream_congs(3)
| null |
?a \<approx> ?b \<Longrightarrow> stream\<cdot>(unstream\<cdot>?a) \<approx> ?b
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 ?H4 (?H5 ?H6 x_1)) x_2
|
[
"StreamFusion.Either_defl",
"Stream.bisimilar",
"StreamFusion.Maybe.match_Nothing",
"Stream.unstream",
"Cfun.cfun.Rep_cfun",
"Stream.stream"
] |
[
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\"",
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.stream_congs(1)
| null |
?a \<approx> ?b \<Longrightarrow> unstream\<cdot>?a = unstream\<cdot>?b
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 ?H3 x_1 = ?H4 ?H5 x_2
|
[
"Cont.cont",
"Stream.unstream",
"StreamFusion.Switch_defl",
"StreamFusion.enumFromToS",
"Stream.bisimilar",
"Cfun.cfun.Rep_cfun"
] |
[
"definition cont :: \"('a::cpo \\<Rightarrow> 'b::cpo) \\<Rightarrow> bool\"\n where \"cont f = (\\<forall>Y. chain Y \\<longrightarrow> range (\\<lambda>i. f (Y i)) <<| f (\\<Squnion>i. Y i))\"",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.stream_congs(2)
| null |
?xs = ?ys \<Longrightarrow> stream\<cdot>?xs \<approx> stream\<cdot>?ys
|
x_1 = x_2 \<Longrightarrow> ?H1 (?H2 ?H3 x_1) (?H2 ?H3 x_2)
|
[
"StreamFusion.filterStep",
"StreamFusion.filterS",
"StreamFusion.Maybe_finite",
"Stream.stream",
"Cfun.cfun.Rep_cfun",
"Stream.Step.Skip",
"Stream.bisimilar"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.isodefl_Maybe
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl (Maybe_map\<cdot>?fa) (Maybe_defl\<cdot>?da)
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 ?H4 x_1) (?H5 ?H6 x_2)
|
[
"StreamFusion.Maybe_defl",
"StreamFusion.Switch.is_S1",
"StreamFusion.Maybe_map",
"Domain.isodefl",
"StreamFusion.L_bisim",
"StreamFusion.appendS",
"Cfun.cfun.Rep_cfun"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.isodefl_L
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl (L_map\<cdot>?fa) (L_defl\<cdot>?da)
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 ?H4 x_1) (?H5 ?H6 x_2)
|
[
"Cfun.cfun.Rep_cfun",
"StreamFusion.L_defl",
"StreamFusion.L_map",
"Domain.isodefl"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToStep_strict(3)
|
lemma enumFromToStep_strict [simp]:
"enumFromToStep\<cdot>\<bottom>\<cdot>x'' = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>\<bottom> = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>\<bottom>) = \<bottom>"
|
enumFromToStep\<cdot>(up\<cdot>?y)\<cdot>(up\<cdot>\<bottom>) = \<bottom>
|
?H1 (?H2 ?H3 (?H4 ?H5 x_1)) (?H6 ?H7 ?H8) = ?H9
|
[
"Up.up",
"Pcpo.pcpo_class.bottom",
"StreamFusion.enumFromToStep",
"Cfun.cfun.Rep_cfun"
] |
[
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.appendStep_strict
|
lemma appendStep_strict [simp]: "appendStep\<cdot>ha\<cdot>hb\<cdot>sb0\<cdot>\<bottom> = \<bottom>"
|
appendStep\<cdot>?ha\<cdot>?hb\<cdot>?sb0.0\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 (?H3 (?H4 ?H5 x_1) x_2) x_3) ?H6 = ?H7
|
[
"Cfun.cfun.Rep_cfun",
"Cfun.cfun.Abs_cfun",
"LazyList.filterL",
"Pcpo.pcpo_class.bottom",
"StreamFusion.appendStep",
"StreamFusion.Either_rep",
"StreamFusion.Both_map"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.zipWithStep_strict
|
lemma zipWithStep_strict [simp]: "zipWithStep\<cdot>f\<cdot>ha\<cdot>hb\<cdot>\<bottom> = \<bottom>"
|
zipWithStep\<cdot>?f\<cdot>?ha\<cdot>?hb\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 (?H3 (?H4 ?H5 x_1) x_2) x_3) ?H6 = ?H7
|
[
"Cfun.cfun.Rep_cfun",
"StreamFusion.zipWithStep",
"Pcpo.pcpo_class.bottom"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToStep_strict(2)
|
lemma enumFromToStep_strict [simp]:
"enumFromToStep\<cdot>\<bottom>\<cdot>x'' = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>\<bottom> = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>\<bottom>) = \<bottom>"
|
enumFromToStep\<cdot>(up\<cdot>?y)\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 (?H4 ?H5 x_1)) ?H6 = ?H7
|
[
"Deflation.deflation",
"StreamFusion.enumFromToStep",
"Stream.unfold",
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"Up.up"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.concatMapStep_strict
|
lemma concatMapStep_strict [simp]: "concatMapStep\<cdot>f\<cdot>ha\<cdot>\<bottom> = \<bottom>"
|
concatMapStep\<cdot>?f\<cdot>?ha\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 (?H3 ?H4 x_1) x_2) ?H5 = ?H6
|
[
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"StreamFusion.concatMapStep"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.deflation_Maybe_map
| null |
deflation ?a \<Longrightarrow> deflation (Maybe_map\<cdot>?a)
|
?H1 x_1 \<Longrightarrow> ?H2 (?H3 ?H4 x_1)
|
[
"Stream.unfold",
"StreamFusion.Maybe_map",
"Deflation.deflation",
"Cfun.cfun.Rep_cfun"
] |
[] |
Stream-Fusion/StreamFusion
|
StreamFusion.deflation_L_map
| null |
deflation ?a \<Longrightarrow> deflation (L_map\<cdot>?a)
|
?H1 x_1 \<Longrightarrow> ?H2 (?H3 ?H4 x_1)
|
[
"StreamFusion.Either.Either_case",
"StreamFusion.L_map",
"Deflation.deflation",
"StreamFusion.Switch_rep",
"Cfun.cfun.Rep_cfun",
"Stream.unstream",
"Stream.Step.Step_case"
] |
[] |
Stream-Fusion/StreamFusion
|
StreamFusion.concatMapS_strict
|
lemma concatMapS_strict [simp]: "concatMapS\<cdot>f\<cdot>\<bottom> = \<bottom>"
|
concatMapS\<cdot>?f\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H5
|
[
"StreamFusion.Maybe_finite",
"Porder.po_class.chain",
"StreamFusion.concatMapS",
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom",
"StreamFusion.Maybe_rep",
"StreamFusion.Both_bisim"
] |
[
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.enumFromToStep_strict(1)
|
lemma enumFromToStep_strict [simp]:
"enumFromToStep\<cdot>\<bottom>\<cdot>x'' = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>\<bottom> = \<bottom>"
"enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>\<bottom>) = \<bottom>"
|
enumFromToStep\<cdot>\<bottom>\<cdot>?x'' = \<bottom>
|
?H1 (?H2 ?H3 ?H4) x_1 = ?H5
|
[
"Stream.stream",
"StreamFusion.enumFromToStep",
"StreamFusion.Either_map",
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom",
"StreamFusion.Maybe_finite"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.enumFromToL_simps'(1)
|
lemma enumFromToL_simps' [simp]:
"x \<le> y \<Longrightarrow>
enumFromToL\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>y) = LCons\<cdot>(up\<cdot>x)\<cdot>(enumFromToL\<cdot>(up\<cdot>(x+1))\<cdot>(up\<cdot>y))"
"\<not> x \<le> y \<Longrightarrow> enumFromToL\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>y) = LNil"
|
?x \<le> ?y \<Longrightarrow> enumFromToL\<cdot>(up\<cdot>?x)\<cdot>(up\<cdot>?y) = LCons\<cdot>(up\<cdot>?x)\<cdot>(enumFromToL\<cdot>(up\<cdot>(?x + 1))\<cdot>(up\<cdot>?y))
|
x_1 \<le> x_2 \<Longrightarrow> ?H1 (?H2 ?H3 (?H4 ?H5 x_1)) (?H4 ?H5 x_2) = ?H6 (?H7 ?H8 (?H4 ?H5 x_1)) (?H1 (?H2 ?H3 (?H4 ?H5 (?H9 x_1 ?H10))) (?H4 ?H5 x_2))
|
[
"LazyList.LList_bisim",
"Groups.plus_class.plus",
"Cfun.cfun.Rep_cfun",
"LazyList.enumFromToL",
"Fixrec.run",
"Domain.defl_set",
"LazyList.LList.LCons",
"Groups.one_class.one",
"Tr.FF",
"Up.up"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition\n run :: \"'a match \\<rightarrow> 'a::pcpo\" where\n \"run = (\\<Lambda> m. sscase\\<cdot>\\<bottom>\\<cdot>(fup\\<cdot>ID)\\<cdot>(Rep_match m))\"",
"definition defl_set :: \"'a::bifinite defl \\<Rightarrow> 'a set\"\nwhere \"defl_set A = {x. cast\\<cdot>A\\<cdot>x = x}\"",
"class one =\n fixes one :: 'a (\"1\")",
"definition FF :: \"tr\"\n where \"FF = Def False\"",
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\""
] |
Stream-Fusion/LazyList
|
LazyList.LList_map_unfold
| null |
LList_map\<cdot>?a = LList_abs oo ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>?a)\<cdot>(u_map\<cdot>(LList_map\<cdot>?a))) oo LList_rep
|
?H1 ?H2 x_1 = ?H3 ?H4 (?H5 (?H6 (?H7 ?H8 ?H9) (?H10 (?H11 ?H12 (?H13 ?H14 x_1)) (?H15 ?H16 (?H1 ?H2 x_1)))) ?H17)
|
[
"LazyList.LList_map",
"Cfun.ID",
"Map_Functions.sprod_map",
"LazyList.appendL",
"Cfun.cfcomp_syn",
"Map_Functions.u_map",
"Map_Functions.ssum_map",
"LazyList.LList_rep",
"LazyList.LList.Rep_LList",
"Cfun.cfun.Rep_cfun",
"LazyList.LList_abs",
"Tr.TT"
] |
[
"definition ID :: \"'a \\<rightarrow> 'a\"\n where \"ID = (\\<Lambda> x. x)\"",
"definition sprod_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<otimes> 'c \\<rightarrow> 'b \\<otimes> 'd\"\n where \"sprod_map = (\\<Lambda> f g. ssplit\\<cdot>(\\<Lambda> x y. (:f\\<cdot>x, g\\<cdot>y:)))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\"",
"definition ssum_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<oplus> 'c \\<rightarrow> 'b \\<oplus> 'd\"\n where \"ssum_map = (\\<Lambda> f g. sscase\\<cdot>(sinl oo f)\\<cdot>(sinr oo g))\"",
"definition TT :: \"tr\"\n where \"TT = Def True\""
] |
Stream-Fusion/LazyList
|
LazyList.enumFromToL_simps'(2)
|
lemma enumFromToL_simps' [simp]:
"x \<le> y \<Longrightarrow>
enumFromToL\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>y) = LCons\<cdot>(up\<cdot>x)\<cdot>(enumFromToL\<cdot>(up\<cdot>(x+1))\<cdot>(up\<cdot>y))"
"\<not> x \<le> y \<Longrightarrow> enumFromToL\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>y) = LNil"
|
\<not> ?x \<le> ?y \<Longrightarrow> enumFromToL\<cdot>(up\<cdot>?x)\<cdot>(up\<cdot>?y) = LNil
|
\<not> x_1 \<le> x_2 \<Longrightarrow> ?H1 (?H2 ?H3 (?H4 ?H5 x_1)) (?H4 ?H5 x_2) = ?H6
|
[
"LazyList.LList_map",
"LazyList.enumFromToL",
"LazyList.LList.LNil",
"Up.up",
"Cfun.cfun.Rep_cfun"
] |
[
"definition up :: \"'a \\<rightarrow> 'a u\"\n where \"up = (\\<Lambda> x. Iup x)\""
] |
Stream-Fusion/LazyList
|
LazyList.zipWithL_strict(2)
|
lemma zipWithL_strict [simp]:
"zipWithL\<cdot>f\<cdot>\<bottom>\<cdot>ys = \<bottom>"
"zipWithL\<cdot>f\<cdot>(LCons\<cdot>x\<cdot>xs)\<cdot>\<bottom> = \<bottom>"
|
zipWithL\<cdot>?f\<cdot>(LCons\<cdot>?x\<cdot>?xs)\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 (?H3 ?H4 x_1) (?H5 (?H6 ?H7 x_2) x_3)) ?H8 = ?H9
|
[
"LazyList.zipWithL",
"Cfun.cfun.Rep_cfun",
"LazyList.LList_take",
"Tr.TT",
"LazyList.LList.LCons",
"Pcpo.pcpo_class.bottom"
] |
[
"definition TT :: \"tr\"\n where \"TT = Def True\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.isodefl_LList
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl (LList_map\<cdot>?fa) (LList_defl\<cdot>?da)
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 ?H4 x_1) (?H5 ?H6 x_2)
|
[
"Fixrec.mplus_syn",
"Cfun.cfun.Rep_cfun",
"LazyList.LList_defl",
"Domain.isodefl",
"LazyList.LList_map"
] |
[
"abbreviation\n mplus_syn :: \"['a match, 'a match] \\<Rightarrow> 'a match\" (infixr \"+++\" 65) where\n \"m1 +++ m2 == mplus\\<cdot>m1\\<cdot>m2\"",
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\""
] |
Stream-Fusion/LazyList
|
LazyList.zipWithL_strict(1)
|
lemma zipWithL_strict [simp]:
"zipWithL\<cdot>f\<cdot>\<bottom>\<cdot>ys = \<bottom>"
"zipWithL\<cdot>f\<cdot>(LCons\<cdot>x\<cdot>xs)\<cdot>\<bottom> = \<bottom>"
|
zipWithL\<cdot>?f\<cdot>\<bottom>\<cdot>?ys = \<bottom>
|
?H1 (?H2 (?H3 ?H4 x_1) ?H5) x_2 = ?H6
|
[
"Porder.below_class.below",
"Cfun.cfun.Rep_cfun",
"LazyList.zipWithL",
"Pcpo.pcpo_class.bottom"
] |
[
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.foldrL_strict
|
lemma foldrL_strict [simp]: "foldrL\<cdot>f\<cdot>z\<cdot>\<bottom> = \<bottom>"
|
foldrL\<cdot>?f\<cdot>?z\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 (?H3 ?H4 x_1) x_2) ?H5 = ?H6
|
[
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"LazyList.foldrL",
"LazyList.LList.LList_case",
"Groups.plus_class.plus"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)"
] |
Stream-Fusion/LazyList
|
LazyList.deflation_LList_map
| null |
deflation ?a \<Longrightarrow> deflation (LList_map\<cdot>?a)
|
?H1 x_1 \<Longrightarrow> ?H2 (?H3 ?H4 x_1)
|
[
"LazyList.LList_map",
"LazyList.LList_defl",
"Cfun.cfun.Rep_cfun",
"Deflation.deflation"
] |
[] |
Stream-Fusion/LazyList
|
LazyList.enumFromToL_strict(2)
|
lemma enumFromToL_strict [simp]:
"enumFromToL\<cdot>\<bottom>\<cdot>y = \<bottom>"
"enumFromToL\<cdot>x\<cdot>\<bottom> = \<bottom>"
|
enumFromToL\<cdot>?x\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H5
|
[
"Groups.plus_class.plus",
"Pcpo.pcpo_class.bottom",
"LazyList.LList.match_LCons",
"LazyList.LList_map",
"Cfun.cfun.Rep_cfun",
"LazyList.enumFromToL"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.enumFromToL_strict(1)
|
lemma enumFromToL_strict [simp]:
"enumFromToL\<cdot>\<bottom>\<cdot>y = \<bottom>"
"enumFromToL\<cdot>x\<cdot>\<bottom> = \<bottom>"
|
enumFromToL\<cdot>\<bottom>\<cdot>?y = \<bottom>
|
?H1 (?H2 ?H3 ?H4) x_1 = ?H5
|
[
"LazyList.LList_map",
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"Porder.below_class.below",
"Groups.one_class.one",
"LazyList.enumFromToL"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"class one =\n fixes one :: 'a (\"1\")"
] |
Stream-Fusion/LazyList
|
LazyList.concatMapL_strict
|
lemma concatMapL_strict [simp]: "concatMapL\<cdot>f\<cdot>\<bottom> = \<bottom>"
|
concatMapL\<cdot>?f\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H5
|
[
"LazyList.concatMapL",
"Cfun.cfun.Rep_cfun",
"Porder.po_class.Lub",
"Pcpo.pcpo_class.bottom",
"Cfun.cfcomp_syn",
"LazyList.LList_defl",
"LazyList.LList.LNil"
] |
[
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/LazyList
|
LazyList.mapL_strict
|
lemma mapL_strict [simp]: "mapL\<cdot>f\<cdot>\<bottom> = \<bottom>"
|
mapL\<cdot>?f\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H5
|
[
"LazyList.mapL",
"Fixrec.fail",
"LazyList.LList_defl",
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"Cfun.ID"
] |
[
"definition\n fail :: \"'a match\" where\n \"fail = Abs_match (sinl\\<cdot>ONE)\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition ID :: \"'a \\<rightarrow> 'a\"\n where \"ID = (\\<Lambda> x. x)\""
] |
Stream-Fusion/LazyList
|
LazyList.appendL_LNil_right
|
lemma appendL_LNil_right: "appendL\<cdot>xs\<cdot>LNil = xs"
|
appendL\<cdot>?xs\<cdot>LNil = ?xs
|
?H1 (?H2 ?H3 x_1) ?H4 = x_1
|
[
"LazyList.appendL",
"LazyList.LList.LNil",
"LazyList.foldrL",
"Fixrec.match_up",
"Porder.below_class.not_below",
"Cfun.cfun.Rep_cfun"
] |
[
"definition\n match_up :: \"'a::cpo u \\<rightarrow> ('a \\<rightarrow> 'c match) \\<rightarrow> 'c match\"\nwhere\n \"match_up = (\\<Lambda> x k. fup\\<cdot>k\\<cdot>x)\"",
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.appendL_strict
|
lemma appendL_strict [simp]: "appendL\<cdot>\<bottom>\<cdot>ys = \<bottom>"
|
appendL\<cdot>\<bottom>\<cdot>?ys = \<bottom>
|
?H1 (?H2 ?H3 ?H4) x_1 = ?H4
|
[
"Pcpo.pcpo_class.bottom",
"LazyList.appendL",
"Cfun.cfun.Rep_cfun"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/LazyList
|
LazyList.filterL_strict
|
lemma filterL_strict [simp]: "filterL\<cdot>p\<cdot>\<bottom> = \<bottom>"
|
filterL\<cdot>?p\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H4
|
[
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom",
"LazyList.filterL"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/Stream
|
Stream.Stream_map_unfold
| null |
Stream_map\<cdot>?a\<cdot>?s = Stream_abs oo sprod_map\<cdot>(u_map\<cdot>(cfun_map\<cdot>?s\<cdot>(Step_map\<cdot>?a\<cdot>?s)))\<cdot>?s oo Stream_rep
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 (?H8 ?H9 (?H10 ?H11 (?H12 (?H13 ?H14 x_2) (?H15 (?H16 ?H17 x_1) x_2)))) x_2) ?H18)
|
[
"Stream.Step_map",
"Map_Functions.sprod_map",
"Cfun.cfcomp_syn",
"Map_Functions.u_map",
"Stream.Stream_map",
"Map_Functions.cfun_map",
"Stream.Stream_abs",
"Cfun.cfun.Rep_cfun",
"Stream.Stream_rep"
] |
[
"definition sprod_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<otimes> 'c \\<rightarrow> 'b \\<otimes> 'd\"\n where \"sprod_map = (\\<Lambda> f g. ssplit\\<cdot>(\\<Lambda> x y. (:f\\<cdot>x, g\\<cdot>y:)))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\"",
"definition cfun_map :: \"('b \\<rightarrow> 'a) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> ('a \\<rightarrow> 'c) \\<rightarrow> ('b \\<rightarrow> 'd)\"\n where \"cfun_map = (\\<Lambda> a b f x. b\\<cdot>(f\\<cdot>(a\\<cdot>x)))\""
] |
Stream-Fusion/Stream
|
Stream.Step_map_unfold
| null |
Step_map\<cdot>?a\<cdot>?s = Step_abs oo ssum_map\<cdot>ID\<cdot>(ssum_map\<cdot>?s\<cdot>(sprod_map\<cdot>(u_map\<cdot>?a)\<cdot>?s)) oo Step_rep
|
?H1 (?H2 ?H3 x_1) x_2 = ?H4 ?H5 (?H6 (?H7 (?H8 ?H9 ?H10) (?H11 (?H12 ?H13 x_2) (?H14 (?H15 ?H16 (?H17 ?H18 x_1)) x_2))) ?H19)
|
[
"Stream.Step_rep",
"Stream.Step_map",
"Map_Functions.u_map",
"Map_Functions.sprod_map",
"LazyList.LList.match_LNil",
"Cfun.cfcomp_syn",
"Fixrec.fail",
"Map_Functions.ssum_map",
"Cfun.ID",
"Cfun.cfun.Rep_cfun",
"Stream.Stream.is_Stream",
"Stream.Step_abs"
] |
[
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\"",
"definition sprod_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<otimes> 'c \\<rightarrow> 'b \\<otimes> 'd\"\n where \"sprod_map = (\\<Lambda> f g. ssplit\\<cdot>(\\<Lambda> x y. (:f\\<cdot>x, g\\<cdot>y:)))\"",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\"",
"definition\n fail :: \"'a match\" where\n \"fail = Abs_match (sinl\\<cdot>ONE)\"",
"definition ssum_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> ('c \\<rightarrow> 'd) \\<rightarrow> 'a \\<oplus> 'c \\<rightarrow> 'b \\<oplus> 'd\"\n where \"ssum_map = (\\<Lambda> f g. sscase\\<cdot>(sinl oo f)\\<cdot>(sinr oo g))\"",
"definition ID :: \"'a \\<rightarrow> 'a\"\n where \"ID = (\\<Lambda> x. x)\""
] |
Stream-Fusion/Stream
|
Stream.unfold_ind
|
lemma unfold_ind:
fixes P :: "('s \<rightarrow> 'a LList) \<Rightarrow> bool"
assumes "adm P" and "P \<bottom>" and "\<And>u. P u \<Longrightarrow> P (unfoldF\<cdot>h\<cdot>u)"
shows "P (unfold\<cdot>h)"
|
adm ?P \<Longrightarrow> ?P \<bottom> \<Longrightarrow> (\<And>u. ?P u \<Longrightarrow> ?P (unfoldF\<cdot>?h\<cdot>u)) \<Longrightarrow> ?P (unfold\<cdot>?h)
|
\<lbrakk>?H1 x_1; x_1 ?H2; \<And>y_0. x_1 y_0 \<Longrightarrow> x_1 (?H3 (?H4 ?H5 x_2) y_0)\<rbrakk> \<Longrightarrow> x_1 (?H6 ?H7 x_2)
|
[
"Fixrec.match_up",
"Stream.Step.Yield",
"Orderings.ord_class.min",
"Adm.adm",
"Stream.unfoldF",
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"Cfun.cfun.Abs_cfun",
"Stream.unfold"
] |
[
"definition\n match_up :: \"'a::cpo u \\<rightarrow> ('a \\<rightarrow> 'c match) \\<rightarrow> 'c match\"\nwhere\n \"match_up = (\\<Lambda> x k. fup\\<cdot>k\\<cdot>x)\"",
"class ord =\n fixes less_eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n and less :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"definition adm :: \"('a::cpo \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"adm P \\<longleftrightarrow> (\\<forall>Y. chain Y \\<longrightarrow> (\\<forall>i. P (Y i)) \\<longrightarrow> P (\\<Squnion>i. Y i))\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/Stream
|
Stream.isodefl_Stream
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl ?fs ?ds \<Longrightarrow> isodefl (Stream_map\<cdot>?fa\<cdot>?fs) (Stream_defl\<cdot>?da\<cdot>?ds)
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_3) (?H7 (?H8 ?H9 x_2) x_4)
|
[
"Domain.isodefl",
"Stream.Step.is_Skip",
"Deflation.deflation",
"Stream.Stream_defl",
"Cfun.cfun.Rep_cfun",
"Stream.Stream_map"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\""
] |
Stream-Fusion/Stream
|
Stream.isodefl_Step
| null |
isodefl ?fa ?da \<Longrightarrow> isodefl ?fs ?ds \<Longrightarrow> isodefl (Step_map\<cdot>?fa\<cdot>?fs) (Step_defl\<cdot>?da\<cdot>?ds)
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_3) (?H7 (?H8 ?H9 x_2) x_4)
|
[
"Stream.Step_defl",
"Cfun.cfun.Rep_cfun",
"Stream.Stream.Stream",
"Domain.isodefl",
"Stream.Step_map",
"Stream.Stream.is_Stream"
] |
[
"definition isodefl :: \"('a \\<rightarrow> 'a) \\<Rightarrow> udom defl \\<Rightarrow> bool\"\n where \"isodefl d t \\<longleftrightarrow> cast\\<cdot>t = emb oo d oo prj\""
] |
Stream-Fusion/Stream
|
Stream.unfoldF
|
lemma unfoldF: "s \<noteq> \<bottom> \<Longrightarrow> unfoldF\<cdot>h\<cdot>u\<cdot>s = unfold2\<cdot>u\<cdot>(h\<cdot>s)"
|
?s \<noteq> \<bottom> \<Longrightarrow> unfoldF\<cdot>?h\<cdot>?u\<cdot>?s = unfold2\<cdot>?u\<cdot>(?h\<cdot>?s)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_2) x_3) x_1 = ?H6 (?H7 ?H8 x_3) (?H9 x_2 x_1)
|
[
"Stream.Stream_abs",
"Stream.unfoldF",
"Fixrec.run",
"Cfun.cfun.Rep_cfun",
"Stream.unfold2",
"Pcpo.pcpo_class.bottom"
] |
[
"definition\n run :: \"'a match \\<rightarrow> 'a::pcpo\" where\n \"run = (\\<Lambda> m. sscase\\<cdot>\\<bottom>\\<cdot>(fup\\<cdot>ID)\\<cdot>(Rep_match m))\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/Stream
|
Stream.unfold
|
lemma unfold: "s \<noteq> \<bottom> \<Longrightarrow> unfold\<cdot>h\<cdot>s = unfold2\<cdot>(unfold\<cdot>h)\<cdot>(h\<cdot>s)"
|
?s \<noteq> \<bottom> \<Longrightarrow> unfold\<cdot>?h\<cdot>?s = unfold2\<cdot>(unfold\<cdot>?h)\<cdot>(?h\<cdot>?s)
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 ?H4 x_2) x_1 = ?H5 (?H6 ?H7 (?H3 ?H4 x_2)) (?H8 x_2 x_1)
|
[
"Stream.Stream_finite",
"Pcpo.pcpo_class.bottom",
"Cfun.cfun.Rep_cfun",
"Stream.bisimilar",
"Stream.Stream.Stream_case",
"Stream.Stream.is_Stream",
"Stream.unfold2",
"Stream.unfold"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/Stream
|
Stream.deflation_Stream_map
| null |
deflation ?a \<Longrightarrow> deflation ?s \<Longrightarrow> deflation (Stream_map\<cdot>?a\<cdot>?s)
|
\<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_2)
|
[
"Deflation.deflation",
"Stream.Stream_map",
"Cfun.cfun.Rep_cfun"
] |
[] |
Stream-Fusion/Stream
|
Stream.deflation_Step_map
| null |
deflation ?a \<Longrightarrow> deflation ?s \<Longrightarrow> deflation (Step_map\<cdot>?a\<cdot>?s)
|
\<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H5 ?H6 x_1) x_2)
|
[
"Deflation.deflation",
"Tr.TT",
"Fixrec.succeed",
"Cfun.cfun.Rep_cfun",
"Stream.Step_map",
"Map_Functions.u_map",
"Stream.Step_defl"
] |
[
"definition TT :: \"tr\"\n where \"TT = Def True\"",
"definition\n succeed :: \"'a \\<rightarrow> 'a match\" where\n \"succeed = (\\<Lambda> x. Abs_match (sinr\\<cdot>(up\\<cdot>x)))\"",
"definition u_map :: \"('a \\<rightarrow> 'b) \\<rightarrow> 'a u \\<rightarrow> 'b u\"\n where \"u_map = (\\<Lambda> f. fup\\<cdot>(up oo f))\""
] |
Stream-Fusion/Stream
|
Stream.stream_unstream_cong
|
lemma stream_unstream_cong:
"a \<approx> b \<Longrightarrow> stream\<cdot>(unstream\<cdot>a) \<approx> b"
|
?a \<approx> ?b \<Longrightarrow> stream\<cdot>(unstream\<cdot>?a) \<approx> ?b
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 ?H4 (?H5 ?H6 x_1)) x_2
|
[
"Stream.unstream",
"Stream.Step_rep",
"Fixrec.mplus_syn",
"Stream.Stream.Rep_Stream",
"Cfun.cfun.Rep_cfun",
"Stream.bisimilar",
"Stream.stream"
] |
[
"abbreviation\n mplus_syn :: \"['a match, 'a match] \\<Rightarrow> 'a match\" (infixr \"+++\" 65) where\n \"m1 +++ m2 == mplus\\<cdot>m1\\<cdot>m2\"",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\"",
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/Stream
|
Stream.unstream_cong
|
lemma unstream_cong:
"a \<approx> b \<Longrightarrow> unstream\<cdot>a = unstream\<cdot>b"
|
?a \<approx> ?b \<Longrightarrow> unstream\<cdot>?a = unstream\<cdot>?b
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 ?H3 x_1 = ?H4 ?H5 x_2
|
[
"Fixrec.succeed",
"Stream.bisimilar",
"Stream.unfoldF",
"Stream.unstream",
"Cfun.cfun.Rep_cfun",
"Domain.defl_set"
] |
[
"definition\n succeed :: \"'a \\<rightarrow> 'a match\" where\n \"succeed = (\\<Lambda> x. Abs_match (sinr\\<cdot>(up\\<cdot>x)))\"",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\"",
"definition defl_set :: \"'a::bifinite defl \\<Rightarrow> 'a set\"\nwhere \"defl_set A = {x. cast\\<cdot>A\\<cdot>x = x}\""
] |
Stream-Fusion/Stream
|
Stream.stream_cong
|
lemma stream_cong:
"xs = ys \<Longrightarrow> stream\<cdot>xs \<approx> stream\<cdot>ys"
|
?xs = ?ys \<Longrightarrow> stream\<cdot>?xs \<approx> stream\<cdot>?ys
|
x_1 = x_2 \<Longrightarrow> ?H1 (?H2 ?H3 x_1) (?H2 ?H3 x_2)
|
[
"Stream.stream",
"Cfun.cfun.Rep_cfun",
"Stream.bisimilar"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/Stream
|
Stream.unfold_eq_fix
|
lemma unfold_eq_fix: "unfold\<cdot>h = fix\<cdot>(unfoldF\<cdot>h)"
|
unfold\<cdot>?h = fix\<cdot>(unfoldF\<cdot>?h)
|
?H1 ?H2 x_1 = ?H3 ?H4 (?H5 ?H6 x_1)
|
[
"Cfun.cfun.Rep_cfun",
"Stream.Stream.match_Stream",
"Fix.fix",
"Stream.unfoldF",
"Stream.unfold"
] |
[
"definition \"fix\" :: \"('a \\<rightarrow> 'a) \\<rightarrow> 'a\"\n where \"fix = (\\<Lambda> F. \\<Squnion>i. iterate i\\<cdot>F\\<cdot>\\<bottom>)\""
] |
Stream-Fusion/Stream
|
Stream.unstream_stream
|
lemma unstream_stream [simp]:
fixes xs :: "'a LList"
shows "unstream\<cdot>(stream\<cdot>xs) = xs"
|
unstream\<cdot>(stream\<cdot>?xs) = ?xs
|
?H1 ?H2 (?H3 ?H4 x_1) = x_1
|
[
"Cfun.cfun.Rep_cfun",
"Stream.stream",
"Cfun.cfun.Abs_cfun",
"Stream.Step.match_Skip",
"Stream.Step_abs",
"Stream.unstream"
] |
[
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
Stream-Fusion/Stream
|
Stream.unfold2_strict
|
lemma unfold2_strict [simp]: "unfold2\<cdot>u\<cdot>\<bottom> = \<bottom>"
|
unfold2\<cdot>?u\<cdot>\<bottom> = \<bottom>
|
?H1 (?H2 ?H3 x_1) ?H4 = ?H5
|
[
"Pcpo.pcpo_class.bottom",
"Stream.unfold2",
"Cfun.cfun.Rep_cfun"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.zeckendorf_unique
|
theorem zeckendorf_unique:
assumes "n > 0"
assumes "n = (\<Sum> i=0..k. fib (c i))" "inc_seq_on c {0..k-1}" "\<forall>i\<in>{0..k}. c i \<ge> 2"
assumes "n = (\<Sum> i=0..k'. fib (c' i))" "inc_seq_on c' {0..k'-1}" "\<forall>i\<in>{0..k'}. c' i \<ge> 2"
shows "k = k' \<and> (\<forall> i \<in> {0..k}. c i = c' i)"
|
0 < ?n \<Longrightarrow> ?n = (\<Sum>i = 0..?k. fib (?c i)) \<Longrightarrow> inc_seq_on ?c {0..?k - 1} \<Longrightarrow> \<forall>i\<in>{0..?k}. 2 \<le> ?c i \<Longrightarrow> ?n = (\<Sum>i = 0..?k'. fib (?c' i)) \<Longrightarrow> inc_seq_on ?c' {0..?k' - 1} \<Longrightarrow> \<forall>i\<in>{0..?k'}. 2 \<le> ?c' i \<Longrightarrow> ?k = ?k' \<and> (\<forall>i\<in>{0..?k}. ?c i = ?c' i)
|
\<lbrakk>?H1 < x_1; x_1 = ?H2 (\<lambda>y_0. ?H3 (x_2 y_0)) (?H4 ?H1 x_3); ?H5 x_2 (?H4 ?H1 (?H6 x_3 ?H7)); \<forall>y_1\<in>?H4 ?H1 x_3. ?H8 (?H9 ?H10) \<le> x_2 y_1; x_1 = ?H2 (\<lambda>y_2. ?H3 (x_4 y_2)) (?H4 ?H1 x_5); ?H5 x_4 (?H4 ?H1 (?H6 x_5 ?H7)); \<forall>y_3\<in>?H4 ?H1 x_5. ?H8 (?H9 ?H10) \<le> x_4 y_3\<rbrakk> \<Longrightarrow> x_3 = x_5 \<and> (\<forall>y_4\<in>?H4 ?H1 x_3. x_2 y_4 = x_4 y_4)
|
[
"Fib.fib",
"Groups.zero_class.zero",
"Set_Interval.ord_class.atLeastAtMost",
"Num.num.One",
"Num.numeral_class.numeral",
"Groups_Big.comm_monoid_add_class.sum",
"Groups.one_class.one",
"Zeckendorf.inc_seq_on",
"Groups.minus_class.minus",
"Num.num.Bit0"
] |
[
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class one =\n fixes one :: 'a (\"1\")",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype num = One | Bit0 num | Bit1 num"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.last_fib_sum_index_constraint
|
lemma last_fib_sum_index_constraint:
assumes "n \<ge> 2" "n = (\<Sum> i=0..k. fib (c i))" "inc_seq_on c {0..k-1}"
assumes "\<forall>i\<in>{0..k}. c i \<ge> 2" "fib i \<le> n" "fib(Suc i) > n"
shows "c k = i"
|
2 \<le> ?n \<Longrightarrow> ?n = (\<Sum>i = 0..?k. fib (?c i)) \<Longrightarrow> inc_seq_on ?c {0..?k - 1} \<Longrightarrow> \<forall>i\<in>{0..?k}. 2 \<le> ?c i \<Longrightarrow> fib ?i \<le> ?n \<Longrightarrow> ?n < fib (Suc ?i) \<Longrightarrow> ?c ?k = ?i
|
\<lbrakk>?H1 (?H2 ?H3) \<le> x_1; x_1 = ?H4 (\<lambda>y_0. ?H5 (x_2 y_0)) (?H6 ?H7 x_3); ?H8 x_2 (?H6 ?H7 (?H9 x_3 ?H10)); \<forall>y_1\<in>?H6 ?H7 x_3. ?H1 (?H2 ?H3) \<le> x_2 y_1; ?H5 x_4 \<le> x_1; x_1 < ?H5 (?H11 x_4)\<rbrakk> \<Longrightarrow> x_2 x_3 = x_4
|
[
"Nat.Suc",
"Num.num.One",
"Groups.one_class.one",
"Fib.fib",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.zero_class.zero",
"Groups.minus_class.minus",
"Num.numeral_class.numeral",
"Groups_Big.comm_monoid_add_class.sum",
"Zeckendorf.inc_seq_on",
"Num.num.Bit0"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\"1\")",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"datatype num = One | Bit0 num | Bit1 num"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_unique_fib_sum
|
lemma fib_unique_fib_sum:
fixes k :: nat
assumes "n \<ge> 2" "inc_seq_on c {0..k-1}" "\<forall>i\<in>{0..k}. c i \<ge> 2"
assumes "n = fib i"
shows "n = (\<Sum>i=0..k. fib (c i)) \<longleftrightarrow> k = 0 \<and> c 0 = i"
|
2 \<le> ?n \<Longrightarrow> inc_seq_on ?c {0..?k - 1} \<Longrightarrow> \<forall>i\<in>{0..?k}. 2 \<le> ?c i \<Longrightarrow> ?n = fib ?i \<Longrightarrow> (?n = (\<Sum>i = 0..?k. fib (?c i))) = (?k = 0 \<and> ?c 0 = ?i)
|
\<lbrakk>?H1 (?H2 ?H3) \<le> x_1; ?H4 x_2 (?H5 ?H6 (?H7 x_3 ?H8)); \<forall>y_0\<in>?H5 ?H6 x_3. ?H1 (?H2 ?H3) \<le> x_2 y_0; x_1 = ?H9 x_4\<rbrakk> \<Longrightarrow> (x_1 = ?H10 (\<lambda>y_1. ?H9 (x_2 y_1)) (?H5 ?H6 x_3)) = (x_3 = ?H6 \<and> x_2 ?H6 = x_4)
|
[
"Groups.minus_class.minus",
"Num.num.One",
"Groups.one_class.one",
"Zeckendorf.inc_seq_on",
"Groups.zero_class.zero",
"Fib.fib",
"Finite_Set.finite",
"Groups_Big.comm_monoid_add_class.sum",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Set_Interval.ord_class.atLeastAtMost"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\"1\")",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.inc_seq_on_aux
|
lemma inc_seq_on_aux: "inc_seq_on c {0..k - 1} \<Longrightarrow> n - fib i < fib (i-1) \<Longrightarrow> fib (c k) < fib i \<Longrightarrow>
(n - fib i) = (\<Sum> i=0..k. fib (c i)) \<Longrightarrow> Suc (c k) < i"
|
inc_seq_on ?c {0..?k - 1} \<Longrightarrow> ?n - fib ?i < fib (?i - 1) \<Longrightarrow> fib (?c ?k) < fib ?i \<Longrightarrow> ?n - fib ?i = (\<Sum>i = 0..?k. fib (?c i)) \<Longrightarrow> Suc (?c ?k) < ?i
|
\<lbrakk>?H1 x_1 (?H2 ?H3 (?H4 x_2 ?H5)); ?H4 x_3 (?H6 x_4) < ?H6 (?H4 x_4 ?H5); ?H6 (x_1 x_2) < ?H6 x_4; ?H4 x_3 (?H6 x_4) = ?H7 (\<lambda>y_0. ?H6 (x_1 y_0)) (?H2 ?H3 x_2)\<rbrakk> \<Longrightarrow> ?H8 (x_1 x_2) < x_4
|
[
"Fib.fib",
"Zeckendorf.inc_seq_on",
"Nat.Suc",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.minus_class.minus",
"Groups.zero_class.zero",
"Groups_Big.comm_monoid_add_class.sum",
"Groups.one_class.one"
] |
[
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class zero =\n fixes zero :: 'a (\"0\")",
"class one =\n fixes one :: 'a (\"1\")"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.one_unique_fib_sum
|
lemma one_unique_fib_sum: "inc_seq_on c {0..k-1} \<Longrightarrow> \<forall>i\<in>{0..k}. c i \<ge> 2 \<Longrightarrow> (\<Sum> i=0..k. fib (c i)) = 1 \<longleftrightarrow> k = 0 \<and> c 0 = 2"
|
inc_seq_on ?c {0..?k - 1} \<Longrightarrow> \<forall>i\<in>{0..?k}. 2 \<le> ?c i \<Longrightarrow> ((\<Sum>i = 0..?k. fib (?c i)) = 1) = (?k = 0 \<and> ?c 0 = 2)
|
\<lbrakk>?H1 x_1 (?H2 ?H3 (?H4 x_2 ?H5)); \<forall>y_0\<in>?H2 ?H3 x_2. ?H6 (?H7 ?H8) \<le> x_1 y_0\<rbrakk> \<Longrightarrow> (?H9 (\<lambda>y_1. ?H10 (x_1 y_1)) (?H2 ?H3 x_2) = ?H5) = (x_2 = ?H3 \<and> x_1 ?H3 = ?H6 (?H7 ?H8))
|
[
"Zeckendorf.inc_seq_on",
"Num.num.One",
"Num.numeral_class.numeral",
"Groups.zero_class.zero",
"Fib.fib",
"Set_Interval.ord_class.atLeastAtMost",
"Num.num.Bit0",
"Set.Collect",
"Groups.minus_class.minus",
"Groups.one_class.one",
"Groups_Big.comm_monoid_add_class.sum"
] |
[
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class one =\n fixes one :: 'a (\"1\")"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_implies_zeckendorf
|
lemma fib_implies_zeckendorf:
assumes "is_fib n" "n > 0"
shows "\<exists> c k. n = (\<Sum> i=0..k. fib(c i)) \<and> inc_seq_on c {0..k-1} \<and> (\<forall> i\<in>{0..k}. c i \<ge> 2)"
|
is_fib ?n \<Longrightarrow> 0 < ?n \<Longrightarrow> \<exists>c k. ?n = (\<Sum>i = 0..k. fib (c i)) \<and> inc_seq_on c {0..k - 1} \<and> (\<forall>i\<in>{0..k}. 2 \<le> c i)
|
\<lbrakk>?H1 x_1; ?H2 < x_1\<rbrakk> \<Longrightarrow> \<exists>y_0 y_1. x_1 = ?H3 (\<lambda>y_2. ?H4 (y_0 y_2)) (?H5 ?H2 y_1) \<and> ?H6 y_0 (?H5 ?H2 (?H7 y_1 ?H8)) \<and> (\<forall>y_3\<in>?H5 ?H2 y_1. ?H9 (?H10 ?H11) \<le> y_0 y_3)
|
[
"Set.empty",
"Set.Collect",
"Set_Interval.ord_class.atLeastAtMost",
"Groups_Big.comm_monoid_add_class.sum",
"Num.num.Bit0",
"Zeckendorf.inc_seq_on",
"Num.numeral_class.numeral",
"Zeckendorf.is_fib",
"Fib.fib",
"Num.num.One",
"Groups.zero_class.zero",
"Groups.one_class.one",
"Groups.minus_class.minus"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"datatype num = One | Bit0 num | Bit1 num",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"class zero =\n fixes zero :: 'a (\"0\")",
"class one =\n fixes one :: 'a (\"1\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.zeckendorf_existence
|
theorem zeckendorf_existence:
assumes "n > 0"
shows "\<exists> c k. n = (\<Sum> i=0..k. fib (c i)) \<and> inc_seq_on c {0..k-1} \<and> (\<forall>i\<in>{0..k}. c i \<ge> 2)"
|
0 < ?n \<Longrightarrow> \<exists>c k. ?n = (\<Sum>i = 0..k. fib (c i)) \<and> inc_seq_on c {0..k - 1} \<and> (\<forall>i\<in>{0..k}. 2 \<le> c i)
|
?H1 < x_1 \<Longrightarrow> \<exists>y_0 y_1. x_1 = ?H2 (\<lambda>y_2. ?H3 (y_0 y_2)) (?H4 ?H1 y_1) \<and> ?H5 y_0 (?H4 ?H1 (?H6 y_1 ?H7)) \<and> (\<forall>y_3\<in>?H4 ?H1 y_1. ?H8 (?H9 ?H10) \<le> y_0 y_3)
|
[
"Num.numeral_class.numeral",
"Num.num.One",
"Groups_Big.comm_monoid_add_class.sum",
"Groups.zero_class.zero",
"Zeckendorf.is_fib",
"Fib.fib",
"Finite_Set.finite",
"Groups.one_class.one",
"Zeckendorf.inc_seq_on",
"Set_Interval.ord_class.atLeastAtMost",
"Num.num.Bit0",
"Groups.minus_class.minus"
] |
[
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype num = One | Bit0 num | Bit1 num",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"class one =\n fixes one :: 'a (\"1\")",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"datatype num = One | Bit0 num | Bit1 num",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_sum_upper_bound
|
lemma fib_sum_upper_bound:
assumes "inc_seq_on c {0..k-1}" "\<forall>i\<in>{0..k}. c i \<ge> 2"
shows "(\<Sum> i=0..k. fib (c i)) < fib (Suc (c k))"
|
inc_seq_on ?c {0..?k - 1} \<Longrightarrow> \<forall>i\<in>{0..?k}. 2 \<le> ?c i \<Longrightarrow> (\<Sum>i = 0..?k. fib (?c i)) < fib (Suc (?c ?k))
|
\<lbrakk>?H1 x_1 (?H2 ?H3 (?H4 x_2 ?H5)); \<forall>y_0\<in>?H2 ?H3 x_2. ?H6 (?H7 ?H8) \<le> x_1 y_0\<rbrakk> \<Longrightarrow> ?H9 (\<lambda>y_1. ?H10 (x_1 y_1)) (?H2 ?H3 x_2) < ?H10 (?H11 (x_1 x_2))
|
[
"Nat.Suc",
"Set.empty",
"Groups.minus_class.minus",
"Num.num.Bit0",
"Num.num.One",
"Zeckendorf.is_fib",
"Groups_Big.comm_monoid_add_class.sum",
"Fib.fib",
"Zeckendorf.inc_seq_on",
"Groups.zero_class.zero",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Set.Collect",
"Set_Interval.ord_class.atLeastAtMost"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class one =\n fixes one :: 'a (\"1\")"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_idx_ge_two_fib_sum_not_zero
|
lemma fib_idx_ge_two_fib_sum_not_zero: "n \<le> m \<Longrightarrow> \<forall>i\<in>{n..m::nat}. c i \<ge> 2 \<Longrightarrow> \<not> (\<Sum> i=n..m. fib (c i)) = 0"
|
?n \<le> ?m \<Longrightarrow> \<forall>i\<in>{?n..?m}. 2 \<le> ?c i \<Longrightarrow> (\<Sum>i = ?n..?m. fib (?c i)) \<noteq> 0
|
\<lbrakk>x_1 \<le> x_2; \<forall>y_0\<in>?H1 x_1 x_2. ?H2 (?H3 ?H4) \<le> x_3 y_0\<rbrakk> \<Longrightarrow> ?H5 (\<lambda>y_1. ?H6 (x_3 y_1)) (?H1 x_1 x_2) \<noteq> ?H7
|
[
"Num.numeral_class.numeral",
"Groups.zero_class.zero",
"Groups_Big.comm_monoid_add_class.sum",
"Num.num.One",
"Zeckendorf.is_fib",
"Num.num.Bit0",
"Set_Interval.ord_class.atLeastAtMost",
"Fib.fib"
] |
[
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype num = One | Bit0 num | Bit1 num",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"datatype num = One | Bit0 num | Bit1 num",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.nat_ge_2_fib_idx_bound
|
lemma nat_ge_2_fib_idx_bound: "2 \<le> n \<Longrightarrow> fib i \<le> n \<Longrightarrow> n < fib (Suc i) \<Longrightarrow> 2 \<le> i"
|
2 \<le> ?n \<Longrightarrow> fib ?i \<le> ?n \<Longrightarrow> ?n < fib (Suc ?i) \<Longrightarrow> 2 \<le> ?i
|
\<lbrakk>?H1 (?H2 ?H3) \<le> x_1; ?H4 x_2 \<le> x_1; x_1 < ?H4 (?H5 x_2)\<rbrakk> \<Longrightarrow> ?H1 (?H2 ?H3) \<le> x_2
|
[
"Groups.zero_class.zero",
"Num.numeral_class.numeral",
"Fib.fib",
"Num.num.One",
"Nat.Suc",
"Groups.minus_class.minus",
"Num.num.Bit0",
"Finite_Set.finite"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype num = One | Bit0 num | Bit1 num",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.ge_two_eq_fib_implies_eq_idx
|
lemma ge_two_eq_fib_implies_eq_idx: "n \<ge> 2 \<Longrightarrow> n = fib i \<Longrightarrow> n = fib j \<Longrightarrow> i = j"
|
2 \<le> ?n \<Longrightarrow> ?n = fib ?i \<Longrightarrow> ?n = fib ?j \<Longrightarrow> ?i = ?j
|
\<lbrakk>?H1 (?H2 ?H3) \<le> x_1; x_1 = ?H4 x_2; x_1 = ?H4 x_3\<rbrakk> \<Longrightarrow> x_2 = x_3
|
[
"Num.numeral_class.numeral",
"Zeckendorf.inc_seq_on",
"Fib.fib",
"Num.num.One",
"Nat.Suc",
"Zeckendorf.is_fib",
"Num.num.Bit0",
"Groups.zero_class.zero"
] |
[
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"datatype num = One | Bit0 num | Bit1 num",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.pos_fib_has_idx_ge_two
|
lemma pos_fib_has_idx_ge_two: "n > 0 \<Longrightarrow> is_fib n \<Longrightarrow> (\<exists> i. i \<ge> 2 \<and> fib i = n)"
|
0 < ?n \<Longrightarrow> is_fib ?n \<Longrightarrow> \<exists>i\<ge>2. fib i = ?n
|
\<lbrakk>?H1 < x_1; ?H2 x_1\<rbrakk> \<Longrightarrow> \<exists>y_0\<ge>?H3 (?H4 ?H5). ?H6 y_0 = x_1
|
[
"Num.num.One",
"Groups.minus_class.minus",
"Groups.zero_class.zero",
"Set.empty",
"Num.numeral_class.numeral",
"Zeckendorf.is_fib",
"Num.num.Bit0",
"Fib.fib",
"Set.Collect"
] |
[
"datatype num = One | Bit0 num | Bit1 num",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"datatype num = One | Bit0 num | Bit1 num",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_sum_zero_equiv
|
lemma fib_sum_zero_equiv: "(\<Sum> i=n..m::nat . fib (c i)) = 0 \<longleftrightarrow> (\<forall> i\<in>{n..m}. c i = 0)"
|
((\<Sum>i = ?n..?m. fib (?c i)) = 0) = (\<forall>i\<in>{?n..?m}. ?c i = 0)
|
(?H1 (\<lambda>y_0. ?H2 (x_1 y_0)) (?H3 x_2 x_3) = ?H4) = (\<forall>y_1\<in>?H3 x_2 x_3. x_1 y_1 = ?H4)
|
[
"Zeckendorf.is_fib",
"Groups.zero_class.zero",
"Groups_Big.comm_monoid_add_class.sum",
"Fib.fib",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.one_class.one"
] |
[
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class one =\n fixes one :: 'a (\"1\")"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.inc_seq_zero_at_start
|
lemma inc_seq_zero_at_start: "inc_seq_on c {0..k-1} \<Longrightarrow> c k = 0 \<Longrightarrow> k = 0"
|
inc_seq_on ?c {0..?k - 1} \<Longrightarrow> ?c ?k = 0 \<Longrightarrow> ?k = 0
|
\<lbrakk>?H1 x_1 (?H2 ?H3 (?H4 x_2 ?H5)); x_1 x_2 = ?H3\<rbrakk> \<Longrightarrow> x_2 = ?H3
|
[
"Fib.fib",
"Set.empty",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.one_class.one",
"Zeckendorf.is_fib",
"Zeckendorf.inc_seq_on",
"Groups.zero_class.zero",
"Groups.minus_class.minus"
] |
[
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"class one =\n fixes one :: 'a (\"1\")",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.ge_two_fib_unique_idx
|
lemma ge_two_fib_unique_idx: "fib i \<ge> 2 \<Longrightarrow> fib i = fib j \<Longrightarrow> i = j"
|
2 \<le> fib ?i \<Longrightarrow> fib ?i = fib ?j \<Longrightarrow> ?i = ?j
|
\<lbrakk>?H1 (?H2 ?H3) \<le> ?H4 x_1; ?H4 x_1 = ?H4 x_2\<rbrakk> \<Longrightarrow> x_1 = x_2
|
[
"Zeckendorf.inc_seq_on",
"Fib.fib",
"Num.numeral_class.numeral",
"Num.num.One",
"Num.num.Bit0",
"Nat.Suc"
] |
[
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_index_strict_mono
|
lemma fib_index_strict_mono : "i \<ge> 2 \<Longrightarrow> j > i \<Longrightarrow> fib j > fib i"
|
2 \<le> ?i \<Longrightarrow> ?i < ?j \<Longrightarrow> fib ?i < fib ?j
|
\<lbrakk>?H1 (?H2 ?H3) \<le> x_1; x_1 < x_2\<rbrakk> \<Longrightarrow> ?H4 x_1 < ?H4 x_2
|
[
"Groups_Big.comm_monoid_add_class.sum",
"Zeckendorf.inc_seq_on",
"Fib.fib",
"Num.num.One",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Num.num.Bit0"
] |
[
"definition inc_seq_on :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat set \\<Rightarrow> bool\" where\n \"inc_seq_on f I = (\\<forall> n \\<in> I. f(Suc n) > Suc(f n))\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class one =\n fixes one :: 'a (\"1\")",
"datatype num = One | Bit0 num | Bit1 num"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.one_fib_idxs
|
lemma one_fib_idxs: "fib i = Suc 0 \<Longrightarrow> i = Suc 0 \<or> i = Suc(Suc 0)"
|
fib ?i = Suc 0 \<Longrightarrow> ?i = Suc 0 \<or> ?i = Suc (Suc 0)
|
?H1 x_1 = ?H2 ?H3 \<Longrightarrow> x_1 = ?H2 ?H3 \<or> x_1 = ?H2 (?H2 ?H3)
|
[
"Nat.Suc",
"Groups.zero_class.zero",
"Groups.minus_class.minus",
"Fib.fib"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.no_fib_betw_fibs
|
lemma no_fib_betw_fibs:
assumes "\<not> is_fib n"
shows "\<exists> i. fib i < n \<and> n < fib (Suc i)"
|
\<not> is_fib ?n \<Longrightarrow> \<exists>i. fib i < ?n \<and> ?n < fib (Suc i)
|
\<not> ?H1 x_1 \<Longrightarrow> \<exists>y_0. ?H2 y_0 < x_1 \<and> x_1 < ?H2 (?H3 y_0)
|
[
"Nat.Suc",
"Fib.fib",
"Zeckendorf.is_fib"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.zero_fib_unique_idx
|
lemma zero_fib_unique_idx: "n = fib i \<Longrightarrow> n = fib 0 \<Longrightarrow> i = 0"
|
?n = fib ?i \<Longrightarrow> ?n = fib 0 \<Longrightarrow> ?i = 0
|
\<lbrakk>x_1 = ?H1 x_2; x_1 = ?H1 ?H2\<rbrakk> \<Longrightarrow> x_2 = ?H2
|
[
"Fib.fib",
"Zeckendorf.is_fib",
"Groups.zero_class.zero",
"Finite_Set.finite"
] |
[
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.finite_fib_ge_two_idx
|
lemma finite_fib_ge_two_idx: "n \<ge> 2 \<Longrightarrow> finite({i. fib i = n})"
|
2 \<le> ?n \<Longrightarrow> finite {i. fib i = ?n}
|
?H1 (?H2 ?H3) \<le> x_1 \<Longrightarrow> ?H4 (?H5 (\<lambda>y_0. ?H6 y_0 = x_1))
|
[
"Set.Collect",
"Num.num.One",
"Num.num.Bit0",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.one_class.one",
"Finite_Set.finite",
"Fib.fib",
"Groups_Big.comm_monoid_add_class.sum",
"Groups.zero_class.zero",
"Num.numeral_class.numeral"
] |
[
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\"1\")",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\""
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.no_fib_implies_le_fib_idx_set
|
lemma no_fib_implies_le_fib_idx_set: "\<not> is_fib n \<Longrightarrow> {i. fib i < n} \<noteq> {}"
|
\<not> is_fib ?n \<Longrightarrow> {i. fib i < ?n} \<noteq> {}
|
\<not> ?H1 x_1 \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 y_0 < x_1) \<noteq> ?H4
|
[
"Nat.Suc",
"Finite_Set.finite",
"Groups.zero_class.zero",
"Set.empty",
"Zeckendorf.is_fib",
"Fib.fib",
"Num.num.One",
"Set.Collect"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition is_fib :: \"nat \\<Rightarrow> bool\" where\n \"is_fib n = (\\<exists> i. n = fib i)\"",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num"
] |
Zeckendorf/Zeckendorf
|
Zeckendorf.fib_strict_mono
|
lemma fib_strict_mono: "i \<ge> 2 \<Longrightarrow> fib i < fib (Suc i)"
|
2 \<le> ?i \<Longrightarrow> fib ?i < fib (Suc ?i)
|
?H1 (?H2 ?H3) \<le> x_1 \<Longrightarrow> ?H4 x_1 < ?H4 (?H5 x_1)
|
[
"Nat.Suc",
"Groups.zero_class.zero",
"Fib.fib",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Num.num.One"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun fib :: \"nat \\<Rightarrow> nat\"\n where\n fib0: \"fib 0 = 0\"\n | fib1: \"fib (Suc 0) = 1\"\n | fib2: \"fib (Suc (Suc n)) = fib (Suc n) + fib n\"",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype num = One | Bit0 num | Bit1 num"
] |
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