hcju/isabelleps huggingface dataset:
vars_change_cons: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap"   and s' :: "('a, 'b) fmap"   and ss :: "('a, 'b) fmap list" assumes "vars_change as vs s = s' # ss" shows "\<exists>as1 act as2. as = as1 @ act # as2 \<and> vars_change as1 vs s = [] \<and> state_succ (exec_plan s as1) act = s' \<and> vars_change as2 vs (state_succ (exec_plan s as1) act) = ss"
casts_casts_eq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and T :: "ty"   and v :: "val"   and v' :: "val"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> T casts v to v "   and "P \<turnstile> T casts v to v' "   and "wf_prog wf_md P" shows "v = v'"
blind_path_def: fixes e :: "'a \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'c" shows "blind_path e h \<equiv> map (blind_path_elem e h)"
simps_2: fixes l :: "('b \<times> 'c) tree"   and a :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree" shows "mset_tree \<langle>l, (a, uu), r\<rangle> = {#a#} + mset_tree l + mset_tree r"
order_emb_def: fixes A :: "('a, 'b) gorder_scheme"   and B :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "order_emb A B f \<equiv> weak_partial_order A \<and> weak_partial_order B \<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. (f x \<sqsubseteq>\<^bsub>B\<^esub> f y) = (x \<sqsubseteq>\<^bsub>A\<^esub> y))"
D_dest: fixes x :: "'c"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "x = ??.Floyd_Warshall.linordered_ab_monoid_add_class.D m i j k" shows "x \<in> {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
y_axis_cmat_def: shows "y_axis_cmat = (0, 1, 1, 0)"
eq_trans: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.Eq_equiv eq"   and "eq\<cdot>x\<cdot>y = TT"   and "eq\<cdot>y\<cdot>z = TT" shows "eq\<cdot>x\<cdot>z = TT"
winding_number_rectpath: fixes z :: "complex"   and a1 :: "complex"   and a3 :: "complex" assumes "z \<in> box a1 a3" shows "winding_number (rectpath a1 a3) z = 1"
ih2: assumes "s'' \<noteq> 0" shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (s'' + length A div 2, l'', r'')"
inject_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x1 :: "'c"   and y1 :: "'c" assumes "antiparallel_edges \<Delta>" shows "(antiparallel_edges.Vertex x1 = antiparallel_edges.Vertex y1) = (x1 = y1)"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'd" shows "norm x = onorm ((*\<^sub>v) x)"
open_Int_closure_subset: fixes S :: "'b set"   and T :: "'b set" assumes "open S" shows "S \<inter> closure T \<subseteq> closure (S \<inter> T)"
lim1: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. f n $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub f $$ (i, j)"
coms_main_inv: fixes G :: "diagram"   and c :: "command" assumes "coms_dia G c" obtains \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" and V :: "node fset" and \<Lambda> :: "node \<Rightarrow> assertion_gadget" and E :: "(node fset \<times> command_gadget \<times> node fset) list" and cs :: "command list" where "G = Graph V \<Lambda> E" and "c = foldr (;;) cs Skip" and "\<pi> \<in> lins (Graph V \<Lambda> E)" and "length cs = length \<pi>" and "\<forall>i<length \<pi>. (case \<pi> ! i of Inl x \<Rightarrow> (coms_ass \<circ> \<Lambda>) x | Inr x \<Rightarrow> (coms_com \<circ> snd3) x) (cs ! i)"
transpose_mat_def: fixes A :: "'b mat" shows "A\<^sup>T \<equiv> mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
no_cs_change_if_no_channel: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "channel i = None" shows "cs c i = cs c' i"
esqrt_tendsto: fixes u :: "'a \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'a filter" assumes "(u \<longlongrightarrow> l) F" shows "((\<lambda>n. esqrt (u n)) \<longlongrightarrow> esqrt l) F"
similar'_take: fixes n :: "nat"   and d :: "Value"   and e :: "CValue" assumes "d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> e" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e"
singleActionChain_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'" shows "P \<Longrightarrow>a<\<nu>x> \<prec> P'"
vreal_inverse_def: shows "vreal_inverse = (\<lambda>x\<in>\<^sub>\<circ>\<real>\<^sub>\<circ>. (inverse (real_of_vreal x))\<^sub>\<real>)"
lookup_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and k :: "'k" shows "lookup (vectorize_poly p) k = proj_poly k p"
le_less_trans: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<preceq> y"   and "y \<prec> z" shows "x \<prec> z"
Suffixes_def: fixes A :: "'a list set" shows "Suffixes A = {w. \<exists>q. q @ w \<in> A}"
additive_freeword_funlift: fixes f :: "'c \<Rightarrow> 'd" shows "UGroupHom (freeword_funlift f)"
clsr_def: fixes T :: "'b set set"   and a :: "'b set" shows "closure\<^bsub>T\<^esub> a = \<Inter> {c. c closed\<^bsub>T\<^esub> \<and> a \<subseteq> c}"
3: shows "minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
Ints_induct: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and q :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.ring_1 one times plus zero minus uminus"   and "q \<in> ring_1.Ints one plus zero minus"   and "\<And>z. P (ring_1.of_int one plus zero minus z)" shows "P q"
subclsRD: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> C \<prec>\<^sub>R D" shows "\<exists>fs ms Bs. class P C = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> Repeats D \<in> set Bs"
distinct_8: fixes x21 :: "'a"   and x22 :: "'a form list" shows "ff \<noteq> Compound x21 x22"
image_subset_iff_subset_vimage: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and B :: "'e set" shows "(f ` A \<subseteq> B) = (A \<subseteq> f -` B)"
d1_2: fixes x :: "'n" assumes "dist a x < d1" shows "onorm (\<lambda>v. f' x v - f' a v) < k"
gen_length_cons_def: fixes g :: "('e, 'f) generator"   and n :: "nat"   and s :: "'f" shows "gen_length_cons g n s = n + length (unstream g s)"
is_basis_columns_Gram_Schmidt_matrix: fixes A :: "((real, 'b) vec, 'a) vec" assumes "is_basis (columns A)"   and "card (columns A) = ncols A" shows "is_basis (columns (Gram_Schmidt_matrix A)) \<and> card (columns (Gram_Schmidt_matrix A)) = ncols A"
Reds_dom_lcl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"
lspasl_magicl_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj" shows "(heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> sep_algebra.sep_impl plus sep_disj A B h2) = (heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> A h1 \<and> sep_algebra.sep_impl plus sep_disj A B h2 \<or> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> B h0 \<and> sep_algebra.sep_impl plus sep_disj A B h2)"
mndet_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "mndet A P \<equiv> if A = {} then STOP else Abs_process (\<Union>x\<in>A. F (x \<rightarrow> P x), \<Union>x\<in>A. D (x \<rightarrow> P x))"
least_pr_weak'_trans: fixes f :: "'f"   and g :: "'f \<times> nat" assumes "least f"   and "kbo'.pr_weak' (f, 0) g" shows "least (fst g) \<and> snd g = 0"
part: fixes p :: "'a"   and xs :: "'a list" shows "part p xs = (rev (filter (\<lambda>x. f x < f p) xs), p # filter (\<lambda>x. f x = f p) xs, rev (filter (\<lambda>x. f p < f x) xs))"
mult_strict_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "(0::'a) < c" shows "a * c < b * c"
G_UN: shows "Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle> = (\<lambda>g. BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {g})) ` FreeGroup S"
set_rev: fixes xs :: "'k list" shows "set (rev xs) = set xs"
nat_approx_posE: fixes e :: "'a" assumes "(0::'a) < e" obtains n :: "nat" where "(1::'a) / of_nat (Suc n) < e"
simps_3: shows "subcomp sA3 = {sA31, sA32}"
fps_XD'_add: fixes a :: "'c fps"   and b :: "'c fps"   and c :: "'c fps" shows "fps_XD' a (b + c) = fps_XD' a b + fps_XD' a c"
seq_compact_closed_subset: fixes s :: "'a set"   and t :: "'a set" assumes "closed s"   and "s \<subseteq> t"   and "seq_compact t" shows "seq_compact s"
ld_asi_user: fixes instr :: "sparc_operation \<times> inst_operand list" assumes "fst instr = load_store_type LDSB \<or> fst instr = load_store_type LDUB \<or> fst instr = load_store_type LDUH \<or> fst instr = load_store_type LD \<or> fst instr = load_store_type LDD" shows "ld_asi instr 0 = 10"
real: shows "connect_obsf (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) (obsf_resource real_resource) = connect_obsf \<A> (obsf_resource (w_adv_real |\<^sub>= w_usr \<rhd> real_resource))"
mono_neutral_right: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'b)" shows "sum g T = sum g S"
high_proc\<^sub>1_def: fixes p :: "prog" shows "high_proc\<^sub>1 p \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma> ''lowIn'') + the (\<sigma> ''highIn'') \<le> the (\<sigma>' ''highOut'')"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "B.seq \<nu> \<mu>"   and "B.seq \<tau> \<sigma>" shows "\<tau> \<cdot> \<sigma> \<star> \<nu> \<cdot> \<mu> = (\<tau> \<star> \<nu>) \<cdot> (\<sigma> \<star> \<mu>)"
div_comp_result_def: fixes i :: "sparc_operation \<times> inst_operand list"   and temp_V :: "word1"   and temp_64bit :: "word64" shows "div_comp_result i temp_V temp_64bit \<equiv> if temp_V = 1 then if fst i = arith_type UDIV \<or> fst i = arith_type UDIVcc then 4294967295 else if fst i \<in> {arith_type SDIV, arith_type SDIVcc} then if 0 < temp_64bit then 2147483647 else word_of_int (0 - uint 2147483648) else ucast temp_64bit else ucast temp_64bit"
simps_12: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PFn x41 x42 x43 \<noteq> PUnit"
prems_2: shows "map dim_vec inputs = input_sizes (Pool m1 m2)"
none: assumes "ioprog a \<sigma> = None"   and "\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S ioprog) P" shows "Q"
weak_reduction_simulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_simulation Rel Cal" shows "weak_reduction_simulation (Rel\<^sup>+) Cal"
map_transfer: fixes Rb :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Sd :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_fun Rb Sd) (rel_fun (rel_option Rb) (rel_option Sd)) map_option map_option"
diff_diff_less: fixes i :: "nat"   and m :: "nat"   and n :: "nat" shows "(i < m - (m - n)) = (i < m \<and> i < n)"
invh_baldL_invc: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "invc r" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight r"
cd: shows "k cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> j"
hull_Un_subset: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set"   and t :: "'b set" shows "S hull s \<union> S hull t \<subseteq> S hull (s \<union> t)"
prems: shows "0 < degree ([:- a, 1:] * p)"
int_pow_closed: fixes x :: "'c"   and i :: "int" assumes "x \<in> carrier H" shows "x [^]\<^bsub>H\<^esub> i \<in> carrier H"
AllowedActs_SKIP: shows "AllowedActs \<bottom> = UNIV"
Prod_any_zero: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "finite {a. f a \<noteq> (1::'d)}"   and "f a = (0::'d)" shows "Prod_any f = (0::'d)"
even_degree: assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n"   and "finite E" shows "\<forall>v\<in>V. even (degree v G)"
class_wf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and c :: "base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "class P C = \<lfloor>c\<rfloor>"   and "wf_prog wf_md P" shows "wf_cdecl wf_md P (C, c)"
right_neutral: fixes a :: "'c \<times> 'd" shows "sup_prod a bot_prod = a"
lock_ok_las'_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and las :: "'a lock_actions" shows "lock_ok_las' ls t las \<equiv> \<forall>l. lock_actions_ok' (ls $ l) t (las $ l)"
rep_in_terms_of_tab: shows "rep r = r.T0.trnr\<^sub>\<epsilon> r r.tab"
Z_good_def: fixes g :: "nat \<Rightarrow> nat" shows "local.Z_good g = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n} \<and> f ` {0..<n} = g ` {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}}"
monom_eq_0_iff: fixes a :: "'a"   and n :: "nat" shows "(Polynomial.monom a n = 0) = (a = (0::'a))"
int_pow_mult: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> (i + j) = x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> i \<otimes>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> j"
runWriterT_def: shows "runWriterT \<equiv> writerT_rep"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_unary a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "wt t \<ge>\<^sub>p wt s" and "\<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s" and "num_args t = 1" and "\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0" and "arg t \<ge>\<^sub>t s"
wfactors_listassoc_cong_l: fixes fs :: "'a list"   and a :: "'a"   and fs' :: "'a list" assumes "wfactors G fs a"   and "fs [\<sim>] fs'"   and "a \<in> carrier G"   and "set fs \<subseteq> carrier G"   and "set fs' \<subseteq> carrier G" shows "wfactors G fs' a"
config_rand_set: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init"
cSUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and M :: "'c multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "\<Squnion> (f ` A) \<subseteq># M"
is_continuous: fixes U :: "'c set" assumes "is_open\<^sub>Y U" shows "is_open\<^sub>X (f \<^sup>\<inverse> X U)"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = sup a b" shows "b \<le> a"
remove_def: fixes x :: "'g"   and A :: "'g set" shows "Set.remove x A = A - {x}"
ex_ncol_cop: fixes D :: "'p"   and E :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "D \<noteq> E" shows "\<exists>F. Coplanar A B C F \<and> \<not> Col D E F"
f: shows "fst \<circ> snd = (\<lambda>(u, w, v). w)"
poly_y_x_pCons: fixes a :: "'a poly"   and p :: "'a poly poly" shows "poly_y_x (pCons a p) = poly_lift a + map_poly (pCons (0::'a)) (poly_y_x p)"
smap_union_empty1: fixes m :: "'c \<Rightarrow> 'd set" shows "{}. \<union>. m = m"
sts_def: shows "sts = scf_list (scf (f, length ts)) ts"
assms_11: shows "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
composition_closed: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<FF> z"   and "b \<in> \<FF> z" shows "+\<^bsub>z\<^esub> a b \<in> \<FF> z"
shared: assumes "s = True" shows "insert_weights s m2 w1 = insert_weights s m2 w2"
diff_induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "\<And>x. P x 0"   and "\<And>y. P 0 (Suc y)"   and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)" shows "P m n"
map_replicate_const: fixes k :: "'c"   and lst :: "'d list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
vec_lambda_eta: fixes g :: "('d, 'e) Finite_Cartesian_Product.vec" shows "(\<chi>i. g $h i) = g"
incident_polar_swap: fixes p :: "proj2"   and q :: "proj2" assumes "proj2_incident p (polar q)" shows "proj2_incident q (polar p)"
simps: fixes a :: "'b tensor" shows "cprank_max1 a = ((\<exists>A. a = A \<and> order A \<le> 1) \<or> (\<exists>A B. a = A \<otimes> B \<and> order A = 1 \<and> cprank_max1 B))"
simple_distributedI: fixes X :: "'a \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> real" assumes "simple_function M X"   and "\<And>x. 0 \<le> P x"   and "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = prob (X -` {x} \<inter> space M)" shows "simple_distributed M X P"
less_Node_dag: fixes l :: "dag"   and a :: "ref"   and r :: "dag"   and x :: "dag" assumes "Node l a r < x" shows "l < x \<and> r < x"
conversion_property: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red {p} f"   and "red {r} p q" shows "is_red {q} f \<or> is_red {r} f"
decode_state_at_initial_state: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<Phi>\<^sub>S\<inverse> \<Pi> \<A> 0 = \<Pi>\<^sub>I"
assms_3: shows "G2 \<in> {G \<times> I |G. G \<in> \<G>}"
quotient_of_inject_eq: fixes a :: "rat"   and b :: "rat" shows "(quotient_of a = quotient_of b) = (a = b)"
nat_pow_eone: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := H'\<rparr>)" shows "x [^]\<^bsub>G\<lparr>carrier := H'\<rparr>\<^esub> 1 = x"
funpow_suffix: fixes k :: "nat"   and j :: "nat" assumes "(f ^^ Suc k) a = (f ^^ k) a" shows "((f ^^ (j + k)) a, (f ^^ k) a) \<in> leq"
Suc: shows "(\<Sum> {0..n})\<^sup>2 = (\<Sum>i = 0..n. i ^ 3)"
poly_poly_lift: fixes p :: "'c poly"   and x :: "'c" shows "poly (poly_lift p) [:x:] = [:poly p x:]"
simE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P" shows "\<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "'b mat" assumes "\<And>j A. P 0 j A"   and "\<And>i j A. (\<And>x xa. \<lbrakk>x = A $$ (i, j); xa = (if A $$ (i, i + 1) = (1::'b) \<and> x \<noteq> (0::'b) then add_col_sub_row (- x) (Suc i) j A else A)\<rbrakk> \<Longrightarrow> P i j xa) \<Longrightarrow> P (Suc i) j A" shows "P a0 a1 a2"
Gromov_product_commute: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "Gromov_product_at e x y = Gromov_product_at e y x"
iT_Plus_iWeakSince_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<B> t2 I \<oplus> k. Q t2) = (P (t1 + k). t1 \<B> t2 I. Q (t2 + k))"
continuous_on_infdist: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "continuous_on S f" shows "continuous_on S (\<lambda>x. infdist (f x) A)"
h2d: shows "(Av\<^sup>2 = Bv\<^sup>2 * (sqrt Cv)\<^sup>2) = (Av\<^sup>2 = Bv\<^sup>2 * Cv)"
prems_1: shows "init_exp_of e0 = \<lfloor>unit\<rfloor>"
matrix_left_invertible_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "(\<exists>B. B ** matrix f = mat (1::'a)) = (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id)"
trans_ParTR: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>t s'" shows "(c1 | c2, s) \<rightarrow>c (c1, s')"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case SO of FO \<Rightarrow> f1 | SO \<Rightarrow> f2) = f2"
cod_cod: fixes f :: "'s" assumes "arr f" shows "cod (cod f) = cod f"
\<tau>rtrancl3p_trans: fixes s :: "'s2"   and tls :: "'tl2 list"   and s' :: "'s2"   and tls' :: "'tl2 list"   and s'' :: "'s2" assumes "s -\<tau>2-tls\<rightarrow>* s'"   and "s' -\<tau>2-tls'\<rightarrow>* s''" shows "s -\<tau>2-tls @ tls'\<rightarrow>* s''"
comp_def: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VV.comp g f = (if (B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)) \<and> (B.arr (fst g) \<and> B.arr (snd g) \<and> src\<^sub>B (fst g) = trg\<^sub>B (snd g)) \<and> B.VxV.cod f = B.VxV.dom g then B.VxV.comp g f else B.VxV.null)"