hcju/isabelleps huggingface dataset:
h_diff_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_diff \<Delta> g i e = flow_network.h_plus \<Delta> g i e - (if cycles \<Delta> \<noteq> {} then flow_network.h_minus \<Delta> g i e else 0)"
zeta_reflect: fixes s :: "complex" assumes "s \<noteq> 0"   and "s \<noteq> 1" shows "rGamma s * zeta (1 - s) = 2 * complex_of_real (2 * pi) powr - s * cos (s * complex_of_real pi / 2) * zeta s"
simps_133: fixes x101 :: "Term.var"   and x102 :: "expr"   and x5 :: "val" shows "x101:=x102 \<noteq> Lit x5"
to_bl_word_rev: fixes w :: "'b word" shows "to_bl (word_reverse w) = rev (to_bl w)"
composition_of_substs: fixes t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" shows "t \<lhd> \<sigma> \<lhd> \<eta> = t \<lhd> \<sigma> \<lozenge> \<eta>"
lookup_OAlist_tc_empty: fixes k :: "'c" shows "OAlist_tc_lookup OAlist_tc_empty k = (0::'b)"
simple_function_indicator_representation: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and x :: "'a" assumes "simple_function M f"   and "x \<in> space M" shows "f x = (\<Sum>y\<in>f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
weak_partial_order_axioms: shows "weak_partial_order (inv_gorder L)"
elims_3: fixes x :: "'a val" assumes "\<not> is_Addr x" obtains "x = Unit"   | "x = Null"   | v :: "bool" where "x = Bool v__"   | v :: "32 word" where "x = Intg v"
intros_12: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and t :: "t"   and bs :: "bytes"   and vs :: "v list"   and a :: "nat" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "load m (nat_of_int k) off (t_length t) = Some bs" shows "\<lparr>s;vs;[$C ConstInt32 k, $Load t None a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C wasm_deserialise bs t]\<rparr>"
check_unsat: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<U> (check s)" shows "minimal_unsat_state_core (check s)"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "of_weak_ranking (weak_ranking_aux (A - M)) x y"   and "of_weak_ranking (weak_ranking_aux (A - M)) y z" shows "of_weak_ranking (weak_ranking_aux (A - M)) x z"
EndPhase1_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
finmap_of_eq_iff: fixes i :: "'d set"   and j :: "'d set"   and m :: "'d \<Rightarrow> 'e"   and n :: "'d \<Rightarrow> 'e" assumes "finite i"   and "finite j" shows "(finmap_of i m = finmap_of j n) = (i = j \<and> (\<forall>k\<in>i. m k = n k))"
lm089: fixes f :: "('b \<times> 'a) set"   and X :: "'b set" shows "Range f - f `` X \<subseteq> Range (Outside' X f)"
1: shows "Rep_phi (Abs_phi stone_phi) = stone_phi"
ChiRel_OneAncestor: fixes S1 :: "'d"   and U :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and S2 :: "'d" assumes "(S1, U) \<in> ChiRel A"   and "(S2, U) \<in> ChiRel A" shows "S1 = S2"
fin_2: shows "ys \<in> UNIV\<^sup>\<star>"
remdups_map_remdups: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "remdups (map f (remdups xs)) = remdups (map f xs)"
scaleC_add_left: fixes a :: "complex"   and b :: "complex"   and x :: "'a" shows "(a + b) *\<^sub>C x = a *\<^sub>C x + b *\<^sub>C x"
K: fixes K :: "ennreal" assumes "K < \<infinity>" shows "emeasure M {x \<in> A. (\<Sum>n. (nn_transfer_operator ^^ n) f x) \<le> K} = 0"
rel_simps_3: fixes m :: "num" shows "(num.Bit1 m \<le> num.One) = False"
unwinding: fixes z :: "complex" shows "of_int (unwinding z) = (z - Ln (exp z)) / (complex_of_real (2 * pi) * \<i>)"
funpow_cycle_offset: fixes k :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and i :: "nat"   and l :: "nat" assumes "(f ^^ k) x = (f ^^ i) x"   and "i \<le> k"   and "i \<le> l" shows "(f ^^ l) x = (f ^^ ((l - i) mod (k - i) + i)) x"
expand: fixes transaction\<^sub>m :: "transaction\<^sub>m"   and transaction\<^sub>m' :: "transaction\<^sub>m" assumes "the_Transaction\<^sub>m transaction\<^sub>m = the_Transaction\<^sub>m transaction\<^sub>m'" shows "transaction\<^sub>m = transaction\<^sub>m'"
less_nat: fixes x :: "nat"   and y :: "nat" shows "(x < y) = less_nat x y"
unify_sound: fixes E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and cs :: "('a \<times> ('b, 'a) Term.term) list" assumes "unify E [] = Some cs" shows "is_imgu (subst_of cs) (set E)"
code: fixes p :: "'c \<Rightarrow> bool"   and g1 :: "'c \<Rightarrow> 'b"   and g21 :: "'c \<Rightarrow> 'a"   and g22 :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "unfold_tllist p g1 g21 g22 a = (if p a then TNil (g1 a) else TCons (g21 a) (unfold_tllist p g1 g21 g22 (g22 a)))"
viewLn_empty: fixes t :: "('a, 'b) FingerTreeStruc" shows "(t \<noteq> Empty) = (FingerTreeStruc.viewLn t \<noteq> None)"
eval_Inv_Can: fixes t :: "'c local.term" assumes "Can t" shows "\<lbrace>Inv t\<rbrace> = local.inv \<lbrace>t\<rbrace>"
simps_6: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and f :: "form" shows "FEval MI e (FEx f) = (\<exists>m\<in>fst MI. FEval MI (\<lambda>y. case y of 0 \<Rightarrow> m | Suc n \<Rightarrow> e n) f)"
h1a: shows "MPoly_Type.degree p var = 1 \<longrightarrow> p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var"
shortest_augmenting_path: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Network c s t"   and "NFlow c s t f"   and "\<exists>p. NPreflow.isAugmentingPath c s t f p" shows "Graph.isShortestPath (residualGraph c f) s (Network.shortest_augmenting_path c s t f) t"
int_div_pos_eq: fixes a :: "int"   and b :: "int"   and q :: "int"   and r :: "int" assumes "a = b * q + r"   and "0 \<le> r"   and "r < b" shows "a div b = q"
t_1: shows "\<Gamma> t = \<Gamma> (Var x)"
nextVertex_in_face: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "f \<bullet> v \<in> \<V> f"
C: shows "{QuoteP (Var i) (Var i'), QuoteP (Var j) (Var j')} \<turnstile> (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
fidx_def: fixes s :: "'b fset"   and x :: "'b" shows "fidx s x = (SOME i. x |\<in>|\<^bsub>i\<^esub> s)"
2: shows "Ind (set w\<^sub>2) (set (b\<^sub>1 @ [a]))"
simps: fixes ly :: "nat list"   and n :: "nat"   and e :: "nat"   and as :: "nat"   and am :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_inv_1 ly n e (as, am) (s, l, r) ires = (let ss = start_of ly as; am' = abc_lm_s am n (abc_lm_v am n - Suc 0); am'' = abc_lm_s am n (abc_lm_v am n) in if s = start_of ly e then inv_stop (as, am'') (s, l, r) ires else if s = ss then False else if s = ss + 2 * n + 1 then inv_locate_b (as, am) (n, l, r) ires else if s = ss + 2 * n + 13 then inv_on_left_moving (as, am'') (s, l, r) ires else if s = ss + 2 * n + 14 then inv_check_left_moving (as, am'') (s, l, r) ires else if s = ss + 2 * n + 15 then inv_after_left_moving (as, am'') (s, l, r) ires else False)"
mndet_distrib: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b process" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "mndet (A \<union> B) P = (mndet A P |-| mndet B P)"
rel_def: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "x \<sim> y \<equiv> (fst x \<in> I \<and> fst y \<in> I) \<and> (snd x \<in> \<FF> (fst x) \<and> snd y \<in> \<FF> (fst y)) \<and> (\<exists>W. W \<in> I \<and> W \<subseteq> fst x \<inter> fst y \<and> \<rho> (fst x) W (snd x) = \<rho> (fst y) W (snd y))"
of_fun_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "IArray.of_fun f n !! i = f i"
bound_compare_defs_2: fixes lt :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and b :: "'a option" shows "\<rhd>\<^sub>u\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> False | Some b' \<Rightarrow> lt b' c"
sumeq: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1} \<union> set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) + (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})))"
non_subgraph_cycle_decomp: fixes a :: "'a"   and b :: "'a" assumes "reaches1 a b"   and "\<not> a \<rightarrow>\<^sup>+ b" shows "\<exists>c d. reaches a c \<and> E c d \<and> \<not> c \<rightarrow> d \<and> reaches d b"
assms: shows "\<exists>k. i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
i_take_first: fixes f :: "nat \<Rightarrow> 'b" shows "f \<Down> Suc 0 = [f 0]"
subring_iff: fixes H :: "'c set" assumes "H \<subseteq> carrier S" shows "subring H S = ring (S\<lparr>carrier := H\<rparr>)"
compatible_def: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.cpo below" shows "cpo.compatible below x y = (\<exists>z. po.is_lub below {x, y} z)"
common_first_summand_overlap: fixes x :: "i"   and y :: "i"   and z :: "i" shows "O (x \<oplus> y) (x \<oplus> z)"
isPath_fwd_cases: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "isPath s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = (s, u) # p'" and "(s, u) \<in> E" and "isPath u p' t"
smult_closed: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme"   and c :: "'h"   and v :: "'j" assumes "LinearCombinations.submodule R N M"   and "c \<in> carrier R"   and "v \<in> N" shows "c \<odot>\<^bsub>M\<^esub> v \<in> N"
RInitFailFinal_4: shows "sh'' = sh'(C \<mapsto> (sfs, Error))"
real_neg_pp_np_help: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "f x \<le> 0" shows "np f x = - f x \<and> pp f x = 0"
groups1: assumes "\<B> index ps = 0" shows "size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 1"
parallel_subspace: fixes A :: "'a set"   and B :: "'a set" assumes "subspace A"   and "subspace B"   and "affine_parallel A B" shows "A = B"
lookup_monom_mult: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x \<Rightarrow>\<^sub>0 nat" shows "lookup (local.punit.monom_mult c t p) v = (if local.punit.adds_pp t v then c * lookup p (local.punit.sminus v t) else (0::'b))"
y_def: shows "y \<equiv> pi - 2 * pi * frac ((pi - x) / (2 * pi))"
assms_3: shows "(x, z) \<in> r\<^sup>*"
tan_of_real: fixes x :: "real" shows "of_real (tan x) = tan (of_real x)"
get_child_nodes_reads: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"
run_reachable: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "run s\<^sub>0 lstep s" shows "system.reachable s\<^sub>0 lstep (s i)"
not_less: fixes x :: "'a word"   and y :: "'a word" shows "(\<not> x <s y) = (y \<le>s x)"
DataPart_def: fixes d :: "'a data"   and n :: "nat" shows "(d !P! n) = DataValue d ! n"
pcp_sat: fixes C :: "'b formula set set"   and S :: "'b formula set" assumes "pcp C"   and "S \<in> C" shows "sat S"
continuous_map_cases_le: fixes X :: "'b topology"   and p :: "'b \<Rightarrow> real"   and q :: "'b \<Rightarrow> real"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_map X euclideanreal p"   and "continuous_map X euclideanreal q"   and "continuous_map (subtopology X {x \<in> topspace X. p x \<le> q x}) Y f"   and "continuous_map (subtopology X {x \<in> topspace X. q x \<le> p x}) Y g"   and "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
fls_lr_inverse_X_power_1: fixes n :: "nat"   and x :: "'a" shows "fls_left_inverse (fls_X ^ n) x = fls_shift (int n) (fls_const x)"
Solutions_def: shows "local.Solutions = {(x, y). a \<bullet> x = b \<bullet> y \<and> length x = m \<and> length y = n}"
globally_sound_modes_compatible: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.globally_sound_mode_use eval (cms, mem)" shows "sifum_security.compatible_modes (map snd cms)"
simps: fixes n :: "'node"   and S :: "'node \<Rightarrow> bool"   and a :: "'node" shows "obsp n S a = (\<exists>as n'. a = n' \<and> n -as\<rightarrow>* n' \<and> (\<forall>nx\<in>set (sourcenodes as). \<not> S nx) \<and> S n')"
assms: fixes x :: "'b"   and d :: "real" assumes "x \<in> S"   and "0 < d" shows "\<exists>i. i \<in> K \<and> x \<in> ball (a i) (r i) \<and> r i < d"
not_Interp_imp_not_INTERP: fixes D' :: "'a literal multiset"   and D :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> Interp N D \<Turnstile> D'" shows "\<not> INTERP N \<Turnstile> D'"
2: fixes j :: "nat" shows "af (af \<phi> (prefix i w)) (prefix j (suffix i w)) = af \<phi> (prefix (i + j) w)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (div_instr instr s1)) \<and> t2 = snd (fst (div_instr instr s2))"
idx_inj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and idx :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and size :: "'a bound \<Rightarrow> nat"   and bound :: "'a bound"   and x :: "'a"   and y :: "'a" assumes "index_locale less_eq less idx size"   and "in_bound less_eq less bound x"   and "in_bound less_eq less bound y"   and "idx bound x = idx bound y" shows "x = y"
min_repU2: fixes w1 :: "32 Word.word"   and y :: "real"   and w2 :: "32 Word.word"   and x :: "real" assumes "w1 \<equiv>\<^sub>L y"   and "w2 \<equiv>\<^sub>L x" shows "wmin w1 w2 \<equiv>\<^sub>L x"
op_norm_eq_0: fixes A :: "(('a, 'b) vec, 'c) vec" shows "(\<parallel>A\<parallel>\<^sub>o\<^sub>p = 0) = (A = 0)"
xt1_6: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
l_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = x \<otimes> a) = (a = \<one>)"
s_Id_inv: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and m :: "mode"   and \<pi>\<^sub>2 :: "Syntax.term list"   and e\<^sub>2 :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>0 \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater>" obtains "\<pi>\<^sub>2 = \<pi>\<^sub>1" and "e\<^sub>2 = e\<^sub>1"
I1: fixes e1 :: "pexpr"   and n :: "nat"   and e :: "pexpr"   and e2 :: "pexpr" assumes "P e1 n e"   and "P e2 n (snd (snd (split_aux e1 n e)))" shows "P (PExpr2 (PMul e1 e2)) n e"
prems: shows "x \<in>\<^sub>\<circ> set (list.set xs)"
assms_2: fixes R :: "('a, 'c) labeled_graph \<times> ('a, 'c) labeled_graph"   and i :: "nat"   and f :: "('c \<times> 'b) set" assumes "R \<in> Rs"   and "graph_homomorphism (fst R) (S i) f" shows "\<exists>j. extensible R (S j) f"
arc_to_ends_to_block: fixes G :: "('a, 'b) pre_digraph"   and e :: "'b"   and x :: "'a"   and y :: "'a" assumes "graph G"   and "graph.arc_to_block G e = {x, y}" shows "arc_to_ends G e = (x, y) \<or> arc_to_ends G e = (y, x)"
comp_fun_commute_hcomp: shows "comp_fun_commute (\<otimes>)"
simps_5: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "Mn_free (Pr n f g) = (Mn_free f \<and> Mn_free g)"
point_axiom: fixes x :: "'a" shows "x \<noteq> 0 \<longrightarrow> (\<exists>y z. point y \<and> point z \<and> y ; z\<^sup>T \<le> x)"
cases: fixes a1 :: "nat \<Rightarrow> 'n"   and a2 :: "nat" assumes "(a1, a2) \<in> scop" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "a1 = \<pi>" and "a2 = n" and "(\<pi>, n) \<in> scp" and "\<pi> n \<in> dom local.att"
snoc_5: shows "ops = before @ (oid, ref) # op_list @ [oper]"
psimps_1: fixes b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and m :: "nat" assumes "dyad_rec_dom (b, l, r, real m)" shows "dyad_rec b l r (real m) = b m"
grounding: shows "{DA} \<union> set CAs \<subseteq> grounding_of_clss M"
newton_seq_fact1: fixes k :: "nat" shows "val_Zp (f' \<bullet> ns k) = val_Zp (f' \<bullet> a)"
reduce_column_preserves: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and i :: "nat" assumes "H \<in> carrier_mat m n"   and "(P, K) = local.reduce_column div_op H"   and "0 < m"   and "i \<in> {0, 1}"   and "i < m" shows "Matrix.row K i = Matrix.row H i"
distinct_13: fixes x2 :: "('e, 'f) Term.term"   and x31 :: "poscheckvariant"   and x32 :: "('e, 'f) Term.term"   and x33 :: "('e, 'f) Term.term" shows "receive\<langle>x2\<rangle> \<noteq> \<langle>x31: x32 \<doteq> x33\<rangle>"
fold_fold: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) RBT.rbt" shows "RBT.fold f t = fold (\<lambda>(x, y). f x y) (RBT.entries t)"
in_eq_rel: fixes n :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm"   and v :: "event_data option list" assumes "wf_tuple n (fv_trm t1 \<union> fv_trm t2) v"   and "is_simple_eq t1 t2"   and "Formula.eval_trm (map the v) t1 = Formula.eval_trm (map the v) t2" shows "v \<in> eq_rel n t1 t2"
success_bind_effectI: fixes f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and x :: "'b"   and g :: "'b \<Rightarrow> 'c Heap" assumes "effect f h h' x"   and "success (g x) h'" shows "success (f \<bind> g) h"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsPB \<Phi>" shows "\<pi>1 x = \<pi>2 x"
\<B>: fixes Vs :: "char list list"   and n :: "nat"   and e :: "char list exp" assumes "length Vs = n" shows "\<B> (compE\<^sub>1 Vs e) n"
simps: fixes l :: "name"   and s :: "tm"   and sl :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sn :: "name"   and k :: "tm"   and t :: "tm" assumes "atom l \<sharp> (s, sl, m, n, sm, sn)"   and "atom sl \<sharp> (s, m, n, sm, sn)"   and "atom m \<sharp> (s, n, sm, sn)"   and "atom n \<sharp> (s, k, sm, sn)"   and "atom sm \<sharp> (s, sn)"   and "atom sn \<sharp> s" shows "PrfP s k t = LstSeqP s k t AND All2 n (SUCC k) (SyntaxN.Ex sn (HPair (Var n) (Var sn) IN s AND (AxiomP (Var sn) OR SyntaxN.Ex m (SyntaxN.Ex l (SyntaxN.Ex sm (SyntaxN.Ex sl (Var m IN Var n AND Var l IN Var n AND HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND (ModPonP (Var sm) (Var sl) (Var sn) OR ExistsP (Var sm) (Var sn) OR SubstP (Var sm) (Var sn)))))))))"
fg: shows "f ` {0..<n} = g ` {0..<n}"
in_homE: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr f" and "D.dom f = a" and "D.cod f = b"
fBex_triv_one_point1: fixes A :: "'a fset"   and a :: "'a" shows "(\<exists>x|\<in>|A. x = a) = (a |\<in>| A)"
simps_7: fixes x3 :: "bool" shows "Bool x3 \<noteq> Unit"
distrib_left: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a *\<^sub>o\<^sub>w (b +\<^sub>o\<^sub>w c) = a *\<^sub>o\<^sub>w b +\<^sub>o\<^sub>w a *\<^sub>o\<^sub>w c"
pre_oldest_configuration_Min: fixes x :: "nat" shows "(x < Min (pre_oldest_tokens q n)) = (\<forall>a\<in>pre_oldest_tokens q n. x < a)"
separable_def: fixes f :: "'c poly" shows "separable f = coprime f (pderiv f)"
induct: fixes P :: "Formula.trm \<times> bool \<times> mconstraint \<times> Formula.trm \<Rightarrow> bool"   and a0 :: "Formula.trm \<times> bool \<times> mconstraint \<times> Formula.trm" assumes "\<And>t1 t2. P (t1, True, MEq, t2)"   and "\<And>t1 t2. P (t1, True, MLess, t2)"   and "\<And>t1 t2. P (t1, True, MLessEq, t2)"   and "\<And>t1 t2. P (t1, False, MEq, t2)"   and "\<And>t1 t2. P (t1, False, MLess, t2)"   and "\<And>t1 t2. P (t1, False, MLessEq, t2)" shows "P a0"
test_shunting_bot_eq: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "(x \<le> y) = (x \<sqinter> ! y = bot)"
simps_2: fixes f :: "'s" shows "arity_hd (Sym f) = arity_sym f"
bottom_edge_explicit: shows "(\<lambda>x. twoC (0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
dens'': shows "has_density (distr (distr M (count_space UNIV) extract_int) (count_space UNIV) f) (count_space UNIV) (g (\<delta> \<circ> IntVal))"
comp_ide_self: fixes a :: "'a" assumes "D.J.ide a" shows "J a a = a"
simps_8: fixes f :: "'e \<Rightarrow> 'd" shows "map f [] = []"
lub_least: fixes Y :: "'a set"   and y :: "'a" assumes "Complete_Partial_Order.chain ord Y"   and "\<And>x. x \<in> Y \<Longrightarrow> ord x y" shows "ord (lub Y) y"
lens_quotient_def: fixes X :: "'c \<Longrightarrow> 'e"   and Y :: "'d \<Longrightarrow> 'e" shows "X /\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. get\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> v. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>) v)\<rparr>"
suffix_appendD: fixes zs :: "'c list"   and xs :: "'c list"   and ys :: "'c list" assumes "suffix (zs @ xs) ys" shows "suffix xs ys"
TI_def: shows "TI = sample_uniform q \<bind> (\<lambda>a. sample_uniform q \<bind> (\<lambda>b. sample_uniform q \<bind> (\<lambda>r. return_spmf ((a, r), b, (a * b + (q - r)) mod q))))"
map_fun_upd: fixes y :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and v :: "'c" assumes "y \<notin> set xs" shows "map (f(y := v)) xs = map f xs"
worec_def: fixes F :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "worec F \<equiv> wfrec (r - Id) F"
limsup_le_extended_Gromov_product_inside': fixes v :: "nat \<Rightarrow> 'a"   and eta :: "'a Gromov_completion"   and e :: "'a"   and x :: "'a" assumes "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> eta" shows "limsup (\<lambda>xa. ereal (Gromov_product_at e (v xa) x)) \<le> extended_Gromov_product_at e eta (to_Gromov_completion x) + ereal (deltaG TYPE('a))"
minus_divide_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "uminus (divide a b) = divide (uminus a) b"
independent_vertices_finite: fixes E :: "'b set set"   and S :: "'b set" assumes "Graph_E E"   and "independent_vertices E S" shows "finite S"
inject: fixes x1 :: "'a"   and x2 :: "'a msstream"   and y1 :: "'a"   and y2 :: "'a msstream" shows "(MSSCons x1 x2 = MSSCons y1 y2) = (x1 = y1 \<and> x2 = y2)"
emeasure_stat_not_C: fixes y :: "'a"   and C :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf" assumes "y \<notin> C" shows "emeasure (MC_syntax.stat K C) {y} = 0"