hcju/isabelleps huggingface dataset:
lock_actions_ok_append: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and Ls :: "lock_action list"   and Ls' :: "lock_action list" shows "lock_actions_ok l t (Ls @ Ls') = (lock_actions_ok l t Ls \<and> lock_actions_ok (upd_locks l t Ls) t Ls')"
theFTOne_sym: fixes cl :: "('g, 'h, 'i) cmd list"   and dl :: "('j, 'k, 'l) cmd list" shows "theFTOne cl dl = theFTOne dl cl"
icdk: shows "i' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k'"
simps: fixes Gcd :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"   and bn :: "'a poly"   and cn :: "'a poly"   and i :: "nat"   and sqr :: "('a poly \<times> nat) list" shows "yun_gcd.yun_factorization_main Gcd bn cn i sqr = (if bn = 1 then sqr else let dn = cn - pderiv bn; an = Gcd bn dn in yun_gcd.yun_factorization_main Gcd (bn div an) (dn div an) (Suc i) ((an, i) # sqr))"
chosenAllocationEquivalence: fixes N :: "'d set"   and \<Omega> :: "'e list"   and b :: "'d \<times> 'e set \<Rightarrow> 'f"   and r :: "integer" assumes "0 < card N"   and "distinct \<Omega>" shows "randomEl (takeAll (\<lambda>x. winningAllocationRel N (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg N \<Omega>)) r = chosenAllocationAlg N \<Omega> b r"
phl_skip: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod x t_one \<subseteq> t_prod t_one x"
r_def: shows "r = (\<lambda>n. r0 (n + N))"
upper_triangular_upt_imp_upper_triangular: fixes A :: "(('d, 'e) vec, 'e) vec" assumes "upper_triangular_upt_k A (nrows A)" shows "upper_triangular A"
assms_5: shows "rk {A, B, P} \<le> 2"
eq_id_iff: fixes f :: "'a \<Rightarrow> 'a" shows "(\<forall>x. f x = x) = (f = id)"
no_taut: shows "\<nexists>A. Pos A \<in># C \<and> Neg A \<in># C"
fiter0: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_iteration.fiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<sqsubseteq> nil"
emb_ssum_def: shows "EMB('e \<oplus> 'f) = ssum_emb oo ssum_map\<cdot>EMB('e)\<cdot>EMB('f)"
simps_2: fixes a :: "Arity"   and x :: "Vars.var"   and e :: "exp" shows "ccTransform a (Lam [x]. e) = Lam [x]. ccTransform (pred\<cdot>a) e"
hoare_partial_complete: fixes dims :: "nat list"   and P :: "complex mat"   and S :: "com"   and Q :: "complex mat" assumes "state_sig.hoare_partial_correct dims P S Q"   and "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P"   and "state_sig.is_quantum_predicate dims Q" shows "state_sig.hoare_partial dims P S Q"
record_simps_25: fixes f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "defined_types (update_defined_types f x) = f (defined_types x)"
triangulize_preserves_dimensions_1: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "triangulize sel_fun mf A = (r', A')" shows "dim_row A' = dim_row A"
up_induct: fixes P :: "'a\<^sub>\<bottom> \<Rightarrow> bool"   and x :: "'a\<^sub>\<bottom>" assumes "P \<bottom>"   and "\<And>x. P (up\<cdot>x)" shows "P x"
pell_power_def: fixes z :: "'b \<times> 'b"   and n :: "nat" shows "pell_power z n = ((\<lambda>z'. pell_mul z' z) ^^ n) (1::'b, 0::'b)"
compat_compX: shows "compatible_modes (restrict_modes mdss (- X))"
simps: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat"   and j :: "nat" shows "gram_schmidt_fs.\<mu> n fs i j = (if j < i then fs ! i \<bullet> gram_schmidt_fs.gso n fs j / \<parallel>gram_schmidt_fs.gso n fs j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
is_NegChecks_def: fixes stateful_strand_step :: "('c, 'd) stateful_strand_step" shows "is_NegChecks stateful_strand_step = (\<exists>x71 x72 x73. stateful_strand_step = \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>)"
select_convs_1: fixes worlds :: "'e set"   and relations :: "'f \<Rightarrow> ('e \<times> 'e) set"   and valuation :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and more :: "'h" shows "worlds \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = worlds"
eq: shows "pdevs_val f (pdevs_of_list xs) = (\<Sum>i<degree (pdevs_of_list xs). f i * r i) *\<^sub>R sum_list xs"
up_injective: fixes x :: "'a"   and y :: "'a" assumes "\<up>x = \<up>y" shows "x = y"
sse_invar_def: fixes \<Sigma>i :: "'d set"   and R :: "('d \<times> 'd) set" shows "sse_invar \<Sigma>i R = {(\<Sigma>, W). W \<subseteq> \<Sigma> \<and> \<Sigma> \<subseteq> R\<^sup>* `` \<Sigma>i \<and> (\<forall>\<sigma>\<in>R\<^sup>* `` \<Sigma>i - \<Sigma>. \<exists>\<sigma>h\<in>W. (\<sigma>h, \<sigma>) \<in> (R - UNIV \<times> \<Sigma>)\<^sup>*)}"
assms: shows "evaluate_list ck env s [e] (s', Rval vs)"
elims: fixes x :: "'a list"   and xa :: "'a list"   and xb :: "(nat \<times> nat list) list"   and y :: "nat" assumes "T\<^sub>p x xa xb = y" obtains s :: "'a list" where "x = s__" and "xa = []" and "xb = []" and "y = 0"   | s :: "'a list" and r :: "'a" and rs :: "'a list" and a :: "nat \<times> nat list" and as :: "(nat \<times> nat list) list" where "x = s" and "xa = r # rs" and "xb = a__ # as" and "y = t\<^sub>p s r a__ + T\<^sub>p (step s r a__) rs as"   | a :: "'a list" and v :: "'a" and va :: "'a list" where "x = aa__" and "xa = v__ # va__" and "xb = []" and "y = undefined"   | a :: "'a list" and v :: "nat \<times> nat list" and va :: "(nat \<times> nat list) list" where "x = a" and "xa = []" and "xb = v # va" and "y = undefined"
vec_to_lpoly_valuate_equiv_dot_prod: fixes y :: "rat vec"   and x :: "rat vec" assumes "dim_vec y = dim_vec x" shows "vec_to_lpoly y \<lbrace> ($) x \<rbrace> = y \<bullet> x"
zero_compare_simps_9: fixes a :: "'a"   and b :: "'a" shows "(a / b \<le> (0::'a)) = ((0::'a) \<le> a \<and> b \<le> (0::'a) \<or> a \<le> (0::'a) \<and> (0::'a) \<le> b)"
oInv_bound2: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "normal F" shows "a < F (oSuc (oInv F a))"
i_set0_ind_i_set: fixes I :: "nat set" assumes "I \<in> i_set_ind" shows "I \<in> i_set0_ind"
fromEnumAlt_def: fixes x :: "'b" shows "fromEnumAlt x \<equiv> THE n. enum_alt n = Some x"
simps_1: shows "nat_to_sch 0 = Base_zero"
cfE_of_finite: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "finite (cfE_of f)"
simps_20: fixes x1 :: "'a"   and x7 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<r>\<^bold>[x7\<^bold>]"
WT_restD_rinit: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and rest :: "('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme" assumes "WT_rest \<I>_adv \<I>_usr I rest" shows "I (rinit rest)"
preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>D (F \<mu>) = map\<^sub>0 (src\<^sub>C \<mu>)"
fa: shows "a = \<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr>"
preserves_iso: fixes f :: "'c" assumes "C.iso f" shows "C.iso (C.map f)"
iter_unfold: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> = nil \<sqinter> c ; c\<^sup>\<omega>"
sorted_rev_iff_nth_mono: fixes xs :: "'a list list" shows "linorder.sorted (\<le>lex) (rev xs) = (\<forall>i j. i \<le> j \<longrightarrow> j < \<^bold>|xs\<^bold>| \<longrightarrow> xs ! j \<le>lex xs ! i)"
apath_nonempty_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "apath u p v"   and "p \<noteq> []" shows "u \<noteq> v"
6: shows "coreflexive ((f \<sqinter> q)\<^sup>T * e\<^sup>T)"
first_edge_hd_tl: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w"   and "tl xs \<noteq> []" shows "first_edge_of_walk xs = (v, hd (tl xs))"
monom_mult_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and t :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair c t p = Abs_poly_mapping (\<lambda>v. if term_powerprod.adds_pp pair_of_term t v then c * lookup p (term_powerprod.sminus pair_of_term term_of_pair v t) else (0::'d))"
proves_eq_intr_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT" shows "\<Theta>,{} \<turnstile> (A \<longmapsto> B) \<longmapsto> (B \<longmapsto> A) \<longmapsto> mk_eq A B"
parZeroRight: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "\<And>Q. (Q, Q \<parallel> \<zero>) \<in> Rel" shows "P \<leadsto>[Rel] P \<parallel> \<zero>"
divide_out_aux_correct: fixes p :: "'a"   and z :: "'a \<times> nat" shows "divide_out_aux p z = (fst z div p ^ multiplicity p (fst z), snd z + multiplicity p (fst z))"
lim_imp_Liminf: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
ope3: shows "openin (top_of_set (U - C)) (U - C - H3)"
get_component_parent_inside: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and node_ptr :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> set c"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r Some parent" shows "parent \<in> set c"
I\<^sub>c_def: shows "I\<^sub>c \<equiv> {(a\<^sub>c, a\<^sub>c), (b\<^sub>c, b\<^sub>c), (b\<^sub>c, c\<^sub>c), (c\<^sub>c, c\<^sub>c), (c\<^sub>c, a\<^sub>c)}"
incl: shows "A - (\<Union>i. (T ^^ i) --` B) \<in> null_sets M"
sign_simps_38: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "am + (bm - ck) = am + bm - ck"
finfun_rec_upd: fixes g :: "'c \<Rightarrow> 'b"   and f :: "('a, 'c) finfun"   and a' :: "'a"   and b' :: "'c" shows "finfun_rec (\<lambda>b. finfun_const (g b)) (\<lambda>a b c. finfun_update c a (g b)) (finfun_update f a' b') = finfun_update (finfun_rec (\<lambda>b. finfun_const (g b)) (\<lambda>a b c. finfun_update c a (g b)) f) a' (g b')"
sum_def: shows "sum \<equiv> comm_monoid_set.F (+) (0::'d)"
rbl_mult: fixes n :: "nat"   and bina :: "int"   and binb :: "int" shows "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = rev (bin_to_bl n (bina * binb))"
clock_jviewIncr_def: shows "clock_jviewIncr \<equiv> \<lambda>a obs' (l, obs). (l + 1, obs')"
af_letter_unfolding: fixes \<phi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (Unf \<phi>) \<nu> \<sim>\<^sub>P af_letter \<phi> \<nu>"
fs'': shows "fs[i := fs ! (i - 1), i - 1 := fs ! i] = fs''"
1: shows "DisjointPathsPlusOne G v0 v1 paths (new_pre @ [x])"
ord_le_eq_subst: fixes a :: "'a::ord"   and b :: "'a::ord"   and f :: "'a::ord \<Rightarrow> 'c::ord"   and c :: "'c::ord" assumes "(a::'a::ord) \<le> (b::'a::ord)"   and "(f::'a::ord \<Rightarrow> 'c::ord) (b::'a::ord) = (c::'c::ord)"   and "\<And>(x::'a::ord) y::'a::ord. x \<le> y \<Longrightarrow> (f::'a::ord \<Rightarrow> 'c::ord) x \<le> f y" shows "(f::'a::ord \<Rightarrow> 'c::ord) (a::'a::ord) \<le> (c::'c::ord)"
assoc_props_6: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<cdot>) prj0 prj1 f g h, Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr> : f \<star> g \<star> h \<Rightarrow> (f \<star> g) \<star> h\<guillemotright>"
row_length_mat: fixes m1 :: "int poly fract list list"   and m2 :: "int poly fract list list" shows "rat_poly.row_length (m1 \<otimes> m2) = rat_poly.row_length m1 *n rat_poly.row_length m2"
maxsimp_vertices: fixes x :: "'b set" assumes "codomain.chamber x" shows "order.greater_eq (\<Union> Y) x"
hull_minimal: fixes s :: "'c set"   and t :: "'c set"   and S :: "'c set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t" shows "S hull s \<subseteq> t"
homogeneous_times: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "homogeneous p"   and "homogeneous q" shows "homogeneous (p * q)"
ivl_disj_un_one_3: fixes l :: "'c"   and u :: "'c" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
fminus_cancel: fixes A :: "'c fset" shows "A |-| A = {||}"
of_complex_one: shows "of_complex 1 = 1\<^sub>h"
sort_oalist_id: fixes x :: "'o"   and xs :: "('a \<times> 'e) list" assumes "oalist_inv_raw x xs" shows "sort_oalist x xs = xs"
borel_eq_halfspace_ge: shows "borel = sigma UNIV ((\<lambda>(a, i). {x. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
intros_10: fixes s :: "'a"   and b :: "'a set"   and c :: "'a com"   and s'' :: "'a option"   and x :: "'a set"   and y :: "'a \<Rightarrow> nat"   and s' :: "'a option" assumes "s \<in> b"   and "Sem c (Some s) s''"   and "Sem (While b x y c) s'' s'" shows "Sem (While b x y c) (Some s) s'"
PTrans: shows "P \<longmapsto>\<^sub>l a<y> \<prec>\<^sub>l P''"
Lambert_W_mono: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "x \<le> y" shows "Lambert_W x \<le> Lambert_W y"
integral_pmf_of_multiset: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> real" assumes "A \<noteq> {#}" shows "measure_pmf.expectation (pmf_of_multiset A) f = (\<Sum>x\<in>set_mset A. real (count A x) * f x) / real (size A)"
presSubstAll_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSubstAll h hA (fromMOD MOD)" shows "presSubstAll (fromIMor h) (fromIMorAbs hA) MOD"
assms: shows "\<forall>x\<in>MPLS_s PROB s. x < k"
opt_less_simps_2: fixes x :: "int option" shows "opt_less x None = True"
less_bot_Bot_is_Value: fixes x :: "'a bot" assumes "Bot < x" shows "\<exists>z. x = Value z"
sup_same_context: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y \<squnion> z"   and "y \<le> x \<squnion> z" shows "x \<squnion> z = y \<squnion> z"
deadlockE: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "deadlock s" obtains "\<forall>t x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> \<not> final x \<and> wset s t = None \<longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s))))" and "\<forall>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<exists>l. 0 < ln $ l) \<and> \<not> waiting (wset s t) \<longrightarrow> (\<exists>l t'. 0 < ln $ l \<and> t \<noteq> t' \<and> thr s t' \<noteq> None \<and> has_lock (locks s $ l) t')" and "\<forall>t x w. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> wset s t \<noteq> \<lfloor>PostWS w\<rfloor>"
expected_utility_weak_preference: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "p \<succeq> q = ((\<Sum>z\<in>outcomes. pmf q z * u z) \<le> (\<Sum>z\<in>outcomes. pmf p z * u z))"
parallel_wiring_inverse: fixes \<I>1 :: "('y, 'ac) \<I>"   and \<I>2 :: "('z, 'ad) \<I>"   and \<I>3 :: "('aa, 'ae) \<I>"   and \<I>4 :: "('ab, 'af) \<I>" shows "(\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<oplus>\<^sub>\<I> (\<I>3 \<oplus>\<^sub>\<I> \<I>4),(\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<oplus>\<^sub>\<I> (\<I>3 \<oplus>\<^sub>\<I> \<I>4) \<turnstile>\<^sub>C parallel_wiring \<odot> parallel_wiring \<sim> 1\<^sub>C"
no_matching_Goto_head: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list" assumes "no_matching_Goto \<gamma> p (r # rs)" shows "no_matching_Goto \<gamma> p [r]"
fps_compose_def: fixes a :: "'a fps"   and b :: "'a fps" shows "a oo b = Abs_fps (\<lambda>n. \<Sum>i = 0..n. fps_nth a i * fps_nth (b ^ i) n)"
substitute_mono1: fixes f :: "'b \<Rightarrow> 'b ttree"   and f' :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" assumes "\<And>x. paths (f x) \<subseteq> paths (f' x)" shows "paths (substitute f T t) \<subseteq> paths (substitute f' T t)"
1: shows "is_derivation (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> derives (item_\<alpha> x) (terminals (drop u p))"
bi_unique_HMA_M_3: shows "right_unique HMA_M"
Set_union_code_9: fixes xs :: "'b list"   and ys :: "'b list" shows "Set_Monad xs \<union> Set_Monad ys = Set_Monad (xs @ ys)"
sT: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and P :: "'s \<Rightarrow> real" assumes "\<And>Q. sound Q \<Longrightarrow> sound (t Q)"   and "sound P" shows "sound (T t P)"
inorder_delete: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (RBT_Map.delete x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
take_Suc_Max_eq_imp_f_join_eq: fixes I :: "nat set"   and xs :: "'b list"   and ys :: "'b list" assumes "finite I"   and "xs \<down> Suc (Max I) = ys \<down> Suc (Max I)" shows "xs \<Join>\<^sub> I = ys \<Join>\<^sub> I"
inj_image_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj f" shows "(f ` A = f ` B) = (A = B)"
rcos_equation: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and ha :: "'c"   and a :: "'c"   and h :: "'c"   and b :: "'c"   and hb :: "'c" assumes "Group.group G"   and "subgroup H G"   and "ha \<otimes>\<^bsub>G\<^esub> a = h \<otimes>\<^bsub>G\<^esub> b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "h \<in> H"   and "ha \<in> H"   and "hb \<in> H" shows "hb \<otimes>\<^bsub>G\<^esub> a \<in> (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> b})"
factor_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" shows "a divides\<^bsub>G\<^esub> b = (\<exists>c\<in>carrier G. b = a \<otimes>\<^bsub>G\<^esub> c)"
backflows_minus_backflows: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "backflows (X - backflows Y) = backflows X - Y"
prime_ideal_cont1Tr: fixes A :: "'a set"   and n :: "nat" assumes "ideal R A" shows "\<forall>P. P \<in> {j. j \<le> n} \<rightarrow> {X. prime_ideal R X} \<and> A \<subseteq> \<Union> (P ` {j. j \<le> n}) \<longrightarrow> (\<exists>i\<le>n. A \<subseteq> P i)"
cscc_invar_ext_axioms: shows "cscc_invar_ext G l D"
m_closed: fixes x :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and y :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "x \<in> carrier (poly_ring M)"   and "y \<in> carrier (poly_ring M)" shows "x \<otimes>\<^bsub>poly_ring M\<^esub> y \<in> carrier (poly_ring M)"
vsubstAbs_preserves_freshAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "freshAbs zs z A"   and "zs \<noteq> ys \<or> z \<noteq> y1" shows "freshAbs zs z (A $[y1 // y]_ys)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "'b" assumes "token_run x xa = y" obtains x :: "nat" and n :: "nat" where "xa__ = x" and "xa = n" and "y = run \<delta> q\<^sub>0 (suffix x w) (n - x)"
preserves_iso: fixes f :: "'c \<times> 'c" assumes "C.CC.iso f" shows "C.CC.iso (C.CC.map f)"
assms_3: fixes \<xi> :: "'a"   and p :: "'b"   and a :: "'c"   and \<xi>' :: "'a"   and p' :: "'b" assumes "(\<xi>, p) \<in> reachable A I"   and "P \<xi> p"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "I a" shows "P \<xi>' p'"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (Ge (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
bool_impl_test_def: shows "bool_impl_test \<equiv> impl_test ifex_of"
user_accessible_annul: fixes new_cpu_reg :: "CPU_register \<Rightarrow> word32"   and new_state_var :: "sparc_state_var"   and s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible (s\<lparr>state_var := new_state_var, cpu_reg := new_cpu_reg\<rparr>) pa = user_accessible s pa"
finsum_rdistr: fixes A :: "'e set"   and a :: "'c"   and f :: "'e \<Rightarrow> 'c" assumes "finite A"   and "a \<in> carrier S"   and "f \<in> A \<rightarrow> carrier S" shows "a \<otimes>\<^bsub>S\<^esub> finsum S f A = (\<Oplus>\<^bsub>S\<^esub>i\<in>A. a \<otimes>\<^bsub>S\<^esub> f i)"
index_nth_id: fixes xs :: "'a list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "index xs (xs ! n) = n"