hcju/isabelleps huggingface dataset:
matrix'_def: fixes X :: "(('b, 'c) vec, 'c) vec"   and Y :: "(('b, 'd) vec, 'd) vec"   and f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" shows "matrix' X Y f = (\<chi>i j. coord Y (f (X $ j)) $ i)"
residue_mult_group_kernel_partition_1: fixes m :: "nat"   and n :: "nat" assumes "1 < m"   and "0 < n"   and "m dvd n" shows "Congruence.partition (carrier (residue_mult_group n)) (rcosets\<^bsub>residue_mult_group n\<^esub> kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m))"
HPhase1or2ReadThen_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv4d s p" shows "HInv4d s' p"
has_as_product_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'a"   and a :: "'a" shows "has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>) D a \<pi>"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = lang n (rexp_of n \<phi>) - {[]}"
rhs: shows "filter_mset Q (add_mset x A) \<times># {#b#} = filter_mset Q A \<times># {#b#} + filter_mset Q {#x#} \<times># {#b#}"
store_packed_size: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and v :: "bytes"   and l :: "nat"   and m' :: "mem" assumes "store_packed m n off v l = Some m'" shows "mem_size m = mem_size m'"
obtuse_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Obtuse Bet Cong A B C" shows "A \<noteq> B \<and> C \<noteq> B \<and> A \<noteq> C"
fresh_2: fixes a :: "name"   and x2 :: "lam"   and x1 :: "lam" shows "a \<sharp> App x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
to_woot_character_keeps_i_lifted: fixes M :: "('c, 'd) message set"   and X :: "('c, 'd) insert_message set" shows "I ` to_woot_character M ` X = I ` X"
lm025: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set" assumes "inj_on f X" shows "inj_on ((`) f) (Pow X)"
less_eq_option_def: fixes x :: "'a option"   and y :: "'a option" shows "(x \<le> y) = option_le (\<le>) x y"
p0_def: shows "p0 = Cring_Poly.truncate R p"
consistent_in_P2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<psi> \<in> \<P>\<^sup>2" shows "consistent \<psi> \<in> \<P>\<^sup>2"
quasicomm_1: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" shows "(y \<otimes> x \<preceq> agg_square_s_algebra_2.comp_inf.tc x) = (mtop \<otimes> x \<preceq> agg_square_s_algebra_2.comp_inf.tc x)"
assms_3: shows "a \<in> hyp2 \<union> S"
assms_1: shows "{x} \<union> set xs = {y} \<union> set ys"
transE: fixes cl :: "'b potype"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "PO cl"   and "(a, b) \<in> order cl"   and "(b, c) \<in> order cl" shows "(a, c) \<in> order cl"
rGamma_series_altdef: fixes z :: "'a"   and n :: "nat" shows "rGamma_series z n = inverse (Gamma_series z n)"
l_def: shows "l \<equiv> \<lambda>f \<omega>. g (shd \<omega>) (f (stl \<omega>))"
total_preorder_on_def: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "total_preorder_on carrier le \<equiv> Order_Predicates.preorder_on carrier le \<and> total_preorder_on_axioms carrier le"
oghoare_Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P Skip Q, A" shows "\<exists>P'. P = AnnExpr P' \<and> P' \<subseteq> Q"
max_prop_1: fixes XS :: "'a list set" assumes "XS \<noteq> {}"   and "finite XS" shows "MAX XS \<in> XS"
sep_list_conj_Cons: fixes x :: "'b \<Rightarrow> bool"   and xs :: "('b \<Rightarrow> bool) list" shows "\<And>* x # xs = (x \<and>* \<And>* xs)"
simplt_def: fixes t :: "tm" shows "simplt t = (let (c, s) = split0 (simptm t) in if c = 0\<^sub>p then lt s else Lt (CNP 0 c s))"
t2_in: shows "flow0 x t2 \<in> {a<--<b}"
simps_5: fixes uu :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and E' :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Es :: "char list set"   and SS :: "char list \<Rightarrow> char list set"   and b :: "bexp"   and C :: "acom" shows "kdeps ({(uu, E', E, Es, SS)} WHILE b DO C) = Es"
domain_subset_valid_pub: shows "Domain Rel \<subseteq> valid_pub"
sep_impl_sep_true: fixes P :: "'a \<Rightarrow> bool" shows "(P \<longrightarrow>* (\<lambda>s. True)) = (\<lambda>s. True)"
nAct_same: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"   and "n' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>" shows "the_enat \<langle>c #\<^bsub>enat n'\<^esub>inf_llist t\<rangle> = the_enat \<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle>"
same: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a" assumes "merge_on A h bo m"   and "a \<in> A" shows "m a a = Some a"
F_sync: fixes P :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "F (P \<lbrakk>A\<rbrakk> Q) = {(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
primitive_part_dvd_primitive_partI: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q" shows "primitive_part p dvd primitive_part q"
separating_weakening: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set"   and S' :: "'a set" assumes "separating_gen G A B S"   and "S \<subseteq> S'" shows "separating_gen G A B S'"
bot_least: fixes a :: "'a" shows "\<bottom> \<le> a"
Inter_anti_mono: fixes B :: "'c set set"   and A :: "'c set set" assumes "B \<subseteq> A" shows "\<Inter> A \<subseteq> \<Inter> B"
simps: fixes cs :: "QDelta ns_constraint list"   and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "checkpoint_simplex (Simplex_State (cs, asi_tv, s)) = Simplex_Checkpoint (checkpoint_s s)"
eval: shows "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')"
case_empty: fixes m :: "'a match_expr" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m Empty # rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
elem_neq_classes_part: shows "is_non_overlapping (P - {Y})"
poly_of_const_def: shows "poly_of_const = (\<lambda>k. local.normalize [k])"
relabel_action_bound': fixes fxl :: "(nat \<times> nat \<Rightarrow> 'capacity) \<times> (nat \<Rightarrow> nat)"   and p :: "pr_operation' list"   and fxl' :: "(nat \<times> nat \<Rightarrow> 'capacity) \<times> (nat \<Rightarrow> nat)" assumes "(fxl, p, fxl') \<in> trcl pr_algo_lts'" shows "length (filter is_RELABEL' p) \<le> 2 * (card V)\<^sup>2"
measurable_If_set: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and M' :: "'b measure"   and g :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "A \<inter> space M \<in> sets M" shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
bdd_lookup_make_bdd: fixes bs :: "bool list"   and n :: "nat"   and f :: "nat list \<Rightarrow> 'a"   and xs :: "nat list" assumes "length bs = n" shows "bdd_lookup (make_bdd f n xs) bs = f (xs @ map of_bool bs)"
iterate: shows "h \<turnstile> iterate_M progs \<rightarrow>\<^sub>h h'"
ipurge_ref_distrib_union: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set"   and Y :: "'b set" shows "ipurge_ref I D u xs (X \<union> Y) = ipurge_ref I D u xs X \<union> ipurge_ref I D u xs Y"
prems: shows "\<forall>i<Suc k. \<not> final (head (f i))"
homogenize_zero: fixes x :: "'b" shows "homogenize x 0 = 0"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> Partition" shows "Class a = A"
assms_2: fixes j :: "'b" assumes "j \<in> B" shows "\<exists>i\<in>A. g j = f i"
sgn_1_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = one) = less zero a"
simps: fixes x :: "'a extC" shows "x\<^sup>T = x"
1: shows "is_offending_flows (set (minimalize_offending_overapprox ff [] G nP)) G nP"
rbd_un_pres: shows "sup_pres bd\<^sub>\<R>"
obtain_elem_zmset_frontier: fixes M :: "'a zmultiset"   and t :: "'a" assumes "0 < zcount M t" obtains s :: "'a" where "s \<le> t \<and> 0 < zcount (zmset_of (mset_set (set_antichain (frontier M)))) s"
fold_min_triv: fixes xs :: "'b list"   and k :: "'b" assumes "\<forall>x\<in>set xs. k \<le> x" shows "fold min xs k = k"
WTrtSFAss: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> C has F,Static:T in D"   and "P \<turnstile> T\<^sub>2 \<le> T" shows "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sF{D} := e\<^sub>2 : Void"
hyps: assumes "is_subobj P (C, C' # rev Cs'')" shows "P \<turnstile> C \<preceq>\<^sup>* last (C' # rev Cs'')"
strengthen_step_4: shows "rv1 \<in> red_vertices prb'"
isolated_singularity_at_add: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "isolated_singularity_at f z"   and "isolated_singularity_at g z" shows "isolated_singularity_at (\<lambda>w. f w + g w) z"
freeword_cases: fixes y :: "('a \<times> bool) list" assumes "y \<in> {as. proper_signed_list as}" obtains x :: "'a freeword" where "y = freeword x"
a1_trans: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "f j \<sqsupset>p f i"
Mp_def: fixes m :: "int" shows "poly_mod.Mp m = map_poly (poly_mod.M m)"
disc_eq_case_1: fixes nat :: "nat" shows "(nat = 0) = (case nat of 0 \<Rightarrow> True | Suc uu_ \<Rightarrow> False)"
opp_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c point" assumes "ell_field R" shows "ell_field.opp R p = (case p of Infinity \<Rightarrow> Infinity | Point x y \<Rightarrow> Point x (\<ominus>\<^bsub>R\<^esub> y))"
simps_2: fixes s :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "exec_plan s (a # as) = exec_plan (state_succ s a) as"
pdivmod_monic_pseudo_divmod: fixes g :: "'a poly"   and f :: "'a poly" assumes "monic g" shows "pdivmod_monic f g = pseudo_divmod f g"
col_transitivity_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "P \<noteq> Q"   and "Tarski_neutral_dimensionless.Col Bet P Q A"   and "Tarski_neutral_dimensionless.Col Bet P Q B" shows "Tarski_neutral_dimensionless.Col Bet Q A B"
contour_integral_bound_part_circlepath: fixes f :: "complex \<Rightarrow> complex"   and c :: "complex"   and r :: "real"   and a :: "real"   and b :: "real"   and B :: "real" assumes "f contour_integrable_on part_circlepath c r a b"   and "0 \<le> B"   and "0 \<le> r"   and "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> cmod (f x) \<le> B" shows "cmod (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
remove_term_def: fixes m0 :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'b mpoly" shows "remove_term m0 p = MPoly (Abs_poly_mapping (\<lambda>m. coeff p m when m \<noteq> m0))"
cod_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.cod (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
not_final_thread_existsE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r2.not_final_thread s t" obtains x :: "'x2" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>"
\<xi>_Inf_many2: shows "\<exists>\<^sub>\<infinity>k. \<bar>\<xi> - real_of_int (p k) / real_of_int (q k)\<bar> < 1 / real_of_int (q k) powr (2 + \<delta>)"
ipresSubstAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'b, 'c, 'h) model_scheme" shows "ipresSubstAbs h hA MOD \<equiv> \<forall>ys Y y us s A. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<longrightarrow> hA (A $[Y / y]_ys) = igSubstAbs MOD ys (h Y) y (hA A)"
levy_continuity1: fixes M :: "nat \<Rightarrow> real measure"   and M' :: "real measure"   and t :: "real" assumes "\<And>n. real_distribution (M n)"   and "real_distribution M'"   and "weak_conv_m M M'" shows "(\<lambda>n. char (M n) t) \<longlonglongrightarrow> char M' t"
strict_mono: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "OrdinalInduct.strict_mono f"   and "A < B" shows "f A < f B"
rcos_assoc_lcos: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "'c set"   and x :: "'c" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x <#>\<^bsub>G\<^esub> K = H <#>\<^bsub>G\<^esub> (x <#\<^bsub>G\<^esub> K)"
tuple_simps_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "partial_magma.dom C (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = partial_magma.dom C h"
nrows_diagonal_to_Smith_i: fixes xs :: "nat list"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "nrows (diagonal_to_Smith_i xs A i bezout) = nrows A"
path_conforms_with_strategy_irrelevant_deadend: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a"   and w :: "'a" assumes "path_conforms_with_strategy p P \<sigma>"   and "deadend v \<or> v \<notin> VV p"   and "valid_path P" shows "path_conforms_with_strategy p P (\<sigma>(v := w))"
prob_space_return: fixes x :: "'a"   and M :: "'a measure" assumes "x \<in> space M" shows "prob_space (return M x)"
rep_eq: fixes x :: "nat_int" shows "|x| = card (Rep_nat_int x)"
simps_6: fixes i :: "String.literal"   and p :: "fol fm"   and q :: "fol fm" shows "free_in i (And p q) = (free_in i p \<or> free_in i q)"
reset_set11: fixes cs :: "'c list"   and d :: "'d"   and u :: "'c \<Rightarrow> 'd" shows "\<forall>c. c \<notin> set cs \<longrightarrow> ([cs\<rightarrow>d]u) c = u c"
mono_def: fixes f :: "'d \<Rightarrow> 'e" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
span_carrier_lin_indpt_card_n: fixes W :: "'b vec set"   and n :: "nat" assumes "W \<subseteq> carrier_vec n"   and "card W = n"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) W" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W = carrier_vec n"
Pred\<^sub>\<alpha>_def: shows "Pred\<^sub>\<alpha> \<equiv> (id ---> abs_Tree\<^sub>\<alpha>) tPred"
Us_def: shows "Us = {u. length u = length ys \<and> u \<le>\<^sub>v ys \<and> 0 < sum_list u \<and> sum_list u \<le> a ! k}"
low_indistinguishable_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>1 :: "'Com"   and c\<^sub>2 :: "'Com" shows "c\<^sub>1 \<sim>\<^bsub>mds\<^esub> c\<^sub>2 = (\<forall>mem\<^sub>1 mem\<^sub>2. mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<longrightarrow> \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>)"
simps_1: fixes lens_get :: "'a \<Rightarrow> 'b"   and lens_put :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and more :: "'c" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
exec1_only1: fixes c :: "com"   and cs :: "com list"   and s :: "state"   and cs' :: "com list"   and t :: "state" assumes "(c # cs, s) \<rightarrow> (cs', t)" shows "\<exists>cs1. cs' = cs1 @ cs"
simps_5: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc No_ics frs sh = exec_instr (instrs_of P C M ! pc) P h stk loc C M pc No_ics frs sh"
assms_2: shows "typ_ok_sig \<Sigma> \<tau>"
simps_2: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "del_min \<langle>l, x, r\<rangle> = merge l r"
runit_char_1: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<r>[a] : a \<otimes> \<I> \<rightarrow> a\<guillemotright>"
map_iterator_finite: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'d \<Rightarrow> 'e option" assumes "map_iterator it m" shows "finite (dom m)"
divideR_apply: fixes c :: "real"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(c *\<^sub>R f) x = c *\<^sub>R f x"
polyn_ext_cf_lo_zero: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and j :: "nat" assumes "pol_coeff S c"   and "0 < j" shows "polyn_expr R X (j - Suc 0) (ext_cf S j c) = \<zero>"
rep_eq: fixes x :: "('b, 'c) mapping_rbt" shows "RBT_Mapping2.keys x = set (RBT_Impl.keys (mapping_rbt.impl_of x))"
deltafun_def: fixes b :: "'d"   and a :: "'c" shows "b \<delta> a = (\<lambda>x. if x = a then b else (0::'d))"
si_empty1: fixes X :: "'c event set" shows "setinterleaving ([], X, []) = {[]}"
toFunction_def: fixes R :: "('c \<times> 'd) set" shows "toFunction R = (,,) R"
emeasure_main_part_UN: fixes S :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range S \<subseteq> sets (completion M)" shows "emeasure M (main_part M (\<Union> (range S))) = emeasure M (\<Union>i. main_part M (S i))"
fr_nStore_nReturn1: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat"   and j :: "nat" assumes "Broadcast n nSend recv"   and "inf_disj n nSend"   and "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "j < n"   and "j \<noteq> k" shows "nStore j t = nReturn k t"
val_is_vD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = val" shows "ip \<in> vD rt"
impl_ded: fixes a :: "'b N"   and b :: "'b N"   and c :: "'b N" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
add_less_cancel_right: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" shows "(a + c \<subset># b + c) = (a \<subset># b)"
fps_lr_inverse_one_one_2: shows "fps_right_inverse 1 (1::'b) = 1"
sup_commute: fixes sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
join_le: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "x \<squnion> y \<sqsubseteq> z"
memb_imp_not_empty: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "S \<noteq> {}"
Uvariation_sym_rel: fixes \<omega> :: "variable \<Rightarrow> real"   and \<nu> :: "variable \<Rightarrow> real"   and U :: "variable set" assumes "Uvariation \<omega> \<nu> U" shows "Uvariation \<nu> \<omega> U"
vread': shows "v \<in> local.reads (path \<sigma>' k')"
nz: fixes n :: "nat" assumes "n0 \<le> n" shows "g n \<noteq> (0::'b)"
primitive_part_def: fixes p :: "'a poly" shows "primitive_part p = map_poly (\<lambda>x. x div content p) p"
append_Cons: fixes x :: "'k"   and xs :: "'k list"   and ys :: "'k list" shows "(x # xs) @ ys = x # xs @ ys"
imp_I: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A H \<turnstile> B" shows "H \<turnstile> A IMP B"
SumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<equiv>\<^sub>s P \<oplus> Q \<oplus> R"
skl1_inv3_derived: shows "skl1_inv1 \<inter> skl1_inv2 \<subseteq> skl1_inv3"
insert_is_Un: fixes a :: "'g"   and A :: "'g set" shows "insert a A = {a} \<union> A"
Lcm_eq_0_I: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "zero \<in> A" shows "Lcm A = zero"
frequently_ex: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "frequently P F" shows "\<exists>x. P x"
x_2: shows "[A*\<^sub>vx]\<le>b"
simActFreeCases: fixes P :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "weakSimAct P (\<alpha> \<prec> Q') P Rel"
real_cond_exp_intA: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "integrable M f"   and "A \<in> sets F" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A (real_cond_exp M F f)"
is_monoid: shows "Group.monoid (add_monoid (R Quot I))"
index_ge_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int" assumes "GDD point_set block_collection groups index" shows "1 \<le> index"