hcju/isabelleps huggingface dataset:
Subst: fixes H :: "fmla set"   and A :: "fmla"   and i :: "name"   and x :: "trm" assumes "H \<turnstile> A"   and "\<forall>B\<in>H. atom i \<sharp> B" shows "H \<turnstile> A(i::=x)"
ordLeq_ordLess_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r \<le>o r'"   and "r' <o r''" shows "r <o r''"
simps_1: fixes x1 :: "'f option"   and y1 :: "'f option" shows "(Resumption.resumption.Done x1 = Resumption.resumption.Done y1) = (x1 = y1)"
acomplete_Node_if_wbal2: fixes l :: "'a tree"   and r :: "'a tree"   and x :: "'a" assumes "acomplete l"   and "acomplete r"   and "\<bar>int (size l) - int (size r)\<bar> \<le> 1" shows "acomplete \<langle>l, x, r\<rangle>"
compact_continuous_image: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "compact s" shows "compact (f ` s)"
Busemann_function_le_dist: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" shows "\<bar>Busemann_function_at xi x y\<bar> \<le> dist x y"
cases: fixes x :: "exp" obtains xa :: "Vars.var" where "x = Var xa__"   | xa :: "Vars.var" and e :: "exp" where "x = Lam [xaa__]. eb__"   | e :: "exp" and xa :: "Vars.var" where "x = App ec__ xa"   | as :: "(Vars.var \<times> exp) list" and e :: "exp" where "x = Terms.Let as e"   | b :: "bool" where "x = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "x = (scrut ? e1 : e2)"
base_case_info_R_def: shows "base_case_info_R = (mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]], [([], []), ([0], []), ([], [0])], [[1], [0], [- 1]])"
discount_factor_nonrandom: fixes r :: "real"   and n :: "nat"   and M :: "'b measure" shows "discount_factor r n \<in> borel_measurable M"
select_convs_3: fixes es :: "'d"   and ps :: "'b \<Rightarrow> 'e"   and pubActs :: "'a \<times> ('b \<Rightarrow> 'c)"   and more :: "'f" shows "pubActs \<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr> = pubActs"
jyzP: fixes i :: "'b" assumes "i \<in> {j}" shows "z \<^bsub>(P i)\<^esub>\<prec> y"
map_def: fixes f :: "'a1 \<times> 'a2" shows "local.map f = (if A1.arr (fst f) \<and> A2.arr (snd f) then (F1 (fst f), F2 (snd f)) else B1xB2.null)"
exhaust: fixes y :: "complex_interval" obtains x1 :: "real interval" and x2 :: "real interval" where "y = Complex_Interval x1 x2"
emb_def: fixes ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" shows "emb ps s = (\<lambda>v. case ps v of None \<Rightarrow> s v | Some r \<Rightarrow> r)"
connected_components_of_subset: fixes C :: "'a set"   and X :: "'a topology" assumes "C \<in> connected_components_of X" shows "C \<subseteq> topspace X"
unique_root_sub_interval_1: fixes p :: "int poly"   and l :: "rat"   and r :: "rat"   and l' :: "rat"   and r' :: "rat" assumes "\<exists>!x. root_cond (p, l, r) x"   and "root_cond (p, l', r') (real_of_1 (p, l, r))"   and "l \<le> l'"   and "r' \<le> r" shows "\<exists>!x. root_cond (p, l', r') x"
inj_img_insertE: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'c"   and B :: "'c set" assumes "inj_on f A"   and "x \<notin> B"   and "insert x B = f ` A" obtains x' :: "'b" and A' :: "'b set" where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
add_decreasing: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> bot"   and "c \<le> b" shows "a \<squnion> c \<le> b"
assms_2: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
lossless: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>" shows "lossless_gpv \<I>' (callee s x)"
assms_1: shows "\<forall>n w. prices Mkt x n w = cls_val_process Mkt pf2 n w"
Red_F_\<G>_def: fixes N :: "('f \<times> 'l) set" shows "local.ord_q_lifting.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F_L_q q C. D \<in> Red_F_q q (\<Union> (\<G>_F_L_q q ` N)) \<or> (\<exists>E\<in>N. False \<and> D \<in> \<G>_F_L_q q E)}"
intro: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "hensel_axioms p f a" shows "hensel p f a"
quietTransition: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and P :: "('d, 'e, 'f) psi"   and \<Psi> :: "'e"   and Rs :: "('d, 'e, 'f) residual" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.quiet substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P Rs" obtains P' :: "('d, 'e, 'f) psi" where "Rs = \<tau> \<prec> P'" and "env.quiet substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P'"
assms_3: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>\<^sub>1"
uncurry_case_stmt: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b \<times> 'c" shows "(case x of (a, b) \<Rightarrow> f a b) = uncurry f x"
distinct_lists_no_parent: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<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 get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"   and "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_child_nodes get_disconnected_nodes h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes" shows "\<nexists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
AboveS_not_empty_in_regularCard: fixes S :: "'c set"   and r :: "('c \<times> 'c) set" assumes "|S| <o r"   and "S \<subseteq> Field r"   and "Card_order r"   and "regularCard r"   and "infinite (Field r)" shows "AboveS r S \<noteq> {}"
invar_insert: fixes q :: "'b tree"   and x :: "'b" assumes "braun q \<and> heap q" shows "braun (insert x q) \<and> heap (insert x q)"
akra_bazzi_p_unique: shows "\<exists>!p. (\<Sum>i<k. as ! i * bs ! i powr p) = 1"
linear_eq_on_span: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and x :: "'d" assumes "clinear f"   and "clinear g"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x"   and "x \<in> cspan B" shows "f x = g x"
abs_nchotomy: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodAbs A" shows "\<exists>xs x X. good X \<and> A = Abs xs x X"
rep_eq: fixes x :: "'a coreflexive" shows "Rep_coreflexive (- x) = Rep_coreflexive x '"
ereal_divide_less_iff: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "0 < c"   and "c < \<infinity>" shows "(a / c < b) = (a < b * c)"
rev_nth: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
prj_chine_assoc_3: shows "\<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> \<nu>\<pi>\<rho>.chine_assoc = \<nu>\<pi>\<rho>.Prj\<^sub>0"
prems_2: fixes s :: "char list \<Rightarrow> int" shows "P' s \<le> P s"
eval_negation: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm" shows "eval i (Neg' p) = (if eval i p = Det False then Det True else if eval i p = Det True then Det False else eval i p)"
simps: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" shows "star r a1 a2 = ((\<exists>x. a1 = x \<and> a2 = x) \<or> (\<exists>x y z. a1 = x \<and> a2 = z \<and> r x y \<and> star r y z))"
Icc_eq_Icc: fixes l :: "'a"   and h :: "'a"   and l' :: "'a"   and h' :: "'a" shows "({l..h} = {l'..h'}) = (l = l' \<and> h = h' \<or> \<not> l \<le> h \<and> \<not> l' \<le> h')"
ask_nondetT_def: shows "ask = nondetM_base.ask_nondet ask"
set_rotate1: fixes xs :: "'c list" shows "set (rotate1 xs) = set xs"
Pfunc_Func_option: fixes A :: "'a set"   and B :: "'b set" shows "Pfunc A B = (\<Union>A'\<in>Pow A. Func_option A' B)"
TESL_interp_decreases_add_head: fixes \<Phi> :: "'b TESL_atomic list"   and \<Phi>' :: "'b TESL_atomic list"   and \<phi> :: "'b TESL_atomic" assumes "set \<Phi> \<subseteq> set \<Phi>'" shows "\<lbrakk>\<lbrakk> \<phi> # \<Phi>' \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<subseteq> \<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
simps: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and least :: "'a \<Rightarrow> bool"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "kbo.kbo w w0 scf least pr_strict pr_weak s t = (if vars_term_ms (scf_term scf t) \<subseteq># vars_term_ms (scf_term scf s) \<and> weight_fun.weight w w0 scf t \<le> weight_fun.weight w w0 scf s then if weight_fun.weight w w0 scf t < weight_fun.weight w w0 scf s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict (f, length ss) (g, length ts) then (True, True) else if pr_weak (f, length ss) (g, length ts) then lex_ext_unbounded (kbo.kbo w w0 scf least pr_strict pr_weak) ss ts else (False, False) else (False, False))"
Ssegment_Worder: fixes D :: "'c Order"   and a :: "'c" assumes "Worder D" shows "Worder (SIod D (Ssegment D a))"
inject_6: fixes x8 :: "'c ltl"   and y8 :: "'c ltl" shows "(G x8 = G y8) = (x8 = y8)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiattractive A (\<lambda>x y. less_eq y x)" shows "attractive_axioms A less_eq"
offs_enum_next_ge: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and i :: "nat"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and k :: "nat" assumes "index_less index key"   and "i < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "offs (enum xs index key n mi ma) k ! i \<le> offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i"
equivalence_of_partition: shows "partition.Equivalence rel.Partition = {(x, y). x \<sim> y}"
new1_finite: fixes \<psi> :: "'d ltlr" shows "finite (new1 \<psi>)"
monad_altc_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_altc return bind altc" shows "monad_altc (return_writer return) (bind_writer return bind) (altc_writer altc)"
keys_simps_1: shows "RBT_Impl.keys rbt.Empty = []"
comp_arr_dom: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and a :: "'d" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a" shows "C f a = f"
Carmichael_pos: fixes n :: "nat" shows "0 < Carmichael n"
IH: fixes y :: "nat" assumes "y < n" shows "finite (set_decode y)"
soundI2: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "sound P"
fimageE: fixes b :: "V"   and r :: "V"   and A :: "V" assumes "b \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A" obtains a :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" and "a \<in>\<^sub>\<circ> A"
mac_def: fixes \<eta> :: "nat"   and r :: "bool list"   and m :: "bool list" shows "mac \<eta> r m \<equiv> return_spmf r"
mahler_measure_poly_ge_1: fixes h :: "int poly" assumes "h \<noteq> 0" shows "1 \<le> mahler_measure h"
ant_tna: fixes x :: "ant" assumes "x \<noteq> \<infinity> \<and> x \<noteq> - \<infinity>" shows "ant (tna x) = x"
uJ_int: fixes n :: "nat" shows "integrable MJ (uJ n)"
octo_of_real_mult: fixes x :: "real"   and y :: "real" shows "octo_of_real (x * y) = octo_of_real x * octo_of_real y"
abs_zero: fixes abs :: "'i \<Rightarrow> 'i"   and plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and uminus :: "'i \<Rightarrow> 'i"   and less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
\<pi>01: shows "\<pi>\<^sub>1 0 = \<pi>\<^sub>1' 0"
addition_halt_lemma: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and lm :: "nat list" assumes "m \<noteq> n"   and "max m n < p"   and "p < length lm" shows "\<forall>na. \<not> (case abc_steps_l (0, lm) (addition m n p) na of (as, lm') \<Rightarrow> \<lambda>(m, p). as = 7) (m, p) \<and> addition_inv (abc_steps_l (0, lm) (addition m n p) na) m n p lm \<longrightarrow> addition_inv (abc_steps_l (0, lm) (addition m n p) (Suc na)) m n p lm \<and> ((abc_steps_l (0, lm) (addition m n p) (Suc na), m, p), abc_steps_l (0, lm) (addition m n p) na, m, p) \<in> addition_LE"
fun_upd_apply: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f"   and y :: "'e"   and z :: "'f" shows "(f(x := y)) z = (if z = x then y else f z)"
atomicity: fixes x :: "'a" assumes "x \<noteq> \<bottom>" shows "\<exists>y. atom y \<and> y \<le> x"
multiset_diff_union_assoc: fixes C :: "'a multiset"   and B :: "'a multiset"   and A :: "'a multiset" assumes "C \<subseteq># B" shows "A + B - C = A + (B - C)"
assms: shows "mopup_aft_erase_a (Suc (Suc (2 * n)), l, []) lm n ires"
epred_max: fixes x :: "enat"   and y :: "enat" shows "epred (max x y) = max (epred x) (epred y)"
diff_0_right: fixes a :: "'b" shows "a - (0::'b) = a"
OO_def: fixes R :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and S :: "'i \<Rightarrow> 'h \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
u_3: shows "u = k' \<otimes> a \<oplus> v'"
upper_pi_float_interval: fixes prec :: "nat" shows "upper (pi_float_interval prec) = ub_pi prec"
points_index_def: fixes B :: "'b set multiset"   and ps :: "'b set" shows "B index ps \<equiv> size (filter_mset ((\<subseteq>) ps) B)"
assms_1: shows "list.has_next (local.filter_generator g) s"
get_add_eq: fixes empty :: "'c"   and get :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and add :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and to_list :: "'c \<Rightarrow> ('d \<times> 'e) list"   and e :: "'c"   and x :: "'d"   and v :: "'e" assumes "env empty get add to_list" shows "get (add e x v) x = Some v"
cop_F_range_inv_def: fixes ds :: "'d set"   and fp :: "'x set" shows "cop_F_range_inv ds fp = (\<forall>x\<in>fp. x \<in> Field (Pd (Xd x)) \<and> Xd x \<in> ds)"
is_tm_dghmI': fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
semifin_chain_on_path: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set" assumes "[f[x ..]X]" shows "\<exists>P\<in>\<P>. X \<subseteq> P"
retraction_def: fixes g :: "'d" shows "Dop.retraction g \<equiv> \<exists>f. Dop.ide (Dop.comp g f)"
exhaust_sel: fixes status :: "status" obtains "status = FAILED"   | "status = SUCCESS"   | "status = FOUND"
word_size_word_size_bits: shows "word_size = 2 ^ word_size_bits"
pl_dvdm_imp_p_dvdm: fixes a :: "int poly"   and b :: "int poly" assumes "pl.dvdm a b" shows "a dvdm b"
concurrent_direct_execution_simulates_store_buffer_history_execution: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "safe_reach_direct safe_free_flowing (ts, m, \<S>)" shows "\<exists>ts' m' \<S>''. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>'') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>'')"
src_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "hseq \<mu> \<nu>" shows "src (\<mu> \<star> \<nu>) = src \<nu>"
matrix_diff_vect_distrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "(A - B) *v v = A *v v - B *v v"
nvalid_SucD: fixes n :: "nat"   and t :: "triple" assumes "\<Turnstile>Suc n: t" shows "\<Turnstile>n: t"
orthogonal_complement_subset: fixes A :: "'a vec set"   and B :: "'a vec set"   and n :: "nat" assumes "A \<subseteq> B" shows "vec_module.orthogonal_complement n B \<subseteq> vec_module.orthogonal_complement n A"
height_insrt: fixes t :: "'a tree\<^sub>0"   and x :: "'a" assumes "is_bal t" shows "height (insrt\<^sub>0 x t) = height t \<or> height (insrt\<^sub>0 x t) = height t + 1"
lfp_exp_bound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "bounded_by 1 (lfp_exp t)"
pop_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<plusminus> b = b \<plusminus> a"
matrix_left_right_inverse: fixes A :: "(('a, 'b) vec, 'b) vec"   and A' :: "(('a, 'b) vec, 'b) vec" shows "(A ** A' = mat (1::'a)) = (A' ** A = mat (1::'a))"
map_aform_err: fixes I :: "real \<times> real pdevs \<Rightarrow> ((real \<times> real pdevs) \<times> real) option"   and i :: "real \<Rightarrow> real"   and e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and x :: "real" assumes "\<And>e X Y. \<lbrakk>e \<in> UNIV \<rightarrow> {- 1..1}; I X = Some Y\<rbrakk> \<Longrightarrow> i (aform_val e X) \<in> aform_err e Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "map_aform_err I p X = Some Y"   and "x \<in> aform_err e X" shows "i x \<in> aform_err e Y"
assms_1: shows "H \<turnstile> HFun_Sigma r"
\<Gamma>_wf_2: fixes x :: "('fun, 'atom) Term.term \<times> nat" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<Gamma> (Var x))"
l1_refines_init_a0i_i: shows "init l1 \<subseteq> R01iai `` init a0i"
trace_nil: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and final :: "'b" shows "traces.trace step final [] final"
lm01: fixes A :: "'c set"   and B :: "'d set" assumes "trivial (A \<times> B)" shows "finite (A \<times> B) \<and> card A * card B \<le> 1"
LetTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and \<tau> :: "'a OCL_Types.type option"   and init :: "'a expr"   and body :: "'a expr"   and \<sigma> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E expr.Let v \<tau> init body : \<sigma>" obtains \<sigma>' :: "'a OCL_Types.type" and \<tau>' :: "'a OCL_Types.type" where "\<tau> = Some \<tau>'" and "\<Gamma> \<turnstile>\<^sub>E init : \<sigma>'" and "\<sigma>' \<le> \<tau>'" and "\<Gamma>(v \<mapsto>\<^sub>f \<tau>') \<turnstile>\<^sub>E body : \<sigma>"
isodefl_def: fixes d :: "'a \<rightarrow> 'a"   and t :: "udom defl" shows "isodefl d t = (cast\<cdot>t = EMB('a) oo d oo PRJ('a))"
mlt_in: fixes q :: "('t \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" assumes "q \<in> rel_dom" shows "mlt q \<in> lt ` {q g \<odot> g |g. g \<in> G \<and> q g \<odot> g \<noteq> 0}"
simps: fixes a1 :: "'b llist"   and a2 :: "'b llist" shows "lprefix a1 a2 = ((\<exists>xs. a1 = LNil \<and> a2 = xs) \<or> (\<exists>xs ys x. a1 = LCons x xs \<and> a2 = LCons x ys \<and> lprefix xs ys))"
existence_ivl_initial_time: fixes x0 :: "'c" assumes "0 \<in> UNIV"   and "x0 \<in> Csafe" shows "0 \<in> existence_ivl0 x0"
jointfun_hom: fixes f :: "nat \<Rightarrow> 'd"   and n :: "nat"   and A :: "'d set"   and g :: "nat \<Rightarrow> 'd"   and m :: "nat"   and B :: "'d set" assumes "f \<in> {i. i \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g \<in> {j. j \<le> Suc (n + m)} \<rightarrow> A \<union> B"
inj_on_image_mem_iff: fixes f :: "'g \<Rightarrow> 'h"   and B :: "'g set"   and a :: "'g"   and A :: "'g set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
a2c_def: shows "a2c \<equiv> \<Lambda> r. lfilter\<cdot>(\<Lambda> v. neg\<cdot>(SetMem\<cdot>v\<cdot>(exceptR\<cdot>r)))\<cdot>(resultR\<cdot>r)"
pgallery_least_length: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D" shows "SimplicialComplex.pmaxsimpchain X (C # (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D])) @ [D])"
valid_path_drop: fixes P :: "'a llist"   and n :: "nat" assumes "valid_path P" shows "valid_path (ldropn n P)"
assms: shows "(c1, s) \<rightarrow>*c (c1', s')"
USdBtrans: fixes d :: "'d" shows "trans \<approx>\<^bsub>d\<^esub>"
linear_sum: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and S :: "'c set" assumes "clinear f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
inf_commute: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = inf y x"