hcju/isabelleps huggingface dataset:
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and t :: "t" shows "h (case t of T_i32 \<Rightarrow> f1 | T_i64 \<Rightarrow> f2 | T_f32 \<Rightarrow> f3 | T_f64 \<Rightarrow> f4) = (case t of T_i32 \<Rightarrow> h f1 | T_i64 \<Rightarrow> h f2 | T_f32 \<Rightarrow> h f3 | T_f64 \<Rightarrow> h f4)"
assms_3: fixes u :: "'a"   and v :: "'a"   and es :: "'a list" assumes "u \<rightarrow>\<^bsub>G\<^esub> v"   and "P (v # es)" shows "P (u # v # es)"
distinct_5: fixes x31 :: "comb"   and x32 :: "comb" shows "S \<noteq> x31 \<bullet> x32"
not_sdom_not_dom: fixes i :: "nat"   and j :: "nat" assumes "\<not> strict_dom i j"   and "i \<noteq> j" shows "\<not> local.dom i j"
simps_2: fixes a :: "('c, 'd) Node" shows "toList (Single a) = nodeToList a"
scalar_equiv: fixes r :: "real"   and x :: "(real, 'a) vec" shows "r *s x = r *\<^sub>R x"
dom_restrict: fixes m :: "'b \<Rightarrow> 'c option"   and A :: "'b set" shows "dom (m |` A) = dom m \<inter> A"
normalize_mult_normalize_right: fixes a :: "'a"   and b :: "'a" shows "normalize (a * normalize b) = normalize (a * b)"
pred_envir_conv: fixes p :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd" shows "pred_envir p f = (\<forall>x. p (f x))"
keys_is_none_rep: fixes k :: "'d"   and m :: "('d, 'e) mapping" shows "(k \<in> Mapping.keys m) = (\<not> Option.is_none (Mapping.lookup m k))"
runWriterT_liftWT: fixes m :: "'e\<cdot>'g" shows "runWriterT\<cdot>(liftWT\<cdot>m) = fmap\<cdot>(Writer\<cdot>mempty)\<cdot>m"
setr_def: shows "setr \<equiv> \<lambda>s. {x. setrp s x}"
side: shows "l \<notin> LID\<^sub>G s\<^sub>1"
simps: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" shows "all_deducible_clauses R S = {C. \<exists>P1 P2. P1 \<in> S \<and> P2 \<in> S \<and> R P1 P2 C}"
lens_indepI: fixes x :: "'b \<Longrightarrow> 'd"   and y :: "'c \<Longrightarrow> 'd" assumes "\<And>u v \<sigma>. put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"   and "\<And>v \<sigma>. get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"   and "\<And>u \<sigma>. get\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) = get\<^bsub>y\<^esub> \<sigma>" shows "x \<bowtie> y"
multiple_point_index: fixes n :: "nat"   and ps :: "'a set" shows "multiple_blocks n index ps = \<B> index ps * n"
0: shows "reflp (\<lambda>a\<^sub>1 a\<^sub>2. g a\<^sub>1 = g a\<^sub>2)"
induct: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "term list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "term list"   and a1 :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "P [] []"   and "\<And>t ts u us. (\<And>a. term.match unapp unconst t u = Some a \<Longrightarrow> P ts us) \<Longrightarrow> P (t # ts) (u # us)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
Group_Gp: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "Group (\<natural>\<^bsub>G\<^esub>H)"
obtain_v1_v2_4: shows "dep PROB v1 v2"
"1_1": assumes "i < j"   and "swap_cols_rows i j A \<in> carrier_mat n n"   and "j < n"   and "Suc i \<le> j" shows "similar_mat (swap_cols_rows_block (Suc i) j (swap_cols_rows i j A)) (swap_cols_rows i j A)"
poly_altdef: fixes p :: "'a poly"   and x :: "'a" shows "poly p x = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"
to_in_S: fixes t :: "transition"   and xb :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "\<exists>to from uid. (uid, (from, to), t) |\<in>| xb \<longrightarrow> to |\<in>| Inference.S xb"
step: shows "P,E,h \<turnstile> e' : T'' \<and> P \<turnstile> T'' \<le> U"
farkas_minkowsky_weyl_theorem_1: fixes X :: "'a vec set"   and Bnd :: "'a" assumes "X \<subseteq> carrier_vec n"   and "finite X" shows "\<exists>nr A. A \<in> carrier_mat nr n \<and> cone X = polyhedral_cone A \<and> (X \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec Bnd \<longrightarrow> A \<in> \<int>\<^sub>m \<inter> Bounded_mat (det_bound (n - 1) (max (1::'a) Bnd)))"
mult_1: fixes a :: "'e" shows "(1::'e) \<cdot> a = a"
segment_2: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
M1_cardeq_rcosetGM1g: fixes g :: "'a" assumes "g \<in> carrier G" shows "card (M1 #> g) = card M1"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>xs. P [] xs"   and "\<And>v va. P (v # va) []"   and "\<And>a xs b ys. \<lbrakk>a = b \<Longrightarrow> P xs (b # ys); \<lbrakk>a \<noteq> b; a < b\<rbrakk> \<Longrightarrow> P xs (b # ys); \<lbrakk>a \<noteq> b; \<not> a < b\<rbrakk> \<Longrightarrow> P (a # xs) ys\<rbrakk> \<Longrightarrow> P (a # xs) (b # ys)" shows "P a0 a1"
powr_real_of_int: fixes x :: "real"   and n :: "int" assumes "0 < x" shows "x powr real_of_int n = (if 0 \<le> n then x ^ nat n else inverse (x ^ nat (- n)))"
instantiate_var_same_typ: fixes a :: "Core.term"   and \<tau> :: "typ"   and B :: "Core.term"   and lev :: "nat"   and x :: "variable" assumes "typ_of a = Some \<tau>"   and "\<not> loose_bvar B lev" shows "subst_bv1 (bind_fv2 (x, \<tau>) lev B) lev a = subst_term [((x, \<tau>), a)] B"
assms_2: shows "j < dim_col (Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)))"
Q: shows "Q \<in> carrier_mat CARD('nc) CARD('nc)"
I_3: fixes i :: "'i" assumes "i \<in> I" shows "integrable M (X i)"
rel_sel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f) nondetT"   and b :: "('e, 'g) nondetT" shows "rel_nondetT' R1 R2 a b = R2 (run_nondet a) (run_nondet b)"
distinct_23: fixes x31 :: "acom"   and x32 :: "acom"   and x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom" shows "x31;; x32 \<noteq> {x61} WHILE x62 DO x63"
succ_def: fixes x :: "hf" shows "succ x \<equiv> x \<triangleleft> x"
at_neq_bot: fixes a :: "'c" shows "at a \<noteq> bot"
approx_mem_monad2: fixes x :: "'a star"   and u :: "'a star" assumes "x \<approx> u" shows "x \<in> monad u"
conseq1: fixes G :: "prog"   and A :: "'b triple set"   and P' :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn"   and P :: "'b assn" assumes "G,A\<turnstile>{P'} t\<succ> {Q}"   and "P \<Rightarrow> P'" shows "G,A\<turnstile>{P} t\<succ> {Q}"
InitNonObject: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and r :: "fdecl list \<times> (char list list \<times> char list exp) mdecl list"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C \<noteq> Object"   and "class P C = \<lfloor>(D, r)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile> \<langle>INIT C' (D # C # Cs,False) \<leftarrow> e,(h, l, sh')\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
grounding_of_cls_ground: fixes C :: "'a literal multiset" assumes "is_ground_cls C" shows "\<G>_F C = {C}"
simps_2: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "remdups_on cmp (x # xs) = (if Bex (set xs) (cmp x) then remdups_on cmp xs else x # remdups_on cmp xs)"
sup_absorb2: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "sup x y = y"
borel_measurable_ereal2: fixes f :: "'c \<Rightarrow> ereal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ereal"   and H :: "ereal \<Rightarrow> ereal \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"   and "(\<lambda>x. H (- \<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"   and "(\<lambda>x. H \<infinity> (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"   and "(\<lambda>x. H (ereal (real_of_ereal (f x))) (- \<infinity>)) \<in> borel_measurable M"   and "(\<lambda>x. H (ereal (real_of_ereal (f x))) \<infinity>) \<in> borel_measurable M" shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
observable_moves_snoc: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')"   and "S,f \<turnstile> (ms',s') -a\<rightarrow> (ms'',s'')" shows "S,f \<turnstile> (ms,s) =as @ [a]\<Rightarrow> (ms'',s'')"
splus_adds_term: fixes t :: "'a"   and u :: "'t"   and v :: "'t" assumes "t \<oplus> u adds\<^sub>t v" shows "u adds\<^sub>t v"
kprecondition_def: fixes s :: "'state_t"   and a :: "'action_t" shows "kprecondition s a \<equiv> invariant s \<and> AS_precondition s (current s) a"
inner_scaleR_left: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and r :: "real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (scaleR r x) y = r * inner x y"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "cnf_to_bool uu [] = True"
ex_y: shows "\<forall>x\<in>A. \<exists>y\<in>A. (y, x) \<in> r"
is_embedding: fixes f :: "'a"   and f' :: "'a" assumes "F.A.arr f"   and "F.A.arr f'"   and "F f = F f'" shows "f = f'"
count_prime_factorization_prime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "count (factorial_semiring.prime_factorization zero times one normalize x) p = factorial_semiring.multiplicity times one p x"
fmlookup_restrict_set: fixes A :: "'d set"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmrestrict_set A m) x = (if x \<in> A then fmlookup m x else None)"
join: fixes a :: "'a"   and b :: "'a" assumes "h a = h b"   and "a \<in> A" shows "\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u)"
R: fixes U :: "ty" assumes "S = T \<rightarrow> U"   and "(x, T) # \<Gamma> \<turnstile> t : U" shows "P"
path_split_second: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "CFG.path sourcenode targetnode valid_edge (sourcenode a) (a # as') n'"
eq_neutr_iff: fixes a :: "'b"   and b :: "'b" shows "(a \<squnion> b = \<bottom>) = (a = \<bottom> \<and> b = \<bottom>)"
F: shows "equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"
mod_by_0: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"
equivalence_of_partition: shows "partition.Equivalence e1.Partition = {(x, y). stk1.rel x y}"
differentiable_chain_within: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and S :: "'b set"   and g :: "'c \<Rightarrow> 'd" assumes "f differentiable at x within S"   and "g differentiable at (f x) within f ` S" shows "g \<circ> f differentiable at x within S"
reaches: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>* b" shows "reaches a b"
finprod_closed: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "f \<in> A \<rightarrow> carrier (G\<lparr>carrier := J\<rparr>)" shows "finprod (G\<lparr>carrier := J\<rparr>) f A \<in> carrier (G\<lparr>carrier := J\<rparr>)"
prems_6: shows "ps (S t i) p \<noteq> None"
cases: fixes a1 :: "pi"   and a2 :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(a1, a2) \<in> bangRel Rel" obtains P :: "pi" and Q :: "pi" where "a1 = !Pa__" and "a2 = !Qa__" and "(Pa__, Qa__) \<in> Rel"   | R :: "pi" and T :: "pi" and P :: "pi" and Q :: "pi" where "a1 = R \<parallel> Pb__" and "a2 = T \<parallel> Qb__" and "(R, T) \<in> Rel" and "(Pb__, Qb__) \<in> bangRel Rel"   | P :: "pi" and Q :: "pi" and a :: "name" where "a1 = <\<nu>a>P" and "a2 = <\<nu>a>Q" and "(P, Q) \<in> bangRel Rel"
fin_succeed_ranks: shows "finite (\<Union> {succeed j |j. j < i})"
pentagon: fixes f :: "'d"   and g :: "'d"   and h :: "'d"   and k :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "D.ide k"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h"   and "src\<^sub>D h = trg\<^sub>D k" shows "(f \<star>\<^sub>D \<a>\<^sub>D[g, h, k]) \<cdot>\<^sub>D \<a>\<^sub>D[f, g \<star>\<^sub>D h, k] \<cdot>\<^sub>D (\<a>\<^sub>D[f, g, h] \<star>\<^sub>D k) = \<a>\<^sub>D[f, g, h \<star>\<^sub>D k] \<cdot>\<^sub>D \<a>\<^sub>D[f \<star>\<^sub>D g, h, k]"
Figure_2: fixes i :: "nat"   and k :: "nat" assumes "i \<le> k" shows "foldl1 f1 (map (h1 k i) [0..<k + 1]) = One"
induct: fixes P1 :: "ref_ty \<Rightarrow> bool"   and P2 :: "ty \<Rightarrow> bool"   and ref_ty :: "ref_ty"   and ty :: "ty" assumes "P1 NullT"   and "\<And>x. P1 (IfaceT x)"   and "\<And>x. P1 (ClassT x)"   and "\<And>x. P2 x \<Longrightarrow> P1 (ArrayT x)"   and "\<And>x. P2 (PrimT x)"   and "\<And>x. P1 x \<Longrightarrow> P2 (RefT x)" shows "P1 ref_ty \<and> P2 ty"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>\<tau>'.map f : F a \<rightarrow>\<^sub>C F' b\<guillemotright>"
assms: shows "\<forall>x\<in>vars\<^sub>t t. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
divides_degree: fixes p :: "'d poly"   and q :: "'d poly" assumes "p dvd q" shows "degree p \<le> degree q \<or> q = 0"
simps_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Deliver x2 = Deliver y2) = (x2 = y2)"
mergesort_remdups_length: fixes as :: "'b list" shows "length (mergesort_remdups as) \<le> length as"
rep_eq: fixes x :: "'c cset" shows "cfinite x = finite (rcset x)"
op_in12: shows "open (inside (path_image c1 \<union> path_image c2))"
hyps_3: fixes ys :: "'a list" assumes "distinct (y # zs)"   and "({y} \<union> set zs) \<inter> set ys = {}" shows "rtrancl_tab r ys y z"
relcomp_unfold: fixes r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
that_3: shows "\<forall>s\<in>S - {t}. f s = (0::'b)"
WSI: fixes ms :: "'node list"   and ms' :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and msx :: "'node list"   and mx :: "'node"   and S :: "'node SDG_node set" assumes "Ball (set ms) valid_node"   and "Ball (set ms') valid_node"   and "length ms = length s"   and "length ms' = length s'"   and "s \<noteq> []"   and "s' \<noteq> []"   and "ms = msx @ mx # tl ms'"   and "get_proc mx = get_proc (hd ms')"   and "\<forall>m\<in>set (tl ms'). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "msx \<noteq> [] \<longrightarrow> (\<exists>mx'. call_of_return_node mx mx' \<and> mx' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)"   and "\<forall>i<length ms'. snd (s ! (length msx + i)) = snd (s' ! i)"   and "Ball (set (tl ms)) return_node"   and "\<forall>i<length ms'. \<forall>V\<in>rv S (CFG_node ((mx # tl ms') ! i)). fst (s ! (length msx + i)) V = fst (s' ! i) V"   and "obs ms \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs ms' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "((ms, s), ms', s') \<in> WS S"
pruned_def: fixes g :: "'g" shows "braun_ssa.pruned g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> braun_ssa.phiDefs g n \<longrightarrow> braun_ssa.liveVal g val)"
fmap_ident: shows "fmap\<cdot>(\<Lambda> x. x) = ID"
ACI_nPlus_flatten_PLUS: fixes X1 :: "'b rexp set"   and X2 :: "'b rexp set" assumes "X1 \<noteq> {}"   and "X2 \<noteq> {}"   and "finite X1"   and "finite X2"   and "\<forall>x\<in>X1 \<union> X2. \<nexists>r s. x = Plus r s" shows "ACI_nPlus (flatten PLUS X1) (flatten PLUS X2) = flatten PLUS (X1 \<union> X2)"
select_convs_1: fixes tdir :: "'b"   and xdir :: "'b"   and ydir :: "'b"   and zdir :: "'b"   and more :: "'c" shows "tdir \<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir, \<dots> = more\<rparr> = tdir"
Can_red_2: fixes a :: "'a local.term" assumes "Ide a" shows "a\<^bold>\<down> \<in> Hom a \<^bold>\<lfloor>a\<^bold>\<rfloor>"
walk_first_edge: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "walk (v # w # xs)" shows "v \<rightarrow> w"
simps_1: fixes x :: "'k"   and uu :: "'m \<Rightarrow> ('k, 'l) result" shows "Error x \<bind> uu = Error x"
prefix_diff1: fixes l :: "'c list"   and m :: "'c list"   and t :: "'c list" shows "(l @ m) @ t - (l @ m) = t"
weakBisimE_1: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "P \<leadsto><weakBisim> Q"
bind_def: shows "(\<bind>) \<equiv> rec_option Map.empty (\<lambda>x f. f x)"
step_wf: shows "wf {(s', s). s \<in> wa_invar (det_wa_wa DWA) \<and> s \<in> wa_cond (det_wa_wa DWA) \<and> (s, s') \<in> wa_step (det_wa_wa DWA)}"
closure_cbox: fixes a :: "'c"   and b :: "'c" shows "closure (cbox a b) = cbox a b"
flowpipe_cont: fixes X0 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and hl :: "real"   and hu :: "real"   and CX :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and h :: "real"   and x0 :: "'a"   and d0 :: "'a \<Rightarrow>\<^sub>L 'a"   and h' :: "real" assumes "flowpipe X0 hl hu CX X1"   and "hl \<le> h"   and "h \<le> hu"   and "(x0, d0) \<in> X0"   and "0 \<le> h'"   and "h' \<le> h" shows "(flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX"
seq_suble: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono f" shows "n \<le> f n"
sublists_def: shows "sublists \<equiv> rec_list [[]] (\<lambda>x xs xsa. xsa @ map ((#) x) (prefixes xs))"
simps_8: fixes x2 :: "t list" shows "Bot \<noteq> Type x2"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "dim_row (mat nr nc f) = nr"
simps_33: fixes vd :: "'a"   and ve :: "bexp"   and vf :: "'a acom"   and vg :: "'a"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "({vd} WHILE ve DO vf {vg} \<le> IF v THEN va ELSE vb {vc}) = False"
sum_mono2: fixes B :: "'a set"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> {#} \<subseteq># f b" shows "comm_monoid_add.sum (+) {#} f A \<subseteq># comm_monoid_add.sum (+) {#} f B"
remdups_wrt_rec: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" shows "remdups_wrt f (x # xs) = (if f x \<in> f ` set xs then remdups_wrt f xs else x # remdups_wrt f xs)"
prems_4: fixes i :: "nat" assumes "i < chainlen \<Pi>" shows "prov_triple (asn (fst3 (nthtriple \<Pi> i)), (c # cs) ! i, asn (thd3 (nthtriple \<Pi> i)))"
beta_eta_appR: fixes t :: "dB"   and t' :: "dB"   and s :: "dB" assumes "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* s \<degree> t'"
smallest_accepting_rank_def: shows "\<MM>.smallest_accepting_rank\<^sub>R \<equiv> if \<MM>.accept\<^sub>R' then Some (LEAST i. finite \<MM>.fail \<and> finite (\<MM>.merge i) \<and> infinite (\<MM>.succeed i)) else None"
goal_sinvar_m: assumes "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> = {}" shows "c_sinvar m (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>)"
Fun: fixes x2a :: "('f, 'a) Term.term"   and t :: "('f, 'a) Term.term"   and u :: "('f, 'a) Term.term" assumes "x2a \<in> set ss" shows "(S x2a t \<longrightarrow> NS t u \<longrightarrow> S x2a u) \<and> (NS x2a t \<longrightarrow> S t u \<longrightarrow> S x2a u) \<and> (NS x2a t \<longrightarrow> NS t u \<longrightarrow> NS x2a u)"
intros_7: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Ghost A L R W # is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is, \<theta>, x, m, \<D>, \<O> \<union> A - R, augment_rels (dom \<S>) R \<R>, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
finite_Prod_UNIV: assumes "finite UNIV"   and "finite UNIV" shows "finite UNIV"
rot_circle_cube_def: shows "rot_circle_cube = prod.swap \<circ> circle_cube \<circ> prod.swap"
atMost_eq_UNIV_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.order_top less_eq less top" shows "(ord.atMost less_eq x = UNIV) = (x = top)"