hcju/isabelleps huggingface dataset:
sup_assoc: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
g: shows "\<guillemotleft>g : b \<rightarrow> a\<guillemotright>"
Hermite_upt_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set" shows "Hermite_upt_row A k associates residues = (Complete_set_non_associates associates \<and> Complete_set_residues residues \<and> echelon_form_upt_row A k \<and> (\<forall>i. mod_type_class.to_nat i < k \<and> \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> associates) \<and> (\<forall>i. mod_type_class.to_nat i < k \<and> \<not> is_zero_row i A \<longrightarrow> (\<forall>j<i. A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> residues (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))))"
fmrel_on_fsetI: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f" assumes "\<And>x. x |\<in>| S \<Longrightarrow> rel_option P (fmlookup m x) (fmlookup n x)" shows "fmrel_on_fset S P m n"
diff_intersect_sym_diff: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A - B) \<inter># (B - A) = {#}"
AC_refines: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> c"   and "b \<sqsubseteq> c" shows "(a \<Squnion> b) \<sqsubseteq> c"
map_pmf_comp: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and M :: "'g pmf" shows "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
B_in_ker: shows "B \<subseteq> {x. f x = (0::'c)}"
1: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0" obtains g :: "'t \<Rightarrow>\<^sub>0 'b" where "g \<in> G" and "lt g adds\<^sub>t lt f"
dns1: fixes x :: "'a" shows "d x \<cdot> x = x"
simps_5: fixes t1 :: "pdf_type"   and t2 :: "pdf_type" shows "countable_type (PRODUCT t1 t2) = (countable_type t1 \<and> countable_type t2)"
set_iterator_intro: fixes it :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord it S0 R" shows "set_iterator it S0"
leq: fixes i :: "nat" assumes "i \<le> Suc n" shows "min i n \<le> k"
P_new_hd_disjoint: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and xs :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new"   and "xs \<in> paths" shows "hd (tl P_new) \<noteq> hd (tl xs)"
add_less_same_cancel2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a + b \<subset># b) = (a \<subset># {#})"
while_unfold: fixes b :: "'b upred"   and P :: "'b hrel" shows "while\<^sup>\<top> b do P od = P ;; while\<^sup>\<top> b do P od \<triangleleft> b \<triangleright>\<^sub>r II"
nn_integral_monotone_convergence_SUP_AE: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. AE x in M. f i x \<le> f (Suc i) x"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Squnion>i. f i x) \<partial>M = (\<Squnion>i. integral\<^sup>N M (f i))"
norm_triangle_le: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (plus x y) \<le> e"
emeasure_pair_measure: fixes M :: "'c measure"   and X :: "('d \<times> 'c) set"   and N :: "'d measure" assumes "sigma_finite_measure M"   and "X \<in> sets (N \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M M) X = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N"
length_0_conv: fixes xs :: "'a list" shows "(length xs = 0) = (xs = [])"
classic_circle: fixes H :: "circline"   and a :: "complex"   and r :: "real" assumes "is_circle H"   and "(a, r) = euclidean_circle H"   and "circline_type H \<le> 0" shows "circline_set H = of_complex ` circle a r"
add_col_sub_row_same_upto: fixes i :: "nat"   and j :: "nat"   and n :: "nat"   and A :: "'b mat"   and v :: "'b" assumes "i < j"   and "j < n"   and "A \<in> carrier_mat n n"   and "inv_upto n ??.Jordan_Normal_Form_Existence.uppert A j" shows "same_upto n j A (add_col_sub_row v i j A)"
tpOfV_pred_def: fixes f :: "Preliminaries.var \<Rightarrow> 'tp" shows "tpOfV_pred f \<equiv> \<forall>\<sigma>. infinite (f -` {\<sigma>})"
scale_scale: fixes a :: "'d"   and b :: "'d"   and x :: "'d" shows "a * (b * x) = a * b * x"
assms_1: shows "0 \<le> fls_subdegree f"
argument_principle: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and poles :: "complex set"   and h :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "open s"   and "connected s"   and "f holomorphic_on s - poles"   and "h holomorphic_on s"   and "valid_path g"   and "pathfinish g = pathstart g"   and "path_image g \<subseteq> s - {w. f w = 0 \<or> w \<in> poles}"   and "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0"   and "finite {w. f w = 0 \<or> w \<in> poles}"   and "\<forall>p\<in>poles. is_pole f p" shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = complex_of_real (2 * pi) * \<i> * (\<Sum>p\<in>{w. f w = 0 \<or> w \<in> poles}. winding_number g p * h p * of_int (zorder f p))"
nhidinv: shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). l \<in> {PToy-:2..PToy-:8} \<longrightarrow> nhid (\<sigma> i) = state.id (\<sigma> i))"
refine2specI: fixes b :: "'b nres"   and a :: "'c nres"   and R :: "('c \<times> 'b) set" assumes "nofail b \<Longrightarrow> a \<le> SPEC (\<lambda>r. \<exists>x. inres b x \<and> (r, x) \<in> R)" shows "a \<le> \<Down> R b"
rel_gpv''_lossless_gpvD2: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'k) gpv"   and \<I>' :: "('i, 'k) \<I>"   and \<I> :: "('h, 'j) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "lossless_gpv \<I>' gpv'"   and "rel_\<I> C R \<I> \<I>'" shows "lossless_gpv \<I> gpv"
assms_3: fixes \<xi> :: "'a"   and \<xi>' :: "'a" assumes "S \<xi> \<xi>'" shows "U \<xi> \<xi>'"
delete_def: fixes a :: "'a"   and t :: "'a bro" shows "delete a t = tree (del a t)"
size1_height: fixes t :: "'b tree" shows "size1 t \<le> 2 ^ height t"
rep_eq: fixes x :: "bool"   and xa :: "bool"   and xb :: "bool"   and xc :: "bool"   and xd :: "bool"   and xe :: "bool"   and xf :: "bool"   and xg :: "String.literal" shows "literal.explode (String.Literal x xa xb xc xd xe xf xg) = Char x xa xb xc xd xe xf False # literal.explode xg"
r_pdec12_prim: shows "prim_recfn 1 r_pdec12"
RPDs_Plus: fixes r :: "mregex"   and s :: "mregex" shows "RPDs (MPlus r s) \<subseteq> {MPlus r s} \<union> RPDs r \<union> RPDs s"
paths: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and xs :: "'c list" assumes "DisjointPaths G v0 v1 paths"   and "xs \<in> paths" shows "v0 \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> v1"
subgame_finite: fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G" shows "finite (\<omega>\<^bsub>ParityGame.subgame G V'\<^esub> ` V\<^bsub>ParityGame.subgame G V'\<^esub>)"
subspace_Inter: fixes f :: "'d set set" assumes "\<forall>s\<in>f. csubspace s" shows "csubspace (\<Inter> f)"
inv_translationD_2: fixes r :: "nat set" assumes "inv_translation r" shows "1 \<in> r"
aux0_def: shows "aux0 = join_msaux args rel1 (add_new_ts_msaux args (\<tau> \<sigma> ne) aux)"
unfold_congs_9: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "nat \<Rightarrow> Literal option"   and f :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option"   and f' :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option" assumes "r = r'"   and "getWatch2 r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "getWatch2_update f r = getWatch2_update f' r'"
load_tmps_fresh_nth_update: fixes i :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "load_tmps is \<inter> dom \<theta> = {}" shows "load_tmps_fresh (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
dg_op_dg_op_dg: shows "op_dg (op_dg \<BB>) = \<BB>"
Pair_vimage_Sigma: fixes x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "Pair x -` Sigma A f = (if x \<in> A then f x else {})"
clm_decompose: fixes c :: "cars" shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1 \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (clm ts) c = lan u \<and> |lan u| = 1)"
list_eq_if: fixes xs :: "'b list"   and ys :: "'b list"   and i :: "nat" assumes "xs = ys"   and "length xs = length ys"   and "i < length xs" shows "xs ! i = ys ! i"
tendsto_in\<^sub>N_iff_limitin: fixes N :: "'a quasinorm"   and x :: "'a"   and u :: "nat \<Rightarrow> 'a" assumes "defect N = 1"   and "x \<in> space\<^sub>N N"   and "\<And>n. u n \<in> space\<^sub>N N" shows "tendsto_in\<^sub>N N u x = limitin (topology\<^sub>N N) u x sequentially"
applyExplain_def: fixes literal :: "Literal"   and state :: "State" shows "applyExplain literal state = (case getReason state literal of None \<Rightarrow> state | Some reason \<Rightarrow> let res = resolve (getC state) (getF state ! reason) (opposite literal) in setConflictAnalysisClause res state)"
bet_out_out_bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "B Out A A'"   and "B Out C C'" shows "Bet A' B C'"
ex1I: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "\<exists>!x. P x"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (\<sigma> f) = A.arr f"
induct: fixes P1 :: "'a exp \<Rightarrow> bool"   and P2 :: "'a exp list \<Rightarrow> bool"   and exp :: "'a exp" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3. P1 (x1\<bullet>\<^sub>sx2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (INIT x1 (x2,x3) \<leftarrow> x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x2; P1 x4\<rbrakk> \<Longrightarrow> P1 (RI (x1,x2) ; x3 \<leftarrow> x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 exp"
nonpos_Reals_def: shows "\<real>\<^sub>\<le>\<^sub>0 = {of_real r |r. r \<le> 0}"
collist: shows "list_all (coll 0 (sum_list xs)) xs"
base: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "rtrancl_tab r xs x x"
simps_1: fixes x :: "'c"   and ya :: "'c" shows "(dual x = dual ya) = (x = ya)"
normalize_smult: fixes c :: "'b"   and p :: "'b poly" shows "normalize (smult c p) = smult (normalize c) (normalize p)"
E'_def: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and V :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "Subgraph_Node_Defs.E' E V x y \<equiv> E x y \<and> V x \<and> V y"
notE_PL: fixes A :: "'a set"   and p :: "'a" assumes "A \<turnstile> .\<not> p" shows "A \<turnstile> p .\<longrightarrow>. \<bottom>"
local: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<zeta> :: "'b"   and R :: "nat set"   and a :: "'c node_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and \<zeta>' :: "'b"   and R' :: "nat set" assumes "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"   and "((\<sigma>, NodeS ii \<zeta> R), a, \<sigma>', NodeS ii \<zeta>' R') \<in> automaton.trans (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o)"   and "S \<sigma> \<sigma>' a"   and "P (\<sigma>, NodeS ii \<zeta> R)" shows "P (\<sigma>', NodeS ii \<zeta>' R')"
source_in_not_cycle: fixes p :: "'v list"   and x :: "'v" assumes "cycle \<Delta> p" shows "(x, source \<Delta>) \<notin> set (cycle_edges p)"
rbt_del_inv1_inv2: fixes lt :: "('a, 'c) rbt"   and x :: "'a" assumes "inv2 lt"   and "inv1 lt" shows "inv2 (rbt_del x lt) \<and> (color_of lt = R \<and> bheight (rbt_del x lt) = bheight lt \<and> inv1 (rbt_del x lt) \<or> color_of lt = B \<and> bheight (rbt_del x lt) = bheight lt - 1 \<and> inv1l (rbt_del x lt))"
pr_key_parts_intro: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Key K \<in> parts (A \<union> spies evs)" shows "Key K \<in> A"
stream_dec_not_Inr: fixes n :: "nat"   and FO :: "nat set"   and s :: "('a \<times> bool list) stream"   and i :: "nat"   and P :: "nat set" assumes "stream_dec n FO s ! i = Inr P"   and "i \<in> FO"   and "i < n" shows "False"
n_preserve1_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "!x \<cdot> y \<le> !x \<cdot> y \<cdot> !x" shows "!x \<cdot> (!x \<cdot> y + t x \<cdot> z)\<^sup>\<dagger> \<le> (!x \<cdot> y)\<^sup>\<dagger> \<cdot> !x"
vprojection_app: fixes x :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V"   and i :: "V" assumes "x \<in>\<^sub>\<circ> vproduct I A" shows "vprojection I A i\<lparr>x\<rparr> = x\<lparr>i\<rparr>"
annot_def: fixes t :: "('d, 'c) FingerTree" shows "annot t \<equiv> FingerTreeStruc.annot (Rep_FingerTree t)"
wffp_last_purgeIdle: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "purgeIdle sl \<noteq> []" shows "local.getGMState (last (purgeIdle sl)) = local.getGMState (last sl)"
IH: fixes x4a :: "('msaux, 'muaux) mformula"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi>' :: "Formula.formula"   and P' :: "String.literal \<Rightarrow> nat option"   and db :: "(String.literal, event_data list set list) mapping" assumes "x4a \<in> set \<phi>s"   and "wf_mformula \<sigma> j P V n R x4a \<phi>'"   and "wf_envs \<sigma> j P P' V db" shows "case meval n (\<tau> \<sigma> j) db x4a of (xs, \<phi>\<^sub>n) \<Rightarrow> wf_mformula \<sigma> (Suc j) P' V n R \<phi>\<^sub>n \<phi>' \<and> list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>')) [Monitor.progress \<sigma> P \<phi>' j..<Monitor.progress \<sigma> P' \<phi>' (Suc j)] xs"
UnderS_subset_Under: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "UnderS r A \<subseteq> Under r A"
card_image: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "card (f ` A) = card A"
contI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>Complete_Partial_Order.chain orda Y; Y \<noteq> {}\<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)" shows "cont luba orda lubb ordb f"
comp_assoc_assoc'_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D[f, g, h] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = f \<star>\<^sub>D g \<star>\<^sub>D h"
def_sel: shows "\<tau> \<Turnstile> \<delta> select_object_any\<^sub>S\<^sub>e\<^sub>t f s_set"
ltl_implies_provable: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "w \<Turnstile> \<phi>" shows "\<exists>j. \<G>\<^sub>F\<^sub>G \<phi> w \<Turnstile>\<^sub>P af\<^sub>G \<phi> (prefix j w)"
af_letter_abs_def: shows "\<up>af \<equiv> af_abs.f_abs"
idx_unidx: fixes i :: "nat" assumes "i < length xs" shows "idx (unidx i) = i"
inject: fixes x1 :: "'a"   and x2 :: "'c"   and y1 :: "'a"   and y2 :: "'c" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
store_alloc_def: fixes v2 :: "'b store_v"   and st :: "'b store_v list" shows "store_alloc v2 st = (st @ [v2], length st)"
bijection_def: fixes f :: "'d \<Rightarrow> 'd" shows "bijection f \<equiv> bij f"
simps_1: fixes uu :: "bool" shows "enumerate_gexp_inputs (Bc uu) = {}"
fmdom_map: fixes f :: "'g \<Rightarrow> 'f"   and m :: "'e \<rightharpoonup>\<^sub>f 'g" shows "fmdom (fmmap f m) = fmdom m"
le_imp_diff_is_add: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b"   and "a \<subseteq># b" shows "(b - a = c) = (b = c + a)"
in_errorD: fixes x :: "nat"   and y :: "'b err"   and n :: "nat" assumes "(x, y) \<in> set (error n)" shows "y = Err"
matrix_to_iarray_fst_Gauss_Jordan_PA: fixes A :: "(('d, 'f) vec, 'e) vec" shows "matrix_to_iarray (fst (Gauss_Jordan_PA A)) = fst (Gauss_Jordan_iarrays_PA (matrix_to_iarray A))"
sup_ident: fixes x :: "'a" shows "x \<squnion> \<bottom>\<bottom> = x"
D_base_not_empty: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "{linordered_monoid.len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs} \<noteq> {}"
inf_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "inf P Q \<equiv> action_of (action.relation_of P \<or> action.relation_of Q)"
faithful_functor_axioms: shows "faithful_functor (\<cdot>) C S.map"
asym: fixes b :: "'t"   and a :: "'t" assumes "lt_of_nat_term_order cmp_term b a"   and "lt_of_nat_term_order cmp_term a b" shows "False"
Suc_numeral: fixes n :: "num" shows "Suc (numeral n) = numeral (n + num.One)"
fds_nth_subseries: fixes P :: "nat \<Rightarrow> bool"   and f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_subseries P f) n = (if P n then fds_nth f n else (0::'b))"
nontriv_solution_power_nat: fixes D :: "nat"   and z :: "nat \<times> nat"   and n :: "nat" assumes "pell D"   and "pell.nontriv_solution D z"   and "0 < n" shows "pell.nontriv_solution D (pell.pell_power D z n)"
intros_26: fixes n :: "i32"   and v1 :: "v"   and v2 :: "v" assumes "\<not> int_eq n 0" shows "\<lparr>[$C v1, $C v2, $C ConstInt32 n, $Select]\<rparr> \<leadsto> \<lparr>[$C v1]\<rparr>"
ball_scale: fixes a :: "real"   and c :: "'b"   and r :: "real" assumes "a \<noteq> 0" shows "(*\<^sub>R) a ` ball c r = ball (a *\<^sub>R c) (\<bar>a\<bar> * r)"
Acc_Sources: fixes X :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "(X \<in> Acc i C) = (C \<in> Sources i X)"
sort_key_simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f (x # xs) = linorder.insort_key less_eq f x (linorder.sort_key less_eq f xs)"
elims: fixes x :: "'b list \<times> ('b \<times> 'b) list"   and y :: "('b \<times> 'b) list list" assumes "grouped_out_arcs x = y" obtains vs :: "'b list" and as :: "('b \<times> 'b) list" where "x = (vs, as)" and "y = grouped_by_fst as"
ring_normalization_rules_2: fixes x :: "'a"   and y :: "'a" shows "x - y = x + - y"
two_comparisons_into_case_order_4: fixes x :: "'a"   and y :: "'a"   and Q :: "'c"   and P :: "'c"   and R :: "'c" shows "(if le x y then if lt x y then Q else P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
coreflexive_dedekind: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "less_eq x one"   and "less_eq y one"   and "less_eq z one" shows "less_eq (inf (times x y) z) (times x (inf y (times x z)))"
monad_commute_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_commute return bind \<equiv> monad return bind \<and> monad_commute_axioms bind"
Neg_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm" assumes "H \<turnstile> A IFF A'" shows "H \<turnstile> Neg A IFF Neg A'"
selectlike_self: fixes \<nu> :: "variable \<Rightarrow> real"   and X :: "(variable \<Rightarrow> real) set"   and V :: "variable set" shows "(\<nu> \<in> selectlike X \<nu> V) = (\<nu> \<in> X)"
INF_inf_const2: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and I :: "'i set"   and f :: "'i \<Rightarrow> 'h"   and x :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}" shows "Inf ((\<lambda>i. inf (f i) x) ` I) = inf (Inf (f ` I)) x"
map_of_Cons_code_2: fixes l :: "'b"   and v :: "'a"   and ps :: "('b \<times> 'a) list"   and k :: "'b" shows "table_of ((l, v) # ps) k = (if l = k then Some v else table_of ps k)"
Prim_Invar1_loc_axioms: shows "Prim_Invar1_loc w g r A"
1: shows "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>) \<Rightarrow>\<^sub>D F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)\<guillemotright>"
freshBoundOutputAction: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "c \<sharp> P" shows "c \<noteq> a"