hcju/isabelleps huggingface dataset:
a_insert_node_def: fixes ptr :: "(_) object_ptr"   and new_child :: "(_) node_ptr"   and reference_child_opt :: "(_) node_ptr option" shows "local.a_insert_node ptr new_child reference_child_opt = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. set_child_nodes ptr (insert_before_list new_child reference_child_opt children))"
disjoint_iff_in_not_in2: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = {}) = (\<forall>x\<in>B. x \<notin> A)"
word_bw_comms_2: fixes x :: "'b word"   and y :: "'b word" shows "x || y = y || x"
Cons: fixes i :: "nat" shows "i < row_length M * row_length M2 \<and> mat (row_length M) (length M) M \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> row (M \<otimes> M2) i = vec_vec_Tensor (row M (i div row_length M2)) (row M2 (i mod row_length M2))"
l_adopt_node_wf_axioms: shows "l_adopt_node_wf DocumentClass.type_wf DocumentClass.known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes DocumentClass.known_ptrs adopt_node"
AE: shows "AE x in M. x \<notin> C \<longrightarrow> (x \<in> A) = (x \<in> B)"
arc_balance_Cons: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and v :: "'a"   and w :: "'a" assumes "trail u (e # es) v" shows "arc_set_balance w (insert e (set es)) = arc_set_balance w {e} + arc_balance w es"
e1e2cond: shows "(e1, e2) \<in> f \<and> e1 = i"
get_owner_document_def: fixes ptr :: "(_) object_ptr" shows "get_owner_document ptr = invoke local.a_get_owner_document_tups ptr ()"
word_upto_cons_front: fixes a :: "'a word"   and b :: "'a word" shows "word_upto a b = word_upto' a b"
assms_3: shows "fst' \<equiv> op_sem Fst"
append_Nil: fixes ys :: "'b list" shows "append [] ys = ys"
hide_sync_D1: fixes A :: "'a set"   and C :: "'a set"   and P :: "'a process"   and Q :: "'a process" assumes "finite A"   and "A \<inter> C = {}" shows "D ((P \<lbrakk>C\<rbrakk> Q) \ A) \<subseteq> D (P \ A \<lbrakk>C\<rbrakk> Q \ A)"
permute_Tree_tConj: fixes p :: "perm"   and tset :: "('a, 'b, 'c, 'd) Tree set['a]" shows "p \<bullet> tConj tset = tConj (p \<bullet> tset)"
simps_11: shows "OUTfromCh data11 = {data2}"
eventually_at_leftI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.order_topology less_eq less open"   and "\<And>x. x \<in> ord.greaterThanLessThan less a b \<Longrightarrow> P x"   and "less a b" shows "eventually P (topological_space.at_within open b (ord.lessThan less b))"
close_eq: shows "close [(p # \<Gamma>, q # closeI \<Delta> j), (q # \<Gamma>, p # closeI \<Delta> j)] (\<Gamma>, \<Delta>) = [(p # \<Gamma>, q # closeI \<Delta> j), (q # \<Gamma>, p # closeI \<Delta> j)]"
IH: shows "Digamma (of_nat n + (1::'a) / (2::'a)) = (\<Sum>k<n. (2::'a) / of_nat (2 * k + 1)) - euler_mascheroni - of_real (2 * ln 2)"
Nil: fixes y :: "'a"   and ys :: "'a list" shows "lexordp [] (y # ys)"
prems_3: shows "i < index (insort a (x # xs)) a"
assms_3: shows "\<forall>J\<subseteq>I. card J \<le> card (\<Union> (A ` J))"
init_lifted: fixes p :: "net_tree" assumes "wf_net_tree p" shows "{(\<sigma>, snd (netgmap sr s)) |\<sigma> s. s \<in> init (pnet np p) \<and> (\<forall>i. if i \<in> net_tree_ips p then \<sigma> i = the (fst (netgmap sr s) i) else \<sigma> i \<in> (fst \<circ> sr) ` init (np i))} \<subseteq> init (opnet onp p)"
cl_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier" shows "cl_in \<E> X \<subseteq> \<E>"
gs_def: shows "gs \<equiv> Mp (\<Prod>\<^sub># fs)"
last_rev: fixes xs :: "'f list" assumes "xs \<noteq> []" shows "last (rev xs) = hd xs"
sum: fixes g :: "'b \<Rightarrow> real"   and A :: "'b set"   and x :: "'a" shows "sum g A *\<^sub>R x = (\<Sum>x\<in>A. g x *\<^sub>R x)"
NRR_validity_good: fixes NRR :: "msg"   and B :: "agent"   and A :: "agent"   and L :: "nat"   and C :: "msg"   and evs :: "event list" assumes "NRR = Crypt (priEK B) \<lbrace>Number f_nrr, Agent A, Nonce L, C\<rbrace>"   and "NRR \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> zg" shows "Says B A \<lbrace>Number f_nrr, Agent A, Nonce L, NRR\<rbrace> \<in> set evs"
sswSts: shows "full_chain (\<leadsto>\<^sub>w) sswSts \<and> emb wSts sswSts \<and> lhd sswSts = lhd wSts \<and> llast sswSts = llast wSts"
get_shadow_root_safe_pure: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "pure (get_shadow_root_safe element_ptr) h"
select_convs_1: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
mod_state_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = s1\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr> \<and> t2 = s2\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>" shows "low_equal t1 t2"
rel_pmf_bindI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d pmf"   and q :: "'e pmf"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f pmf"   and g :: "'e \<Rightarrow> 'g pmf" assumes "rel_pmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" shows "rel_pmf P (p \<bind> f) (q \<bind> g)"
wary_sub: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat" assumes "sub s t"   and "arity.wary arity_sym arity_var t" shows "arity.wary arity_sym arity_var s"
indRelTEQ_impl_TRel_weakly_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelTEQ TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
indets_poly_eval_focus_subset: fixes a :: "'b \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (poly_eval a (focus X p)) \<subseteq> \<Union> (indets ` a ` X) \<union> (indets p - X)"
bij: fixes f :: "'b \<Rightarrow> 'c" assumes "bijective f" shows "bij f"
qe_eps_def: shows "qe_eps = R.lift_nnf_qe qe_eps\<^sub>1"
star_ext: fixes x :: "'a" shows "x \<le> qstar x"
rev_swap: fixes xs :: "'g list"   and ys :: "'g list" shows "(rev xs = ys) = (xs = rev ys)"
lunit_in_hom_1: fixes f :: "'a" assumes "ide f" shows "arr \<l>[f] \<and> src f \<in> sources \<l>[f] \<and> trg f \<in> targets \<l>[f]"
simps_2: fixes k :: "'c"   and ks :: "'c list"   and f :: "'d option \<Rightarrow> 'd"   and v :: "'d option"   and ps :: "('c \<times> ('c, 'd) trie) list" shows "update_with_trie (k # ks) f (Trie v ps) = Trie v (AList.update_with_aux empty_trie k (update_with_trie ks f) ps)"
rel_count_def: fixes u :: "nat"   and u' :: "nat"   and x :: "bool"   and y :: "nat" shows "rel_count u u' x y = ((x \<longrightarrow> u' = Suc u) \<and> (\<not> x \<longrightarrow> u' = u) \<and> y = u')"
mat_det_congruence: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (congruence M H) = cor ((cmod (mat_det M))\<^sup>2) * mat_det H"
N_vector_top_pp: fixes x :: "'a" shows "- (- (x \<sqinter> top) \<sqinter> top) \<sqinter> top \<sqinter> top = - - (x \<sqinter> top)"
assms_3: shows "isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M)"
Well_order_iso_copy: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "well_order_on A r"   and "bij_betw f A A'" shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
kruskal: fixes P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and A :: "'b set" assumes "wqo_on P (A \<times> UNIV)" shows "wqo_on (kruskal_tree_tree.emb A P) (trees A)"
validTransition_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "valid P" shows "valid P'"
characteristic_weak_formula_is_characteristic': fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and Q :: "'c"   and P :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "nominal_ts.valid satisfies transition Q (indexed_weak_nominal_ts.characteristic_weak_formula \<tau> satisfies transition P) = indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q"
simps_2: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and as :: "'c list"   and bs :: "'b list" shows "map_tailrec_rev f (a # as) bs = map_tailrec_rev f as (f a # bs)"
rel_sel: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e option"   and b :: "'f option" shows "rel_option R a b = ((a = None) = (b = None) \<and> (a \<noteq> None \<longrightarrow> b \<noteq> None \<longrightarrow> R (the a) (the b)))"
all_countings: fixes a :: "nat"   and b :: "nat" shows "all_countings a b = a + b choose a"
disj: shows "\<not> isVal e \<or> safe_hd S' = safe_hd (restr_stack (- set r) S')"
distinct_300: fixes x18 :: "float"   and x151 :: "floatarith"   and x152 :: "nat" shows "Num x18 \<noteq> x151 ^\<^sub>e x152"
circline_set_of_ocircline: fixes H :: "ocircline" shows "circline_set (of_ocircline H) = ocircline_set H"
Int_interval: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "cbox a b \<inter> cbox c d = cbox (\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i)"
comp_anti: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (- y \<le> - x)"
ind: fixes i :: "int" assumes "\<And>j. \<lbrakk>k \<le> j; j < i\<rbrakk> \<Longrightarrow> P j" shows "P i"
simps_5: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
oth_class_taut_4_a: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<phi> in v]"
inline1_in_sub_gpvs: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and x :: "'ret"   and s' :: "'s"   and input :: "'ret'" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "(x, s') \<in> results_gpv \<I>' (callee' input)"   and "input \<in> responses_\<I> \<I>' out"   and "\<I> \<turnstile>g gpv \<surd>" shows "rpv' x \<in> sub_gpvs \<I> gpv"
homotopic_paths_imp_subset: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "homotopic_paths s p q" shows "path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
strictly_maximal_wrt_subst: fixes A :: "'a"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "strictly_maximal_wrt (A \<cdot>a \<sigma>) (C \<cdot> \<sigma>)" shows "strictly_maximal_wrt A C"
StoreBuffer: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'sb \<times> 'dirty \<times> 'owns \<times> 'rels) list"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'sb"   and \<D> :: "'dirty"   and \<O> :: "'owns"   and \<R> :: "'rels"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'shared"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'sb"   and \<O>' :: "'owns"   and \<R>' :: "'rels"   and \<S>' :: "'shared" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (m', sb', \<O>', \<R>', \<S>')" shows "(ts, m, \<S>) \<Rightarrow> (ts[i := (p, is, \<theta>, sb', \<D>, \<O>', \<R>')], m', \<S>')"
measurable_simple_function: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_function M f" shows "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
simps_1: fixes at :: "('c, 'd) atm" shows "nvL (Pos at) = nvA at"
\<ll>'_ide_simp: fixes a :: "'c local.term set" assumes "ide a" shows "\<ll>' a = lunit' a"
cap_positive: fixes e :: "nat \<times> nat" assumes "e \<in> E" shows "(0::'a) < c e"
a_d_add_closure: fixes x :: "'a"   and y :: "'a" shows "d (ad x + ad y) = ad x + ad y"
negligible_le: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "Negligible.negligible g"   and "\<And>\<eta>. \<bar>f \<eta>\<bar> \<le> g \<eta>" shows "Negligible.negligible f"
new_tv_def: fixes n :: "nat"   and ts :: "'a" shows "new_tv n ts = (\<forall>m. m \<in> free_tv ts \<longrightarrow> m < n)"
ext_method_simp: fixes m :: "qtname \<times> 'b" shows "(declclass m, mthd m) = m"
red1': shows "s1 -1-stls1\<rightarrow>*t \<infinity>"
tendsto_log: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "0 < a"   and "a \<noteq> 1"   and "0 < b" shows "((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
res_def: fixes \<eta> :: "nat" shows "MAC.res \<eta> = 1\<^sub>C |\<^sub>= MAC.enm \<eta> |\<^sub>= MAC.dem \<eta> \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> MAC.RO.res \<eta> \<parallel> INSEC.res"
\<PP>_induct: fixes p :: "(symbol \<times> character list) list"   and P :: "(symbol \<times> character list) list \<Rightarrow> bool" assumes "p \<in> \<PP>"   and "P []"   and "\<And>p k u. \<lbrakk>\<And>q. q \<in> \<P> k u \<Longrightarrow> P q; p \<in> \<P> k (Suc u)\<rbrakk> \<Longrightarrow> P p" shows "P p"
absolutely_integrable_sum: fixes T :: "'c set"   and S :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite T"   and "\<And>a. a \<in> T \<Longrightarrow> f a absolutely_integrable_on S" shows "(\<lambda>x. \<Sum>a\<in>T. f a x) absolutely_integrable_on S"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<mu> \<star> \<nu> \<star> \<pi>)"
measure_Union': fixes F :: "'b set set"   and M :: "'b measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M"   and "disjoint F" shows "Sigma_Algebra.measure M (\<Union> F) = sum (Sigma_Algebra.measure M) F"
simps_21: shows "OUTfromCh data21 = {data18, data19}"
dbm_lt_asym: fixes e :: "'d DBMEntry"   and f :: "'d DBMEntry" assumes "e \<prec> f" shows "\<not> f \<prec> e"
sum_eq_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "finite F" shows "(comm_monoid_add.sum plus zero f F = zero) = (\<forall>a\<in>F. f a = zero)"
tagged_fun_relD_both: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'f"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'f) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, Let x' f') \<in> B"
simps: fixes xs :: "'d list" shows "IArray.list_of (IArray xs) = xs"
lossless_R1: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool" shows "lossless_spmf (R1 M \<sigma>)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<noteq> (0::'b)"
strongBisimWeakEq: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "P \<simeq> Q"
3: shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q x) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) - set X"
abort_5: fixes x :: "'b"   and a\<^sub>1 :: "'a"   and a\<^sub>2 :: "'a" assumes "x \<in> S'"   and "f x a\<^sub>1 a\<^sub>2" shows "r a\<^sub>1 a\<^sub>2"
H: shows "\<forall>bs. Ipoly bs p = Ipoly bs q"
c_traces: fixes xps :: "('e \<times> 'f) list"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "(xps \<in> traces (c_process step out s\<^sub>0)) = (\<exists>X. (xps, X) \<in> c_failures step out s\<^sub>0)"
closed_atLeastAtMost: fixes a :: "'c"   and b :: "'c" shows "closed {a..b}"
v1: shows "atom v1 \<sharp> (z, X0, X2, X3)"
cases: fixes a1 :: "'g LDCFG_node"   and a2 :: "'h"   and Use :: "'g \<Rightarrow> 'h set"   and E :: "'g"   and X :: "'g"   and H :: "'h set"   and L :: "'h set" assumes "(a1, a2) \<in> lift_Use_set Use E X H L" obtains V :: "'h" and n :: "'g" where "a1 = Node n" and "a2 = V__" and "V__ \<in> Use n"   | V :: "'h" where "a1 = Node E" and "a2 = Va__" and "Va__ \<in> H"   | V :: "'h" where "a1 = Node X" and "a2 = V" and "V \<in> L"
diff_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
matpow_sum_one_mono: fixes n :: "nat"   and i :: "'n"   and j :: "'n"   and k :: "nat" shows "matpow (A + mat (1::'a)) n $h i $h j \<le> matpow (A + mat (1::'a)) (n + k) $h i $h j"
sw_commute: fixes zs :: "'a"   and zs' :: "'a"   and x :: "'b"   and y :: "'b"   and x' :: "'b"   and y' :: "'b"   and us :: "'a"   and u :: "'b" assumes "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}" shows "u @us[x \<and> y]_zs @us[x' \<and> y']_zs' = u @us[x' \<and> y']_zs' @us[x \<and> y]_zs"
import: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}" shows "\<turnstile>\<^sub>3\<^sub>b {P} c { Q}"
maps_def: fixes f :: "'e \<Rightarrow> 'd list"   and xs :: "'e list" shows "List.maps f xs = concat (map f xs)"
cterms_def': fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>" shows "cterms \<Gamma> = {uu_. \<exists>p pn. uu_ = p \<and> p \<in> ctermsl (\<Gamma> pn) \<and> not_call p}"
simps_1: fixes x :: "'g" shows "remove1 x [] = []"
I_def: shows "I = (\<lambda>m. exp (- (x + real m)\<^sup>2) / (- 2 * (x + real m) ^ (2 * n + 1)) - exp (- x\<^sup>2) * inverse (- 2 * x ^ (2 * n + 1)) - real (2 * n + 1) / 2 * integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * Suc n)))"
maximal_literal_def: fixes L :: "'a literal"   and C :: "'a literal set" shows "maximal_literal L C = (\<forall>x. x \<in> C \<longrightarrow> (L, x) \<notin> lit_ord)"
is_natural_transformation: shows "natural_transformation hom'.comp hom'.comp FG.map hom'.map \<psi>.map"
coinduct: fixes X :: "'b llist \<Rightarrow> 'b llist \<Rightarrow> bool"   and x1 :: "'b llist"   and x2 :: "'b llist" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> (\<exists>xs. x1 = LNil \<and> x2 = xs \<and> lfinite xs) \<or> (\<exists>xs ys x zs. x1 = LCons x xs \<and> x2 = prepend zs (LCons x ys) \<and> (X xs ys \<or> emb xs ys))" shows "emb x1 x2"
i_aggregate_assoc: fixes a :: "nat"   and b :: "nat"   and ag :: "'b list \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b" assumes "0 < a"   and "0 < b"   and "\<And>xs. length xs mod a = 0 \<Longrightarrow> ag (f_aggregate xs a ag) = ag xs" shows "i_aggregate (i_aggregate f a ag) b ag = i_aggregate f (a * b) ag"
sets_exponential: fixes l :: "real" shows "sets (exponential l) = sets borel"
cond_def: fixes I :: "nat set"   and I' :: "nat set"   and l :: "'a vec \<Rightarrow> 'a"   and c :: "'a vec"   and h :: "nat"   and k :: "nat" shows "cond I I' l c h k \<equiv> b = lincomb l (a ` I) \<and> h \<in> I \<and> l (a h) < (0::'a) \<and> (\<forall>h'. h' \<in> I \<longrightarrow> h' < h \<longrightarrow> (0::'a) \<le> l (a h')) \<and> c \<in> carrier_vec n \<and> local.span (a ` (I - {h})) = {x \<in> carrier_vec n. c \<bullet> x = (0::'a)} \<and> c \<bullet> b < (0::'a) \<and> k < m \<and> c \<bullet> a k < (0::'a) \<and> (\<forall>k'<k. (0::'a) \<le> c \<bullet> a k') \<and> I' = insert k (I - {h})"
map_code_def: fixes l :: "'c list"   and f :: "'c \<Rightarrow> 'b" shows "map_code l f = map f l"
subst_TVar_scheme_list: fixes A :: "type_scheme list" shows "$ TVar A = A"
disc_eq_case_1: fixes list :: "'a list" shows "(list = []) = (case list of [] \<Rightarrow> True | uu_ # uua_ \<Rightarrow> False)"
distinct_nthsI: fixes xs :: "'a list"   and I :: "nat set" assumes "distinct xs" shows "distinct (nths xs I)"
dghm_cn_comp_is_dghm: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<FF> :: "V" assumes "digraph \<alpha> \<AA>"   and "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
H_elem_map_eq: fixes H1 :: "'a set" assumes "H1 \<in> rcosets H" shows "H #> (SOME x. x \<in> carrier G \<and> H #> x = H1) = H1"
step_sim: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme"   and a :: "'h"   and a' :: "'h"   and b :: "'i" assumes "simulation R A B"   and "(a, a') \<in> g_E A"   and "(a, b) \<in> R" shows "\<exists>b'. (b, b') \<in> g_E B \<and> (a', b') \<in> R"
bilinear_r_add1: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m :: "'g"   and m21 :: "'i"   and m22 :: "'i"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m \<in> carrier M1"   and "m21 \<in> carrier M2"   and "m22 \<in> carrier M2"   and "bilinear_map f R M1 M2 N" shows "f (m, m21 \<plusminus>\<^bsub>M2\<^esub> m22) \<plusminus>\<^bsub>N\<^esub> -\<^sub>a\<^bsub>N\<^esub> (f (m, m21) \<plusminus>\<^bsub>N\<^esub> f (m, m22)) = \<zero>\<^bsub>N\<^esub>"
Expr: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n\<rightarrow> s1"
b'_def: shows "b' \<equiv> \<lambda>f. \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"
simps_24: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case INTEG of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f3"
seskey_spied: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "SesKey SK \<in> spied s" shows "\<exists>S A C. fst SK = Some S \<and> A \<in> fst (snd SK) \<and> C \<in> snd (snd SK) \<and> {PriKey S, PriKey A, PriKey C} \<subseteq> spied s"
wcount_union: fixes f :: "'b \<Rightarrow> nat"   and M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" shows "wcount f (M + N) a = wcount f M a + wcount f N a"
F1set2_bd_incr: fixes x :: "('e, 'd, 'f) F1" shows "|F1set2 x| \<le>o F1bd'"
diff_add_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c + b - a = c + (b - a)"