hcju/isabelleps huggingface dataset:
resolution_rule: fixes C\<^sub>1 :: "fterm literal set"   and Cs :: "fterm literal set set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "C\<^sub>1 \<in> Cs"   and "C\<^sub>2 \<in> Cs"   and "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>" shows "resolution_step Cs (Cs \<union> {resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>})"
nth_sinit: fixes i :: "nat"   and n :: "nat"   and xs :: "'b seq" assumes "i < n" shows "sinit n xs ! i = xs !\<^sub>s i"
seq_Sup_distr: fixes A :: "('a, 'c) urel set"   and Q :: "('c, 'b) urel" shows "\<Or>A ;; Q = \<Or>((\<lambda>P. P ;; Q) ` A)"
P: fixes i :: "enat" shows "Measurable.pred (F i) (P i)"
NSsums_NSsummable: fixes f :: "nat \<Rightarrow> real"   and l :: "real" assumes "f NSsums l" shows "NSsummable f"
cycle_free_0_0: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and n :: "nat" assumes "cycle_free M n" shows "\<one> \<le> M 0 0"
equality: fixes r :: "('a, 'b, 'c, 'd) configuration_scheme"   and r' :: "('a, 'b, 'c, 'd) configuration_scheme" assumes "states r = states r'"   and "msgs r = msgs r'"   and "ps r = ps r'"   and "cs r = cs r'"   and "more r = more r'" shows "r = r'"
bhvks4: fixes t :: "nat \<Rightarrow> cnf"   and p :: "PROB"   and P :: "PROB set"   and kId :: "'kid"   and t' :: "nat \<Rightarrow> 'KS" assumes "t \<in> arch"   and "p \<in> P" shows "sb.eval kId t t' 0 (\<box>\<^sub>b[\<lambda>ks. sub P = ksrp ks]\<^sub>b \<longrightarrow>\<^sup>b (\<not>\<^sup>b (\<exists>\<^sub>bP'. sb.pred (p \<in> P') \<and>\<^sup>b [\<lambda>ks. unsub P' = ksrp ks]\<^sub>b)) \<WW>\<^sub>b [\<lambda>ks. (p, solve p) \<in> kscs ks]\<^sub>b)"
nonsingular_def: fixes a :: "'a"   and b :: "'a" shows "nonsingular a b = (\<guillemotleft>4\<guillemotright> \<otimes> a [^] 3 \<oplus> \<guillemotleft>27\<guillemotright> \<otimes> b [^] 2 \<noteq> \<zero>)"
sy0: shows "sy \<cdot> (s2 \<cdot> r - s \<cdot> r2) = \<zero>"
rel_interior_interior: fixes S :: "'c set" assumes "affine hull S = UNIV" shows "rel_interior S = interior S"
negative_zless: fixes n :: "nat"   and m :: "nat" shows "- int (Suc n) < int m"
FWdelay_bisimulation_obs_flip: shows "FWdelay_bisimulation_obs final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) (flip bisim_wait) \<tau>move2 \<tau>move1"
le: fixes i :: "'a"   and j :: "'n" shows "0 \<le> B $h i $h j"
filter_mset_empty_conv: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset" shows "(filter_mset P M = {#}) = (\<forall>L\<in>#M. \<not> P L)"
llist_all2_conv_all_lnth: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b llist"   and ys :: "'c llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>n. enat n < llength ys \<longrightarrow> P (xs ?! n) (ys ?! n)))"
exp_of_nat_mult: fixes n :: "nat"   and x :: "'a" shows "exp (of_nat n * x) = exp x ^ n"
assms: shows "formulaEntailsLiteral (removeAll clause formula) literal"
snapshotted_and_not_done_implies_marker_in_channel: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "trace init t final"   and "ps (S t i) p \<noteq> None"   and "snd (cs (S t i) cid) \<noteq> Done"   and "i \<le> length t"   and "channel cid = Some (p, q)" shows "Marker \<in> set (msgs (S t i) cid)"
l8_12: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt C D A B "
Phase1or2ReadThen_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "Phase1or2ReadThen s s' p d q = (d \<in> disksWritten s p \<and> mbal (disk s d q) < mbal (dblock s p) \<and> blocksRead s' = (blocksRead s)(p := (blocksRead s p)(d := blocksRead s p d \<union> {\<lparr>block = disk s d q, proc = q\<rparr>})) \<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> disk s' = disk s \<and> phase s' = phase s \<and> dblock s' = dblock s \<and> disksWritten s' = disksWritten s)"
R: fixes U :: "'a set"   and V :: "'a set" assumes "U open"   and "A \<subseteq> U"   and "V open"   and "B \<subseteq> V"   and "U \<inter> V = {}" shows "R"
guarded_def: fixes sys :: "'a \<Rightarrow> ('b, 'c) process_term" shows "guarded sys \<equiv> \<forall>X Y. sys X \<noteq> VAR Y"
gauss_cnj_uminus: fixes a :: "gauss_int" shows "gauss_cnj (- a) = - gauss_cnj a"
leg0_commutes: shows "\<epsilon>.cod.leg0 \<cdot> \<epsilon>.chine = \<epsilon>.dom.leg0"
converse_Un: fixes r :: "('d \<times> 'c) set"   and s :: "('d \<times> 'c) set" shows "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
distinct_53: fixes x6 :: "Arraytype" shows "NullT \<noteq> ArrT x6"
incseq_SucD: fixes A :: "nat \<Rightarrow> 'd"   and i :: "nat" assumes "incseq A" shows "A i \<le> A (Suc i)"
INF_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq (Inf (f ` A)) x = (\<forall>y. less x y \<longrightarrow> (\<exists>i\<in>A. less (f i) y))"
prems_3: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
HH: shows "H = H \<inter> payload \<union> H \<inter> valid \<union> H \<inter> range LtK"
congruentI: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z" shows "f respects r"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "getM state = M @ [(opposite literal, decision)]"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantConsistent (getM state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "uniq Wl'"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M" shows "let state' = notifyWatches_loop literal Wl' newWl state in (\<forall>l. l \<in> set (getQ state') - set (getQ state) \<longrightarrow> (\<exists>clause. clause el getF state \<and> literal el clause \<and> isUnitClause clause l (elements (getM state)))) \<and> (\<forall>clause. clause \<in> set Wl' \<longrightarrow> (\<forall>l. isUnitClause (getF state ! clause) l (elements (getM state)) \<longrightarrow> l \<in> set (getQ state')))"
interval_of': fixes s :: "'b word \<times> 'b word" shows "interval_of s = (case s of (s, e) \<Rightarrow> {s..e})"
quotient_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "A // r = (\<Union>x\<in>A. {r `` {x}})"
fc: shows "(\<lambda>p\<in>prime_factors n. 1) \<in> carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"
F4: shows "\<forall>x\<^sub>1. analz x\<^sub>1 \<subseteq> analz (synth x\<^sub>1)"
set: fixes x :: "'a" shows "set_delete_message (DeleteMessage x) = {x}"
prefix_trie_Lf: fixes xs :: "bool list"   and t :: "trie" shows "(prefix_trie xs t = Lf) = (xs = [] \<and> t = Lf)"
case_prodI: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
absorb2: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># a" shows "a \<inter># b = b"
rsum_def: fixes D :: "(real \<times> real \<times> real) list"   and f :: "real \<Rightarrow> real" shows "rsum D f = (\<Sum>(u, x, v)\<leftarrow>D. f x * (v - u))"
if_multithreaded_axioms: shows "if_multithreaded final r"
PEI_7: fixes fs :: "nat \<Rightarrow> ValD \<rightarrow> ValD" assumes "\<And>xs. xs \<in> unlr PE.delta \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr PE.delta" shows "(\<lambda>i. ValF\<cdot>(fs i)) \<in> unlr PE.delta"
TOTALS: shows "\<forall>a\<in>Field s. \<forall>b\<in>Field s. (a, b) \<in> s \<or> (b, a) \<in> s"
has_vector_derivative: fixes f :: "real \<Rightarrow> 'd"   and f' :: "'d"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'e"   and g' :: "'e" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. prod (f x) (g x)) has_vector_derivative prod (f x) g' + prod f' (g x)) (at x within s)"
Step_G: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat" shows "step_result (SG, C) (i, G) = (case SG ! i of (x, [! (\<langle> q \<rangle> ! p)]) \<Rightarrow> (([], [p]) # closeI SG i, C))"
codomains_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.codomains C f = {partial_magma.cod C f}"
additive_def: fixes f :: "'a \<Rightarrow> 'b" shows "additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
unit_is_iso: shows "D.iso (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
left_commute: fixes b :: "'c option"   and a :: "'c option"   and c :: "'c option" shows "least b (least a c) = least a (least b c)"
fo_interpretation_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "fo_interpretation x = congruence x"
proj_finite: fixes t :: "cnf llist"   and c :: "'id" assumes "lfinite t" shows "lfinite (\<pi>\<^bsub>c\<^esub>t)"
rel_witness_fun2: fixes A :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "rel_fun (A OO A') B f g"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_fun A' (\<lambda>(x, y') y. y = y' \<and> B x y') (rel_witness_fun A A' (f, g)) g"
edom_Aheap: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "ArityAnalysisHeapSafe Aexp Aheap" shows "edom (Aheap \<Gamma> e\<cdot>a) \<subseteq> domA \<Gamma>"
wf_nFOr: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" assumes "local.wf idx (FOr \<phi> \<psi>)" shows "local.wf idx (nFOr \<phi> \<psi>)"
JE: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f"   and "\<I> p"   and "\<I> q" shows "(p, q) \<in> b \<union> m \<union> ov \<union> s \<union> d \<union> f\<inverse> \<union> e \<union> f \<union> s\<inverse> \<union> d\<inverse> \<union> ov\<inverse> \<union> m\<inverse> \<union> b\<inverse>"
MkArr_Map: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" shows "A_B.MkArr (A_B.Dom f) (A_B.Cod f) (A_B.Map f) = f"
create_document_def: shows "create_document \<equiv> l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_document"
square_part_0_iff: fixes n :: "'b" shows "(square_part n = (0::'b)) = (n = (0::'b))"
simps_2: fixes n :: "nat" shows "ord_of_nat (Suc n) = ZFC_in_HOL.succ (ord_of_nat n)"
intros_7: fixes X :: "freemsg"   and Y :: "freemsg" assumes "msgrelp X Y" shows "msgrelp Y X"
u_eq_v_AE: fixes n :: "nat" shows "AE x in M. v n x = u n x"
F_def: shows "F = (\<lambda>f. \<Sum>i\<in>Basis. (E * real (f i)) *\<^sub>R i) ` (Basis \<rightarrow>\<^sub>E {0..<n})"
sel_1: fixes x1 :: "('d \<times> 'e \<times> 'e) set"   and x2 :: "'e set" shows "edges (LG x1 x2) = x1"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y ys. P (opr x y # ys) \<Longrightarrow> P (x # y # ys)" shows "P a0"
mod_add_cong: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and c :: "'c"   and a' :: "'c"   and b :: "'c"   and b' :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (plus a b) c = modulo (plus a' b') c"
minus_1_inverse_pow: fixes x :: "'b"   and n :: "nat" shows "x / (- (1::'b)) ^ n = x * (- (1::'b)) ^ n"
H'_Suc: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "T.prob x (H' y (Suc n)) = U x y * T.prob y (H' y n)"
is_extensional: fixes f :: "'s" assumes "\<not> S.arr f" shows "\<Phi> f = S'.null"
mono_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat"   and A :: "'a set" shows "Dom_Kildall.mono r step n A = (\<forall>\<tau> p \<tau>'. \<tau> \<in> A \<and> p < n \<and> \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>' \<longrightarrow> set (step p \<tau>) {\<sqsubseteq>\<^bsub>r\<^esub>} set (step p \<tau>'))"
fv_sub: shows "fv \<phi> \<subseteq> fv \<psi>"
weakPsiCongParPres: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi" assumes "\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> P \<parallel> R \<doteq> Q \<parallel> R"
triangle_in_hom_4: shows "\<guillemotleft>\<r>[f\<^sup>*] \<cdot> (f\<^sup>* \<star> \<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<eta> \<star> f\<^sup>*) \<cdot> \<l>\<^sup>-\<^sup>1[f\<^sup>*] : f\<^sup>* \<Rightarrow> f\<^sup>*\<guillemotright>"
inv_unique: fixes y :: "'c"   and x :: "'c"   and y' :: "'c" assumes "y \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"   and "x \<otimes>\<^bsub>H\<^esub> y' = \<one>\<^bsub>H\<^esub>"   and "x \<in> carrier H"   and "y \<in> carrier H"   and "y' \<in> carrier H" shows "y = y'"
Pi_mono: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and C :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi A B \<subseteq> Pi A C"
l2_child: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and dir :: "dir"   and ds :: "nat set" assumes "d < length b"   and "p \<in> grid (child b dir d) ds" shows "l2_\<phi> (p ! d) (b ! d) = (1 - real_of_int (Grid_Point.sgn dir) * (real_of_int (ix p d) / 2 ^ (lv p d - lv b d) - real_of_int (ix b d))) / 2 ^ (lv p d + 1)"
rep_eq: fixes x :: "'b" shows "bounds_of_interval (interval_of x) = (x, x)"
nonneg: fixes x :: "'alt" shows "0 \<le> pmf p x + \<epsilon> * (pmf q' x - pmf p' x)"
axioms_1: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" assumes "open_cover_of_open_subset X is_open U I C" shows "open_cover_of_subset X is_open U I C"
measurable_piecewise_restrict2: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure"   and N :: "'b measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>n. A n \<in> sets M"   and "space M = \<Union> (range A)"   and "\<And>n. \<exists>h\<in>M \<rightarrow>\<^sub>M N. \<forall>x\<in>A n. f x = h x" shows "f \<in> M \<rightarrow>\<^sub>M N"
mcont2mcont: fixes lub' :: "'a set \<Rightarrow> 'a"   and ord' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b llist"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'a" assumes "mcont lub' ord' lSup lprefix f"   and "mcont lub ord lub' ord' t" shows "mcont lub ord lSup lprefix (\<lambda>x. f (t x))"
LIMSEQ_power_zero: fixes x :: "'a" assumes "norm_class.norm x < 1" shows "(^) x \<longlonglongrightarrow> (0::'a)"
Un_ac_4: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
root_2: shows "([:- a, 1::'a:] * p) \<circ>\<^sub>p q \<noteq> 0"
freeword_funlift_def: fixes f :: "'c \<Rightarrow> 'd" shows "freeword_funlift f \<equiv> THE g. g \<circ> Abs_freeletter = f \<and> UGroupHom g"
rec_by_index_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "rec_by_index x1 y1 = rec_by_index x2 y2" shows "x1 = x2 \<and> y1 = y2"
hom: shows "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) (f \<circ> g) id"
simps_9: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "pnPlus (Plus r s) (Plus v va) = pnPlus r (pnPlus s (Plus v va))"
assms_2: shows "v11 (g \<restriction>\<^sup>l\<^sub>\<circ> f `\<^sub>\<circ> A)"
mat_multI_def: fixes ze :: "'a"   and pl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ti :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and nr :: "nat"   and m1 :: "'a list list"   and m2 :: "'a list list" shows "mat_multI ze pl ti nr m1 m2 \<equiv> map (matT_vec_multI ze pl ti (Matrix_Legacy.transpose nr m1)) m2"
image_diff_subset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" shows "f ` A - f ` B \<subseteq> f ` (A - B)"
td_bl: shows "type_definition to_bl of_bl {bl. length bl = LENGTH('b)}"
hoare3_sound: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3 {P} c { Q}" shows "\<Turnstile>\<^sub>3 {P} c { Q}"
tauPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "\<tau>.P \<leadsto>\<^sup>^<Rel> \<tau>.Q"
lm035: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "(Domain P \<subseteq> Domain Q) = (P +< Q = Q)"
Im_divide_Reals: fixes r :: "complex"   and z :: "complex" assumes "r \<in> \<real>" shows "Im (z / r) = Im z / Re r"
SD_efficient_lottery_exists: fixes p :: "'alt pmf" assumes "p \<in> lotteries_on alts" obtains q :: "'alt pmf" where "q \<in> lotteries_on alts" and "Pareto (SD \<circ> R) p q" and "SD_efficient R q"
xor_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(x \<oplus> y) \<oplus> z = x \<oplus> y \<oplus> z"
b_def: shows "b \<equiv> \<lambda>k. let (dk, ak, bk, M) = DF (Suc k) in bk"
dim_hyperplane: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
repr_independenceD: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and y :: "'c"   and x :: "'c" assumes "Group.group G"   and "subgroup H G"   and "y \<in> carrier G"   and "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y" shows "y \<in> H #>\<^bsub>G\<^esub> x"
extended_simps_10: fixes x :: "'b extended" shows "(\<stileturn> \<le> x) = (x = \<stileturn>)"
circline_type_neg_card_gt3_diag: fixes H :: "circline" assumes "circline_type H < 0"   and "circline_diag H" shows "\<exists>A B C. A \<noteq> B \<and> A \<noteq> C \<and> B \<noteq> C \<and> {A, B, C} \<subseteq> circline_set H"
set_mset_subst_cls_mset_subst_clss: fixes CC :: "'a literal multiset multiset"   and \<mu> :: "'s" shows "set_mset (CC \<cdot>cm \<mu>) = set_mset CC \<cdot>cs \<mu>"
r: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "r a b = r' a b"
assms_2: shows "\<guillemotleft>f' : a \<rightarrow> a'\<guillemotright>"
bind_cong: fixes x :: "'d option"   and y :: "'d option"   and f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "Option.bind x f = Option.bind y g"
firstInNeta: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "\<not> member DenyAll a" shows "first_srcNet a \<in> set (net_list_aux (a # p))"
S3: shows "ls = c_unfold (c_len u) (c_snd (u - 1))"
not_le: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(\<not> lex_pm x y) = lex_pm_strict y x"
R12_a: shows "pmf (sds R12) a = pmf (sds R9) a"
prems_1: shows "\<Gamma>\<turnstile>Catch c1' c2' \<down> Normal s"
d_absorb1: fixes x :: "'a"   and y :: "'a" shows "d x + d x \<cdot> d y = d x"
Nil2_notin_lex: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set" shows "(xs, []) \<notin> lex r"
b: shows "u \<cdot> w \<cdot> y \<le> w \<cdot> y \<cdot> z"
fvi_trm_plus: fixes x :: "nat"   and b :: "nat"   and c :: "nat"   and t :: "Formula.trm" shows "(x \<in> Formula.fvi_trm (b + c) t) = (x + c \<in> Formula.fvi_trm b t)"
Cons_1: fixes L' :: "(real mpoly \<times> nat) list"   and a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list" assumes "\<forall>a\<in>set L'. case a of (p, i) \<Rightarrow> p = 0"   and "(a, b) = chophelper L L'" shows "\<forall>a\<in>set a. case a of (p, i) \<Rightarrow> p = 0"
igWlsBinp_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'm option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
summedBidVectorCharacterization: fixes N :: "'c set"   and G :: "'d set"   and bids :: "'c \<times> 'd set \<Rightarrow> 'e" shows "N \<times> (Pow G - {{}}) = Domain (summedBidVectorRel bids N G)"
eligible: fixes DA :: "'a literal multiset"   and As :: "'a list" assumes "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0) DA" shows "eligible As DA"
in_snd_sym_preproc_addnewE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" assumes "p \<in> set (snd (sym_preproc_addnew gs vs fs v))" obtains "p \<in> set fs"   | g :: "'t \<Rightarrow>\<^sub>0 'c" and s :: "'a" where "g \<in> set gs" and "p = monom_mult (1::'c) s g"
assms_2: shows "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>"
orderE: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "semilattice_order f less_eq less"   and "less_eq a b" obtains "a = f a b"
pr_simps_5: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (pr1 a b) = HF'.prod a b"
iso_digraph_is_iso_dghm: shows "\<exists>\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
big_d: fixes a :: "nat list"   and b :: "nat list"   and ys :: "nat list"   and i :: "nat" shows "set (big_d a b ys i) = hlde_ops.Di a b i ys"
contour_integrable_on: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" shows "(f contour_integrable_on g) = ((\<lambda>t. f (g t) * vector_derivative g (at t)) integrable_on {0..1})"
closed_orbit_\<omega>_limit_set: fixes x :: "'a" assumes "closed_orbit x" shows "range (flow0 x) = \<omega>_limit_set x"
fminus_finter2: fixes A :: "'c fset"   and C :: "'c fset"   and B :: "'c fset" shows "A |\<inter>| C |-| (B |\<inter>| C) = A |\<inter>| C |-| B"