hcju/isabelleps huggingface dataset:
inf_Inf_fold_inf: 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 A :: "'b set"   and B :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "inf (Inf A) B = Finite_Set.fold inf B A"
g'_4: shows "path_image g' \<subseteq> s - {p} - pts"
closed_open_segment_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "closed_segment a b \<noteq> open_segment c d"
isolated_singularity_at_uminus: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z" shows "isolated_singularity_at (\<lambda>w. - f w) z"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "PdXXX.Cd d X = greatest (Pd' d) (X \<inter> Field (Pd' d))"
tm_in: shows "flow0 x tm \<in> {a<--<b}"
hd_replicate: fixes n :: "nat"   and x :: "'b" assumes "n \<noteq> 0" shows "hd (replicate n x) = x"
RedInitRInit: fixes C' :: "char list"   and C :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>RI (C,C\<bullet>\<^sub>sclinit([])) ; Cs \<leftarrow> e,(h, l, sh),b\<rangle>"
assms_3: fixes x :: "'a" shows "u 1 x \<le> 0"
insort_key_remove1: fixes a :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> set xs"   and "sorted (map f xs)"   and "hd (filter (\<lambda>x. f a = f x) xs) = a" shows "insort_key f a (remove1 a xs) = xs"
Recv_given_channel: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   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" assumes "distributed_system channel"   and "isRecv ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "distributed_system.can_occur channel trans send recv ev c" shows "\<exists>s s' m. ev = Recv cid q p s s' m"
fmchoice': fixes S :: "'a set"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite S"   and "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>m. fmdom' m = S \<and> fmpred Q m"
chain_ncrit_def: fixes data :: "nat \<times> 'e"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and q_in_bs :: "bool"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" shows "chain_ncrit data gs bs hs q_in_bs ps p q = (let v = lt (fst p); l = term_of_pair (lcs (pp_of_term v) (lp (fst q)), component_of_term v); i = fst (snd p); j = fst (snd q) in (\<exists>r\<in>set gs. let k = fst (snd r) in k \<noteq> i \<and> k \<noteq> j \<and> lt (fst r) adds\<^sub>t l \<and> pair_in_list ps i k \<and> (q_in_bs \<or> pair_in_list ps j k) \<and> fst r \<noteq> 0) \<or> (\<exists>r\<in>set bs. let k = fst (snd r) in k \<noteq> i \<and> k \<noteq> j \<and> lt (fst r) adds\<^sub>t l \<and> pair_in_list ps i k \<and> (q_in_bs \<or> pair_in_list ps j k) \<and> fst r \<noteq> 0) \<or> (\<exists>h\<in>set hs. let k = fst (snd h) in k \<noteq> i \<and> k \<noteq> j \<and> lt (fst h) adds\<^sub>t l \<and> pair_in_list ps i k \<and> pair_in_list ps j k \<and> fst h \<noteq> 0))"
simps_2: fixes n :: "int" shows "pos_arctic (Num_arc n) = (0 \<le> n)"
proj_nAct: fixes n :: "nat"   and t :: "cnf llist"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" assumes "enat n < llength t" shows "dynamic_component.proj tCMP active c (ltake (enat n) t) = ltake (dynamic_component.nAct tCMP active c (enat n) t) (dynamic_component.proj tCMP active c t)"
abs_eq: fixes x :: "'a list" shows "fset_from_list x = Abs_fset (set x)"
no_effectless_proj: fixes as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and vs :: "'d set" shows "no_effectless_act (as_proj as vs)"
middle_2: fixes x :: "'b"   and y :: "'b" assumes "complete_space_with_middle TYPE('b)" shows "dist (complete_space_with_middle.middle x y) y = dist x y / 2"
old_infer_of_def: fixes \<iota> :: "'a literal multiset inference" shows "old_infer_of \<iota> = old_Infer (mset (side_prems_of \<iota>)) (main_prem_of \<iota>) (concl_of \<iota>)"
vc': shows "vc C' ((I \<and>* $ (Suc 0)) \<and>* F)"
point_diff_def: fixes A :: "point"   and B :: "point" shows "A - B = A + - B"
geom_proc_real_cond_exp_AE: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "real_cond_exp N (fct_gen_subalgebra N (stream_space borel) (proj_stoch_proc geom_proc n)) (geom_proc (Suc n)) w = expl_cond_expect N (proj_stoch_proc geom_proc n) (geom_proc (Suc n)) w"
sxalloc_elim_cases_2: fixes G :: "prog"   and j :: "jump"   and s :: "st"   and s' :: "state" assumes "G\<turnstile>(Some (Jump j), s) \<midarrow>sxalloc\<rightarrow> s'" obtains "s' = (Some (Jump j), s)"
lfp_induct: fixes f :: "'a \<Rightarrow> 'a"   and P :: "'a" assumes "mono f"   and "f (lfp f \<sqinter> P) \<le> P" shows "lfp f \<le> P"
inject_2: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm"   and y21 :: "('c, 'd) tm"   and y22 :: "('c, 'd) tm" shows "(App x21 x22 = App y21 y22) = (x21 = y21 \<and> x22 = y22)"
val_0_n_val_0: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(v x = 0) = (n_val K v x = 0)"
take_rows_and_take_cols: fixes M :: "'a mat"   and r :: "nat list" shows "take_rows_from_matrix M r = (take_cols_from_matrix M\<^sup>T r)\<^sup>T"
t1_space_openin_delete: fixes X :: "'c topology" shows "t1_space X = (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (U - {x}))"
strong_exhaust: fixes y :: "Lambda_Z.term"   and c :: "'a" obtains x1 :: "name" where "y = Var x1"   | x21 :: "Lambda_Z.term" and x22 :: "Lambda_Z.term" where "y = App x21 x22"   | x31 :: "name" and x32 :: "Lambda_Z.term" where "{atom x31} \<sharp>* c" and "y = Abs x31 x32"
i: shows "(\<lambda>n. integral (cbox (sum ((*\<^sub>R) (- real n)) Basis) (sum ((*\<^sub>R) (real n)) Basis)) (\<lambda>x. if x \<in> s then f x else (0::'a))) \<longlonglongrightarrow> i"
value_def: fixes tree :: "'a tree" shows "value tree = (case tree of \<langle>x21, x22, x23\<rangle> \<Rightarrow> x22)"
assms_6: shows "SIDE_PRECOND (X \<subseteq> a sctn)"
0: fixes ys :: "'varSort" shows "|{y. (rho [x \<leftarrow> X]_xs) ys y \<noteq> None}| <o |UNIV|"
ex_redchainTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "d_gchain G n f"   and "D_gchain G (card (f ` {i. i \<le> n}) - Suc 0) g"   and "g ` {i. i \<le> card (f ` {i. i \<le> n}) - Suc 0} = f ` {i. i \<le> n}" shows "g (card (f ` {i. i \<le> n}) - Suc 0) = f n"
ideal_generated_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.ring plus zero minus uminus times" shows "ring.ideal_generated plus zero uminus times {zero} = {zero}"
fun_upd_idem: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c" assumes "f x = y" shows "f(x := y) = f"
finite_is_class: fixes G :: "prog" shows "finite {C. is_class G C}"
IH: shows "map (local.coeff p) (rev [0..<length p]) = p"
resActLeft: fixes x :: "name"   and \<alpha> :: "act"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "x \<sharp> \<alpha>"   and "Id \<subseteq> Rel" shows "\<lparr>\<nu>x\<rparr>\<alpha>.P \<leadsto>[Rel] \<alpha>.(\<lparr>\<nu>x\<rparr>P)"
hext_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "h \<unlhd> h' \<equiv> \<forall>a C fs. h a = \<lfloor>(C, fs)\<rfloor> \<longrightarrow> (\<exists>fs'. h' a = \<lfloor>(C, fs')\<rfloor>)"
common_reparam_exists_imp_eq_line_integral_2: fixes basis :: "(real \<times> real) set"   and one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "finite basis"   and "finite one_chain1"   and "finite one_chain2"   and "boundary_chain one_chain1"   and "boundary_chain one_chain2"   and "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"   and "common_reparam_exists one_chain1 one_chain2"   and "\<forall>(k, \<gamma>)\<in>one_chain1. valid_path \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain2. valid_path \<gamma>" shows "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
call: fixes s :: "s"   and vs :: "v list"   and j :: "nat"   and i :: "nat" shows "\<lparr>s;vs;[$Call j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Callcl (sfunc s i j)]\<rparr>"
elims: fixes x :: "nat rexp set"   and y :: "nat list set" assumes "L x = y" obtains S :: "nat rexp set" where "x = S" and "y = \<Union> (lang ` S)"
mem_simps_2: fixes c :: "'b" shows "(c \<in> {}) = False"
inf_sup_aci_3: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
is_regular_one_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring.is_regular R addition multiplication zero unit (comm_ring.one_sheaf_spec R addition multiplication zero unit U) U"
face_cycle_succ_neq: fixes a :: "'b \<times> 'b" assumes "a \<in> arcs (with_proj (list_digraph G_list))"   and "tail (with_proj (list_digraph G_list)) a \<noteq> head (with_proj (list_digraph G_list)) a" shows "face_cycle_succ a \<noteq> a"
emeasure_bind_pmf: fixes M :: "'b pmf"   and N :: "'b \<Rightarrow> 'a pmf"   and X :: "'a set" shows "emeasure (measure_pmf (M \<bind> N)) X = \<integral>\<^sup>+ x. emeasure (measure_pmf (N x)) X \<partial>measure_pmf M"
prefixes_not_Nil: fixes xs :: "'a list" shows "prefixes xs \<noteq> []"
cdj': shows "k' + n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> j'"
local_steps_def: fixes T :: "((('a \<Rightarrow> 'b) \<times> 'c) \<times> 'd \<times> ('a \<Rightarrow> 'e) \<times> 'f) set"   and J :: "'a set" shows "local_steps T J \<equiv> \<forall>\<sigma> \<zeta> s a \<sigma>' s'. ((\<sigma>, s), a, \<sigma>', s') \<in> T \<and> (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<longrightarrow> (\<exists>\<zeta>'. (\<forall>j\<in>J. \<zeta>' j = \<sigma>' j) \<and> ((\<zeta>, s), a, \<zeta>', s') \<in> T)"
gt_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and u :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "epo.gt ground_heads_var gt_sym extf u t"   and "epo.gt ground_heads_var gt_sym extf t s" shows "epo.gt ground_heads_var gt_sym extf u s"
sublinear_wp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "sublinear (wp (Apply f))"
Py_nn: fixes y :: "'c" assumes "y \<in> space T" shows "0 \<le> Py y"
tnth_TNil: fixes b :: "'b"   and n :: "nat" shows "tnth (TNil b) n = undefined n"
Map_comp: fixes t' :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and t :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and a' :: "'a2"   and a :: "'a2" assumes "A2_B.seq t' t"   and "A2.seq a' a" shows "A2_B.Map (t' \<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>] t) (a' \<cdot>\<^sub>A\<^sub>2 a) = A2_B.Map t' a' \<cdot>\<^sub>B A2_B.Map t a"
ex_mulDisj_maximal: fixes S :: "'a set"   and T :: "'a set set" assumes "mul_closed_set R S"   and "\<zero> \<notin> S"   and "1\<^sub>r \<in> S"   and "T = {I. ideal R I \<and> S \<inter> I = {}}" shows "\<exists>mx. maximal_set T mx"
dims0_def: shows "c.dims0 = nths dims c.vars0"
enc_eqI: fixes \<Sigma> :: "'b list"   and x :: "('b \<times> bool list) list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and w' :: "'b list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "x \<in> formula.enc \<Sigma> (w, I)"   and "x \<in> formula.enc \<Sigma> (w', I')"   and "formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>"   and "formula.wf_interp_for_formula \<Sigma> (w', I') \<phi>"   and "length I = length I'" shows "formula.enc \<Sigma> (w, I) = formula.enc \<Sigma> (w', I')"
min_eqL: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "min a b = a"
CP: shows "CP \<equiv> \<lambda>g. (*\<cdot>) (- g) \<circ> P \<circ> (*\<cdot>) g"
ldropWhile_trans_fusion: fixes P :: "'c \<Rightarrow> bool"   and g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lunstream (dropWhile_raw P g) (True, s) = ldropWhile P (lunstream g s)"
weak_equality_spec_weak_equality: fixes p :: "(char list multiset \<times> int) multiset"   and p' :: "int mpoly"   and r :: "(char list multiset \<times> int) multiset"   and r' :: "int mpoly" assumes "(p, p') \<in> mset_poly_rel"   and "(r, r') \<in> mset_poly_rel" shows "weak_equality_spec p r \<le> weak_equality p' r'"
invariant_def: fixes M :: "'a set" shows "rev.invariant M = (\<forall>x\<in>M. rev.trapped x M)"
j_finite: fixes x :: "'v option"   and y :: "'v option" shows "j (x, y) < \<top>"
4: fixes k :: "nat" shows "\<exists>l\<ge>k. P l = {}"
FL_valid_Act: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('a, 'pred, 'act, 'effect) FL_Formula.formula" shows "P \<Turnstile> FL_Formula.Act f \<alpha> x = (\<exists>\<alpha>'' x' P'. FL_Formula.Act f \<alpha> x = FL_Formula.Act f \<alpha>'' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
mset_pos_uminus: fixes A :: "'a zmultiset" shows "mset_pos (- A) = mset_neg A"
Path_upd: fixes u :: "nat"   and as :: "nat list"   and f :: "nat \<Rightarrow> nat option"   and v :: "nat option"   and x :: "nat"   and y :: "nat" assumes "u \<notin> set as" shows "Path (f(u := v)) x as y = Path f x as y"
MapsToOp: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "f maps\<^bsub>C\<^esub> X to Y" shows "f maps\<^bsub>Op C\<^esub> Y to X"
cl_invI: fixes p :: "pState"   and g :: "'b gState_scheme" assumes "length (pState.channels p) \<le> length (gState.channels g)" shows "cl_inv (g, p)"
exit_coind: fixes K :: "'state \<Rightarrow> bool"   and s :: "'state"   and v :: "'value" assumes "K s"   and "\<And>trn. \<lbrakk>K (srcOf trn); validTrans trn; \<not> T trn\<rbrakk> \<Longrightarrow> (\<phi> trn \<longrightarrow> f trn \<noteq> v) \<and> K (tgtOf trn)" shows "exit s v"
is_empty_rep_def: fixes p :: "int extended \<times> int extended" shows "is_empty_rep p = (let (l, h) = p in h < l \<or> l = \<infinity> \<and> h = \<infinity> \<or> l = -\<infinity> \<and> h = -\<infinity>)"
closed_eucl_atLeastAtMost: fixes a :: "'c"   and b :: "'c" shows "closed {a..b}"
smult_inj: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inj (Polynomial.smult a)"
empty_interior_convex_hull: fixes S :: "'a set" assumes "finite S"   and "card S \<le> DIM('a)" shows "interior (convex hull S) = {}"
dom_char: fixes f :: "'a" shows "Left_a.dom f = (if Left_a.arr f then local.dom f else null)"
isLub_upper: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b"   and y :: "'b" assumes "CL cl"   and "isLub S cl L"   and "y \<in> S" shows "(y, L) \<in> order cl"
Proper_Roots_def: fixes M :: "node list" shows "Proper_Roots M \<equiv> Roots \<noteq> {} \<and> Roots \<subseteq> \<lbrace>\<acute>(<) (length M)\<rbrace>"
nth_eucl_def: fixes x :: "'b"   and i :: "nat" shows "x $\<^sub>e i = x \<bullet> Basis_list ! i"
mk_release_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_release x y \<equiv> case x of true\<^sub>n \<Rightarrow> y | false\<^sub>n \<Rightarrow> mk_globally y | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> x R\<^sub>n y"
surjective: fixes r :: "'a obj_scheme" shows "r = \<lparr>tag = tag r, values = values r, \<dots> = obj.more r\<rparr>"
part_witness_subsets_pow: fixes X :: "part_witness set set" assumes "\<forall>s\<in>X. s \<in> {{}, {a}, {b, c}, {a, b, c}}" shows "X \<in> set ` set (subseqs [{}, {a}, {b, c}, {a, b, c}])"
init_red_sync_end: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>e\<rfloor>"   and "e\<^sub>1 = e"   and "throw_of e = None"   and "\<not> sub_RI e'" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e',s\<^sub>1,icheck P (the (INIT_class e\<^sub>0)) e'\<rangle>"
card1_tr0: fixes A :: "'c set"   and a :: "'c" assumes "finite A"   and "card A = Suc 0"   and "a \<in> A" shows "{a} = A"
bij_betw_comp_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and f' :: "'b \<Rightarrow> 'd"   and A'' :: "'d set" assumes "bij_betw f A A'" shows "bij_betw f' A' A'' = bij_betw (f' \<circ> f) A A''"
suc_cong: fixes H :: "fmla set"   and t :: "trm"   and t' :: "trm" assumes "H \<turnstile> t EQ t'" shows "H \<turnstile> suc t EQ suc t'"
harm_nonneg: fixes n :: "nat" shows "(0::'a) \<le> harm n"
update_simps_3: fixes up :: "(nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and s :: "('a, 'b) state" shows "\<T> (\<B>\<^sub>i\<^sub>u_update up s) = \<T> s"
measure_negligible_finite_Union: fixes \<F> :: "'c set set" assumes "finite \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> S \<in> lmeasurable"   and "pairwise (\<lambda>S T. negligible (S \<inter> T)) \<F>" shows "Sigma_Algebra.measure lebesgue (\<Union> \<F>) = sum (Sigma_Algebra.measure lebesgue) \<F>"
elims_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "xb__ = []"   | x :: "'b" and ys :: "'b list" where "xb__ = x # ys" and "Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys"
norm_fact: fixes n :: "nat" shows "norm (fact n) = fact n"
case_2: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
rep_eq: fixes x :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "pmf (wt_pmf x) = (case x of (c, s) \<Rightarrow> \<lambda>i. if proper c then if i < brn c then wt c s i else 0 else if i = 0 then 1 else 0)"
asprod_n_0: fixes n :: "int" shows "n *\<^sub>a 0 = 0"
T_pass2': fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>2' hs = Pairing_Heap_Tree_Analysis.T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>2 (homs hs)"
Pd'_tax_Pd': fixes d :: "'b" shows "Pd'_tax d \<subseteq> Pd' d"
monad_state_altc_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and altc :: "'e cset \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_state_altc return bind get put altc" shows "monad_state_altc (return_writer return) (bind_writer return bind) (get_writer get) (put_writer put) (altc_writer altc)"
Nil_genPrefix: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set" shows "([], xs) \<in> genPrefix r"
safetyD: fixes A :: "'b set"   and P :: "'b llist set"   and t :: "'b llist" assumes "safety A P"   and "t \<in> A\<^sup>\<infinity>"   and "\<And>r. r \<in> finpref A t \<Longrightarrow> \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P" shows "t \<in> P"
reduces_evaluates_to_values: fixes t :: "trm"   and t' :: "trm" assumes "t \<Down> t'" shows "val t'"
even_set_bit_iff: fixes m :: "nat"   and a :: "'a" shows "even (set_bit m a) = (even a \<and> m \<noteq> 0)"
poly_eq_0_iff_dvd: fixes p :: "'b poly"   and c :: "'b" shows "(poly p c = (0::'b)) = ([:- c, 1::'b:] dvd p)"
ns_constraints_ops: shows "Incremental_NS_Constraint_Ops init_nsc_code assert_nsc_code check_nsc_code solution_nsc_code checkpoint_nsc_code backtrack_nsc_code invariant_nsc checked_nsc"
conjugate_iff_residuated: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "conjugation_pair f g = residuated_pair f (\<lambda>x. - g (- x))"
Rep_real_int_inject: fixes x :: "real_int"   and y :: "real_int" shows "(Rep_real_int x = Rep_real_int y) = (x = y)"
properties_def: fixes lambda :: "nat"   and mu :: "nat" shows "properties lambda mu \<equiv> 0 < lambda \<and> inj_on seq {0..<mu + lambda} \<and> (\<forall>i\<ge>mu. \<forall>j. seq (i + j * lambda) = seq i)"
f_def: shows "f \<equiv> vs \<odot> 0\<^sup>\<infinity>"
slg_rel_def_internal: fixes R :: "('b \<times> 'c) set" shows "slg_rel R \<equiv> {(succs, G). \<forall>v. (succs v, G `` {v}) \<in> \<langle>R\<rangle>list_set_rel}"
assms: assumes "Im z = 0" shows "\<bar>Re z\<bar> < 1"
assms_3: shows "InvariantCnCharacterization (getCn state) (getC state) (getM state)"
inducts: fixes \<I> :: "('b, 'd) \<I>"   and callee :: "'b \<Rightarrow> ('d \<times> 'e) spmf" assumes "\<I> \<turnstile>c callee \<surd>" obtains "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call"
td_gal: fixes c :: "nat"   and b :: "nat"   and a :: "nat" assumes "0 < c" shows "(b * c \<le> a) = (b \<le> a div c)"
call_return_node_edge: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = sourcenode a \<and> targetnode a'' = targetnode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
cod_pr1: 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 "cod (pr1 a b) = a"
simps_5: fixes Vs :: "String.literal list"   and v :: "'a val" shows "compE1 Vs (Val v) = Val v"