hcju/isabelleps huggingface dataset:
trimmed_pos_imp_trimmed: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and x :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "multiseries.trimmed_pos trimmed dominant_term x" shows "trimmed x"
evalS_models: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and \<phi> :: "('a, 'p) Kform"   and s :: "'s" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t"   and "subjective a \<phi>"   and "s \<in> toSet aec" shows "evalS (clock_evalProp envVal) (clock_knowledge envObs cec) (clock_commonKnowledge envObs cec) aec \<phi> = clock_repMC (toSet cec), s \<Turnstile> \<phi>"
safe_orig: shows "\<forall>x<k. safe_delayed (c (n + x))"
is_zero_def: fixes p :: "'b poly" shows "is_zero p = List.null (coeffs p)"
fun_eq_on_sym: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "fun_eq_on f g A" shows "fun_eq_on g f A"
less_option_None_is_Some: fixes x :: "'a option" assumes "None < x" shows "\<exists>z. x = Some z"
S_def: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> ('a \<Rightarrow> ('b, 'd) cconstraint)" assumes "Probabilistic_Timed_Automaton A" shows "Probabilistic_Timed_Automaton.S A \<equiv> {(l, u). l \<in> Probabilistic_Timed_Automaton.L A \<and> (\<forall>x\<in>Probabilistic_Timed_Automaton.\<X> A. (0::'d) \<le> u x) \<and> u \<turnstile> PTA.inv_of A l}"
rec: fixes f :: "'b option \<Rightarrow> ('c \<times> ('c, 'b) Trie.trie \<times> 'a) list \<Rightarrow> 'a"   and x1 :: "'b option"   and x2 :: "('c \<times> ('c, 'b) Trie.trie) list" shows "rec_trie f (Trie.trie.Trie x1 x2) = f x1 (map (map_prod id (\<lambda>trie. (trie, rec_trie f trie))) x2)"
SPLIT_1: shows "(([u], {#}), ee, sh, ch) \<in> ntrp fg"
correct_state_heap_change: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and hconf :: "'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and spurious_wakeups :: "bool"   and t :: "'b"   and xcp :: "'a option"   and h :: "'c"   and frs :: "'a frame list"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and xcp' :: "'a option"   and h' :: "'c"   and frs' :: "'a frame list"   and t'' :: "'b"   and xcp'' :: "'a option"   and frs'' :: "'a frame list" assumes "JVM_conf_read addr2thread_id thread_id2addr empty_heap allocate typeof_addr heap_read heap_write hconf P"   and "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "JVM_heap_base.exec_1_d addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t (Normal (xcp, h, frs)) ta (Normal (xcp', h', frs'))"   and "JVM_heap_conf_base.correct_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi> t (xcp, h, frs)"   and "JVM_heap_conf_base.correct_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi> t'' (xcp'', h, frs'')" shows "JVM_heap_conf_base.correct_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi> t'' (xcp'', h', frs'')"
division_of_union_self: fixes p :: "'a set set"   and s :: "'a set" assumes "p division_of s" shows "p division_of \<Union> p"
\<Psi>_sel_3: shows "source \<Psi> = None"
es_perm: shows "edge_succ (rev_swap a b) permutes arcs G"
simps: fixes x :: "real"   and y :: "real" shows "Rmin x + Rmin y = Rmin (min x y)"
not_distinct_decomp: fixes ws :: "'g list" assumes "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ [y] @ ys @ [y] @ zs"
valid_abs_goal_2: fixes v :: "nat" assumes "abs_ast_goal v \<noteq> None" shows "the (abs_ast_goal v) \<in> set (the (abs_range_map v))"
filter_id_conv: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
good1: fixes ws :: "letter list list"   and w :: "letter list" assumes "good ws" shows "good (w # ws)"
coincidence_term_cor: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<theta> :: "trm"   and I :: "interp" assumes "Uvariation \<omega> \<omega>' U"   and "FVT \<theta> \<inter> U = {}" shows "term_sem I \<theta> \<omega> = term_sem I \<theta> \<omega>'"
image_cong_simp: fixes M :: "'e set"   and N :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "M = N"   and "\<And>x. x \<in> N =simp=> f x = g x" shows "f ` M = g ` N"
unbox_typeof: fixes \<tau> :: "Unboxed.type"   and d :: "'dyn"   and blob :: "('dyn, 'ubx1, 'ubx2) unboxed" assumes "unbox \<tau> d = Some blob" shows "typeof blob = Some \<tau>"
exec_mbindFPurge_failure: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = None" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e (a # S) ioprog) M) = (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S ioprog) M)"
path_state_closure: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and s :: "'a"   and \<tau> :: "'b list"   and s' :: "'a" assumes "Unwinding SES \<V>"   and "s \<tau>\<Longrightarrow>\<^bsub>SES\<^esub> s'"   and "s \<in> S\<^bsub>SES\<^esub>" shows "s' \<in> S\<^bsub>SES\<^esub>"
less_eq_hempty: fixes u :: "hf" shows "(u \<le> 0) = (u = 0)"
p: shows "p = (\<Sum>a\<in>A. q a *s a)"
primitive_part_idemp: fixes f :: "'b poly" shows "primitive_part (primitive_part f) = primitive_part f"
L2_Boiler: fixes x :: "nat \<Rightarrow> bit list"   and s :: "nat \<Rightarrow> nat list"   and y :: "nat \<Rightarrow> nat list" assumes "SteamBoiler x s y"   and "ts x" shows "ts y"
lm05: fixes P :: "nat \<Rightarrow> bool"   and m :: "nat" shows "concat (map (\<lambda>n. if P n then [n] else []) [0..<m]) = concat (map (\<lambda>n. if n \<in> set [0..<m] then if P n then [n] else [] else []) [0..<m])"
rSim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto><(weakBisim O X O bisim)> Q"
Invariants_points: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and x :: "'b" assumes "qmpt M T"   and "A \<in> sets (qmpt.Invariants M T)"   and "x \<in> A" shows "T x \<in> A"
D'': shows "[abstr g (\<lambda>i j. \<infinity>) v]\<^bsub>v,n\<^esub> = {u. u \<turnstile> g}"
poly_different_sign_imp_root: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "sgn (poly p a) \<noteq> sgn (poly p b)" shows "\<exists>x\<ge>a. x \<le> b \<and> poly p x = 0"
v_def: shows "v = (\<lambda>x. x powr (real n + 1 / 2))"
submanifold_axioms: shows "submanifold charts1 k (U x)"
saturated_def: fixes S :: "(('c, 'd) fm list \<times> 'd) set" shows "saturated S \<equiv> \<forall>p i. (\<^bold>\<diamond> p) at i in' S \<longrightarrow> (\<nexists>a. p = Nom a) \<longrightarrow> (\<exists>j. (\<^bold>@ j p) at i in' S \<and> (\<^bold>\<diamond> Nom j) at i in' S)"
zeta_partial_sum_neg_asymp_equiv: fixes s :: "real" assumes "0 < s" shows "sum_upto (\<lambda>n. real n powr s) \<sim>[at_top] (\<lambda>x. x powr (1 + s) / (1 + s))"
ext_inject: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a"   and id' :: "nat"   and no' :: "nat"   and nhid' :: "nat"   and msg' :: "msg"   and num' :: "nat"   and sid' :: "nat"   and more' :: "'a" shows "(\<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id', no = no', nhid = nhid', msg = msg', num = num', sid = sid', \<dots> = more'\<rparr>) = (id = id' \<and> no = no' \<and> nhid = nhid' \<and> msg = msg' \<and> num = num' \<and> sid = sid' \<and> more = more')"
bind_transfer: fixes R0 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "(crel_vs R0 ===> (R0 ===>\<^sub>T R1) ===>\<^sub>T R1) (\<lambda>v f. f v) (\<bind>)"
eq_arctan_pos: fixes z :: "complex" assumes "0 < Re z" shows "Im (Ln z) = arctan (Im z / Re z)"
var_closed_1: fixes K :: "'a set" assumes "subring K R" shows "X \<in> carrier (K [X])"
Lim_transform2: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'c)) F" shows "(g \<longlongrightarrow> a) F"
simps_15: fixes Vs :: "char list list"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (if (e) e\<^sub>1 else e\<^sub>2) = if (compE\<^sub>1 Vs e) compE\<^sub>1 Vs e\<^sub>1 else compE\<^sub>1 Vs e\<^sub>2"
step: fixes r :: "('a + 'b + 'c, 'd) Old_Datatype.node set"   and r' :: "('a + 'b + 'c, 'd) Old_Datatype.node set" assumes "X r r'" shows "(\<exists>a. r = TERMINAL a \<and> r' = TERMINAL a) \<or> (\<exists>R R' b. r = LINEAR b R \<and> r' = LINEAR b R' \<and> (X R R' \<or> Eq_RESUMPTION R R')) \<or> (\<exists>rs rs' c. r = BRANCH c rs \<and> r' = BRANCH c rs' \<and> (\<forall>d. X (rs d) (rs' d) \<or> Eq_RESUMPTION (rs d) (rs' d)))"
assms: shows "evaluate_dec False mn env (update_clock (\<lambda>_. count1) s) d (update_clock (\<lambda>_. count1) s', r)"
sound_weaken_gen: fixes A :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and B :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "sublist A B"   and "sound (A, C)" shows "sound (B, C)"
pivot_rvars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "rvars (\<T> (pivot x\<^sub>i x\<^sub>j s)) = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}"
triangular_column_def: fixes j :: "nat"   and A :: "'a mat" shows "local.triangular_column j A \<equiv> \<forall>i>j. i < dim_row A \<longrightarrow> A $$ (i, j) = (0::'a)"
power_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'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"   and n :: "nat" assumes "class.field divide inverse times one plus zero minus uminus" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
abs_if_raw: shows "abs \<equiv> \<lambda>a. if a < (0::'c) then - a else a"
pair_pmf_def: fixes A :: "'a pmf"   and B :: "'b pmf" shows "pair_pmf A B = A \<bind> (\<lambda>x. B \<bind> (\<lambda>y. return_pmf (x, y)))"
A1: shows "(p + 1) * (q + 1) = r + 1"
all: shows "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S"
HEndPhase2_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4d s p" shows "HInv4d s' p"
min_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
simps_2: fixes v :: "variable"   and T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubst (Fv v T) \<rho> = Fv v (tsubstT T \<rho>)"
prems_2: fixes w :: "'a" assumes "w \<in> set (awalk_verts u (e # es))" shows "P w \<or> Q w"
add_sheaf_spec_in_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and t :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "s \<in> \<O> U"   and "t \<in> \<O> U"   and "U \<subseteq> Spec" shows "add_sheaf_spec U s t \<in> \<O> U"
ipresIGSwapAll_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'n, 'o, 'p, 'q, 'h, 'j, 'r) model_scheme"   and MOD' :: "('s, 't, 'n, 'u, 'v, 'q, 'i, 'k, 'w) model_scheme" shows "ipresIGSwapAll h hA MOD MOD' \<equiv> ipresIGSwap h MOD MOD' \<and> ipresIGSwapAbs hA MOD MOD'"
not_residue_primroot_0_right: fixes n :: "nat" shows "residue_primroot n 0 = (n = 1)"
split_beta: fixes f :: "'s \<Rightarrow> 't \<Rightarrow> 'r"   and prod :: "'s \<times> 't" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
start_heap_nStart: fixes P :: "'a prog"   and a :: "nat"   and obj :: "obj" assumes "start_heap P a = \<lfloor>obj\<rfloor>" shows "fst obj \<noteq> Start"
lappend_llist_of_llist_of: fixes xs :: "'a list"   and ys :: "'a list" shows "lappend (llist_of xs) (llist_of ys) = llist_of (xs @ ys)"
Par_Sbis: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "properL dl"   and "SbisL cl dl" shows "Par cl \<approx>s Par dl"
hensel_lifting_monic_def: fixes p :: "int"   and n :: "nat"   and u :: "int poly"   and vs :: "int poly list" shows "hensel_lifting_monic p n u vs = (if vs = [] then [] else let pn = p ^ n; C = poly_mod.Mp pn u in Let (product_factor_tree p (create_factor_tree vs)) (hensel_lifting_main p n C))"
SN_on_mono: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('a \<times> 'a) set" assumes "SN_on r A"   and "s \<subseteq> r" shows "SN_on s A"
a_inv_op_closed2: fixes x :: "'a"   and h :: "'a" assumes "x \<in> carrier R"   and "h \<in> I" shows "x \<oplus> h \<oplus> \<ominus> x \<in> I"
has_terminal: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category_with_terminal_object C" shows "\<exists>t. category.terminal C t"
scale_eq_0_iff: fixes a :: "'d"   and x :: "'d" shows "(a * x = (0::'d)) = (a = (0::'d) \<or> x = (0::'d))"
T_thm_1_lub: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "CLF cl f" shows "lub (fix f (pset cl)) cl = lub {x \<in> pset cl. (x, f x) \<in> order cl} cl"
a_2: fixes uu :: "('a \<times> 'c) list"   and k :: "'a" assumes "distinct (map fst (rev uu))"   and "k \<notin> fst ` set uu" shows "None = fold (\<lambda>(x, v) m. m(f x \<mapsto> v)) uu Map.empty (f k)"
assoc_end_max_def: shows "assoc_end_max \<equiv> fst \<circ> snd \<circ> snd"
Joints_singleton_image: fixes x :: "'b \<times> 'b pdevs" shows "Joints [x] = (\<lambda>x. [x]) ` Affine x"
\<sigma>_def: shows "\<sigma> = (\<lambda>v. Var (\<sigma>' v))"
add_nonneg_nonneg: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
that_2: assumes "r \<cdot> u \<and>\<^sub>p r' \<cdot> v = r \<and>\<^sub>p r'" shows "thesis"
Busemann_function_at_def: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" shows "Busemann_function_at xi x y = real_of_ereal (Sup {limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) |u. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi})"
tab\<^sub>1_def: fixes r :: "'a" shows "tab\<^sub>1 r \<equiv> SOME g. isomorphic_rep (VVV.iso_class_rep \<lbrakk>r\<rbrakk>) (tab\<^sub>0 r) g"
simps_1: fixes \<mu> :: "'a" shows "Dom \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
A: shows "of_int_hom.mat_hom A \<in> carrier_mat (length fs) (length gs)"
IH_2: shows "\<not> t2 \<prec> t2"
mod_exp_nat_aux_code: fixes m :: "nat"   and y :: "nat"   and x :: "nat"   and n :: "nat" shows "mod_exp_nat_aux m y x n = (if n = 0 then y else if n = 1 then x * y mod m else if even n then mod_exp_nat_aux m y (x * x mod m) (n div 2) else mod_exp_nat_aux m (x * y mod m) (x * x mod m) (n div 2))"
merge_conv_merge_it: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "Common.merge f xs ys = merge_it f xs ys"
Distinct: fixes i :: "nat" assumes "distinct procList"   and "i < j"   and "procList ! i = procList ! j" shows "False"
le_sqrt_iff: fixes x :: "nat"   and y :: "nat" shows "(x \<le> local.sqrt y) = (x\<^sup>2 \<le> y)"
balanceR_def: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "balanceR l k v r = (if cl r = R then let lr = lsub r; rr = rsub r in if cl lr = R then Node (Node l B k v (lsub lr)) R (key lr) (val lr) (Node (rsub lr) B (key r) (val r) rr) else if cl rr = R then Node (Node l B k v lr) R (key r) (val r) (Node (lsub rr) B (key rr) (val rr) (rsub rr)) else Node l B k v r else Node l B k v r)"
exec_move_BlockSomeI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and V :: "nat"   and T :: "ty"   and v :: "'addr val" assumes "exec_move ci P t e h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t {V:T=\<lfloor>v\<rfloor>; e} h (stk, loc, Suc (Suc pc), xcp) ta h' (stk', loc', Suc (Suc pc'), xcp')"
intros: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "has_bochner_integral M f x" shows "integrable M f"
o_inv_o_cancel: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" assumes "inj f" shows "(g \<circ>\<circ>\<circ> inv_into) UNIV f \<circ> f = g"
S: shows "S \<subseteq> {A - {a} |A. A \<in> F \<and> a \<in> A}"
eq: shows "{(y, f). y \<in> S \<and> y \<noteq> x \<and> f \<in> derangements (S - {x, y})} = {(y, f). y \<in> S - {x} \<and> f \<in> derangements (S - {x, y})}"
AbstFormP_Mem: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstFormP v i (Q_Mem x y) (Q_Mem x' y')"
Sum2_1: fixes Q :: "pi"   and Rs :: "residual"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>l Rs" shows "P \<oplus> Q \<Longrightarrow>\<^sub>l Rs"
gg': fixes z :: "'a" assumes "z \<in> carrier C" shows "g (g' z) = z"
AD_3: shows "A =m \<Prod>\<^sub># (factors_of_factor_tree l)"
induct: fixes x :: "'b"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> \<langle>S\<rangle>"   and "P (0::'b)"   and "\<And>s. s \<in> S \<Longrightarrow> P s"   and "\<And>w w'. \<lbrakk>w \<in> \<langle>S\<rangle>; P w; w' \<in> \<langle>S\<rangle>; P w'\<rbrakk> \<Longrightarrow> P (w - w')" shows "P x"
f_Exec_State_Idle_replicate_NoMsg_gr_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "nat"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>m\<^esup> c))"   and "m < n" shows "output_fun (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg"
def_hashmap_size_lexlist_def: shows "def_hashmap_size = (\<lambda>_. 2 * def_hashmap_size TYPE('a))"
dist_eqs_2: fixes discr :: "bool discr"   and cfun' :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "CB\<cdot>discr \<noteq> CFn\<cdot>cfun'"
sprefix_same_imp_eq: fixes xs :: "'c list"   and ys :: "'c stream"   and xs' :: "'c list" assumes "sprefix xs ys"   and "sprefix xs' ys"   and "length xs = length xs'" shows "xs = xs'"
\<pi>_floor_cancel: fixes y :: "real" shows "\<pi> (real_of_int \<lfloor>y\<rfloor>) = \<pi> y"
sup_bot_left: fixes x :: "'a upred" shows "(false \<or> x) = x"
power_one_right: fixes a :: "'a" shows "a ^ 1 = a"
index_together_alt_ss: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set"   and G :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> G"   and "G \<in> groups"   and "card ps = 2" shows "block_collection index ps = 0"
rel_fun_mono': fixes Y :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<longrightarrow> B x y" shows "(X ===> A) f g \<longrightarrow> (Y ===> B) f g"
L: fixes n :: "nat" shows "L n \<in> I \<and> n \<le> L n"
FAssThrow1: fixes e :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>throw e\<bullet>F{D} := e\<^sub>2,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
glob_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<box>\<^sub>b\<gamma> \<equiv> \<lambda>t n. \<forall>n'\<ge>n. \<gamma> t n'"
chine_hcomp_trg_ide: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "chine_hcomp (trg f) f = C.cod (Leg1 (Dom f)) \<down>\<down> Leg1 (Dom f)"
Subset_fresh_iff: fixes a :: "atom"   and t :: "tm"   and u :: "tm" shows "a \<sharp> t SUBS u = (a \<sharp> t \<and> a \<sharp> u)"
index_set_lm_2: fixes A :: "nat set" assumes "index_set A" shows "index_set (- A)"
simps_5: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre ({I} WHILE b DO C) Q = (\<lambda>s. I s + 1)"
non_trivial_combine_imp_dependent: fixes K :: "'a set"   and Ks :: "'a list"   and Us :: "'a list" assumes "subfield K R"   and "set Ks \<subseteq> K"   and "R.combine Ks Us = \<zero>"   and "\<not> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "R.dependent K Us"
mgu_always_unifies: fixes M :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and N :: "('a, 'b) Term.term" assumes "Unifier \<theta> M N" shows "\<exists>\<delta>. mgu M N = Some \<delta>"
assms: shows "h \<Turnstile> (\<exists>\<^sub>Ax. P x)"
unique: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "P x y"   and "P x z" shows "y = z"