hcju/isabelleps huggingface dataset:
Category_axioms_def: fixes C :: "('g, 'h, 'i) Category_scheme" shows "Category_axioms C \<equiv> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>X. X \<in> obj\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X)) \<and> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f)) \<and> (\<forall>f g h. f \<approx>>\<^bsub>C\<^esub> g \<longrightarrow> g \<approx>>\<^bsub>C\<^esub> h \<longrightarrow> f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)) \<and> (\<forall>f X Y g Z. f maps\<^bsub>C\<^esub> X to Y \<longrightarrow> g maps\<^bsub>C\<^esub> Y to Z \<longrightarrow> f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z)"
vals_add_distinct: fixes xs :: "('d, 'c) bintree option list"   and x :: "('d, 'c) bintree option" assumes "distinct (vals xs)"   and "distinct (vals [x])"   and "set (vals xs) \<inter> set (vals [x]) = {}" shows "distinct (vals (add x xs))"
sorted_sorted_domain: fixes ko :: "'b nat_term_order"   and xs :: "('b, 'c) oalist_ntm" shows "sorted_wrt (ko.lt (key_order_of_nat_term_order_inv ko)) (OAlist_sorted_domain_ntm ko xs)"
hm_delete_rule': fixes m :: "'a \<Rightarrow> 'b option"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and k :: "'a" shows "<is_hashmap' m l ht> hm_delete k ht <is_hashmap (m |` (- {k}))>"
rd_star: fixes p :: "assertion"   and q :: "assertion" shows "rd_ass (p \<star> q) = rd_ass p \<union> rd_ass q"
mult_1_right: fixes a :: "'b" shows "a + (0::'b) = a"
monom_inv_ConsD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c \<times> nat"   and xs :: "('c \<times> nat) list" assumes "class.linorder less_eq less"   and "linorder.monom_inv less_eq (x # xs)" shows "linorder.monom_inv less_eq xs"
adopt_node_is_component_unsafe: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> adopt_node document_ptr node_ptr \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 document_ptr)|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (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)|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 |h \<turnstile> get_owner_document (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)|\<^sub>r)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
assms: assumes "deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))" shows "to1 u v = to2 u v"
submodule_def: fixes H :: "'b set"   and R :: "('c, 'd) ring_scheme"   and M :: "('c, 'b, 'e) module_scheme" shows "Module.submodule H R M \<equiv> subgroup H (add_monoid M) \<and> submodule_axioms H R M"
jb: shows "j_to_jb_index n_as i = (bi, li)"
oap: shows "order a p = Suc (order a (pderiv p))"
cont_g: shows "continuous (at (f a) within f ` S) g"
complex_roots_of_rat_poly_def: fixes p :: "rat poly" shows "complex_roots_of_rat_poly p = complex_roots_of_int_poly (snd (rat_to_int_poly p))"
even_odd_mirror_pure: fixes even :: "bool"   and x :: "'a" shows "even_odd_mirror even (pure x) = pure x"
ft: shows "(\<lambda>x. f (x + t)) absolutely_integrable_on {- pi..pi}"
real_atLeastGreaterThan_eq: fixes N :: "ereal"   and M :: "ereal" shows "real_of_ereal ` {N<..<M} = (if N = \<infinity> then {} else if N = - \<infinity> then if M = \<infinity> then UNIV else if M = - \<infinity> then {} else {..<real_of_ereal M} else if M = - \<infinity> then {} else if M = \<infinity> then {real_of_ereal N<..} else {real_of_ereal N<..<real_of_ereal M})"
return_sets_cong: fixes M :: "'a measure"   and N :: "'a measure" assumes "sets M = sets N" shows "return M = return N"
slice2_is_SDG_path_slice2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n'' \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n" shows "n'' \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'"
finite_subset_Liminf_llist_imp_exists_index: fixes Xs :: "'a set llist"   and X :: "'a set" assumes "\<not> lnull Xs"   and "finite X"   and "X \<subseteq> Liminf_llist Xs" shows "\<exists>i. enat i < llength Xs \<and> X \<subseteq> \<Inter> (lnth Xs ` {j. i \<le> j \<and> enat j < llength Xs})"
int_pow_mult: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier R" shows "[(i + j)] \<cdot> x = [i] \<cdot> x \<oplus> [j] \<cdot> x"
wf_pnorm: fixes wf_atom :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and r :: "'a rexp" assumes "alphabet.wf wf_atom n r" shows "alphabet.wf wf_atom n (pnorm r)"
2: shows "App (apps (Hd (head t)) (butlast (args t))) (last (args t)) = t"
lookup_times: fixes p :: "'c \<Rightarrow>\<^sub>0 'b"   and q :: "'c \<Rightarrow>\<^sub>0 'b"   and u :: "'c" shows "lookup (p * q) u = (\<Sum>t\<in>keys p. lookup p t * (\<Sum>v\<in>keys q. lookup q v when u = t + v))"
agents_eq: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "agents = {i. R i \<noteq> (\<lambda>_ _. False)}"
the_apartment_isoD_1: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and A' :: "'b set set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "A' \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "C \<in> A \<inter> A'" shows "ChamberComplexIsomorphism A A' (ChamberComplexWithApartmentSystem.the_apartment_iso A A')"
distinct_5: fixes x2 :: "bool"   and x31 :: "'a"   and x32 :: "'b" shows "<\<bottom>, outM x2> \<noteq> <x31, x32>"
shufflesE: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs \<in> shuffles xs ys" obtains "zs = xs" and "ys = []"   | "zs = ys" and "xs = []"   | x :: "'a" and xs' :: "'a list" and z :: "'a" and zs' :: "'a list" where "xs = x # xs'" and "zs = z__ # zs'__" and "x = z__" and "zs'__ \<in> shuffles xs' ys"   | y :: "'a" and ys' :: "'a list" and z :: "'a" and zs' :: "'a list" where "ys = y # ys'" and "zs = z # zs'" and "y = z" and "zs' \<in> shuffles xs ys'"
measurable_pmf_measure1: fixes M :: "'a pmf"   and N :: "'b measure" shows "measure_pmf M \<rightarrow>\<^sub>M N = UNIV \<rightarrow> space N"
join_equality: fixes x :: "'c"   and y :: "'c"   and sup :: "'c" assumes "is_sup x y sup" shows "x \<squnion> y = sup"
phis_addN_def: fixes g :: "'g"   and n :: "'node \<times> 'val" shows "phis_addN g n = fold (\<lambda>v. Mapping.map_default v {} (insert n)) (case phis g n of None \<Rightarrow> [] | Some x \<Rightarrow> id x)"
sum_list_nlex_eq_sum_list_inl: fixes X :: "(real \<times> real) pdevs" shows "sum_list (map snd (list_of_pdevs (nlex_pdevs X))) = sum_list (inl X)"
index_lfp: fixes f :: "'a set acom \<Rightarrow> 'a set acom"   and i :: "com" shows "Collecting.lfp f i \<in> {c'. strip c' = i}"
polymul_properties_2: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
HInit_Inv2c: fixes s :: "state" shows "HInit s \<longrightarrow> Inv2c s"
frag_cmul_def: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * poly_mapping.lookup a x)"
sorted_list_of_A: fixes t :: "'b tree" assumes "heap t" shows "sorted (list_of_A t)"
af_append_false: fixes \<phi> :: "'a ltln"   and w :: "'a set list"   and w' :: "'a set list" assumes "af \<phi> w \<sim> false\<^sub>n" shows "af \<phi> (w @ w') \<sim> false\<^sub>n"
wf_dbfm_trans_fm: fixes A :: "fm" shows "wf_dbfm (trans_fm [] A)"
neg_less_divide_eq: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less a (divide b c) = less b (times a c)"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<sim> P \<parallel> Q \<parallel> R"
cccFix_choose_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "cccFix_choose \<Gamma> = (if nonrec \<Gamma> then case hd \<Gamma> of (x, xa) \<Rightarrow> cccFix_nonrec x xa else cccFix \<Gamma>)"
read_only_reads_unowned_tl: fixes t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "read_only_reads_unowned (t # ts)" shows "read_only_reads_unowned ts"
list_dnf_to_dnf_def: fixes X :: "'b list set" shows "list_dnf_to_dnf X = fset_of_list ` X"
no_mem: fixes x :: "'a" assumes "\<And>a i. (a, i) \<in> set bs \<Longrightarrow> poly a x \<noteq> (0::'a)" shows "order x p = 0"
at_least_as_goodD: fixes z :: "'b"   and y :: "'b"   and B :: "'b set"   and Pr :: "('b \<times> 'b) set" assumes "z \<in> at_least_as_good y B Pr" shows "z \<succeq>[Pr] y"
diagonal_step_PQ_invertible_Q: fixes P :: "(('a, 'b) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and p :: "'a"   and q :: "'a"   and u :: "'a"   and v :: "'a"   and d :: "'a" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)"   and "i \<noteq> k"   and "i < min (nrows A) (ncols A)"   and "k < min (nrows A) (ncols A)" shows "invertible Q"
get_scdom_component_subset_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "set c \<subseteq> set sc"
isomorphic_def: fixes a :: "'d"   and a' :: "'d" shows "D'.isomorphic a a' = (\<exists>f. D'.in_hom f a a' \<and> D'.iso f)"
seql_def: fixes i :: "'g"   and P :: "'h \<times> 'i \<Rightarrow> bool" shows "seql i P \<equiv> \<lambda>(\<sigma>, p). P (\<sigma> i, p)"
phiArg_distinct_nodes: fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q"   and "p \<noteq> q"   and "p \<in> allVars g" shows "defNode g p \<noteq> defNode g q"
dghm_inc_ArrMap_app: fixes x :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "x \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "dghm_inc \<BB> \<CC>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = x"
coprime_n_idealsTr4: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "(\<forall>k\<le>Suc n. ideal R (J k)) \<and> (\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. i \<noteq> j \<longrightarrow> coprime_ideals R (J i) (J j))" shows "coprime_ideals R (i\<Pi>\<^bsub>R,n\<^esub> J) (J (Suc n))"
PhiWhilePOp_def: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhilePOp A b \<Phi> = (\<lambda>\<Psi> (r, u). (evalB b u \<longrightarrow> (\<exists>z. \<Phi> (z, u) \<and> (\<forall>w. A z w \<longrightarrow> \<Psi> (r, w)))) \<and> (\<not> evalB b u \<longrightarrow> A r u))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<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 known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_known_ptrs known_ptr known_ptrs"   and "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_child_nodes_locs get_parent get_parent_locs" shows "l_get_parent\<^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 known_ptrs get_parent get_parent_locs"
rel_Class_iff: fixes x :: "'c set \<times> 'd"   and y :: "'c set \<times> 'd" shows "stfx.rel x y = (y \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y \<and> x \<in> eqY.Class y)"
simulation_axioms: shows "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R"
rep_number_on_set_def: fixes B :: "'a set set"   and x :: "'a" assumes "finite B" shows "mset_set B rep x = int (card {b \<in> B. x \<in> b})"
subst_empty: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "subst t fmempty = t"
F_1: fixes n :: "nat" shows "uf n = eNorm N1 (f1 n) + eNorm N2 (f2 n)"
inj: shows "inj_on (\<lambda>b. (y + x * b) mod q) {..<q}"
strongdB_aux: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set"   and i :: "nat"   and V :: "'com list"   and V' :: "'com list"   and m1 :: "'id \<Rightarrow> 'val"   and W :: "'com list"   and m2 :: "'id \<Rightarrow> 'val"   and m1' :: "'id \<Rightarrow> 'val" assumes "Strong_d_Bisimulation d R"   and "i < length V"   and "(V, V') \<in> R"   and "\<langle>V ! i,m1\<rangle> \<rightarrow> \<langle>W,m2\<rangle>"   and "m1 =\<^bsub>d\<^esub> m1'" shows "\<exists>W' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow> \<langle>W',m2'\<rangle> \<and> (W, W') \<in> R \<and> m2 =\<^bsub>d\<^esub> m2'"
intermediate_point_theorem: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a" assumes "point p"   and "point r" shows "(p \<le> x ; y ; r) = (\<exists>q. point q \<and> p \<le> x ; q \<and> q \<le> y ; r)"
disjE: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "sb.eval cid t t' n (\<gamma> \<or>\<^sup>b \<gamma>')" shows "sb.eval cid t t' n \<gamma> \<or> sb.eval cid t t' n \<gamma>'"
maximal_def: fixes S :: "('c, 'd) fm set" shows "maximal S \<equiv> \<forall>p. p \<notin> S \<longrightarrow> \<not> consistent ({p} \<union> S)"
FOREACHc_filter_deforestation: fixes S :: "'c set"   and P :: "'c \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and s :: "'d" assumes "finite S" shows "FOREACH\<^sub>C {x \<in> S. P x} c f s = FOREACH\<^sub>C S c (\<lambda>x s. if P x then f x s else RETURN s) s"
fls_subdegree_fls_X_intpow: fixes i :: "int" shows "fls_subdegree (fls_X_intpow i) = i"
concurrentD1: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'c"   and s2 :: "'c" assumes "happens_before hb_weak hb"   and "happens_before.concurrent hb s1 s2" shows "\<not> hb s1 s2"
rep_eq: fixes x :: "natural"   and xa :: "natural" shows "nat_of_natural (x - xa) = nat_of_natural x - nat_of_natural xa"
safe_max_Some_intro: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "\<exists>y. safe_max X = Some y"
div_mult_self_is_m: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "m * n div n = m"
set_to_ns: shows "set (to_ns cs) = {uu_. \<exists>i n c. uu_ = (i, n) \<and> (i, c) \<in> set cs \<and> n \<in> set (constraint_to_qdelta_constraint c)}"
update_induct: fixes P :: "Store \<Rightarrow> bool"   and s :: "Store"   and l :: "Location"   and v :: "Value" assumes "P s"   and "\<lbrakk>alive (ref l) s; alive v s; \<tau> v \<le> ltype l\<rbrakk> \<Longrightarrow> P (Abs_Store (Rep_Store s\<lparr>vals := (vals (Rep_Store s))(l := v)\<rparr>))" shows "P (s\<langle>l := v\<rangle>)"
Gt_lt_conv: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" assumes "comparator comp" shows "(comp x y = Gt) = lt_of_comp comp y x"
rel_dds_Grp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'e" shows "rel_dds (Grp UNIV f)\<inverse>\<inverse> (Grp UNIV g) = Grp UNIV (map_dds f g)"
rel_restrict_sub: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" shows "rel_restrict R A \<subseteq> R"
index_mat_swapcols_3: fixes k :: "nat"   and l :: "nat"   and A :: "'d mat" shows "dim_col (swapcols k l A) = dim_col A"
QXcarr: shows "Q \<union> X \<subseteq> carrier_vec n"
eucl_truncate_up_eq_eucl_truncate_down: fixes p :: "nat"   and x :: "'a" shows "eucl_truncate_up p x = - eucl_truncate_down p (- x)"
simps_9: fixes x3 :: "error"   and x2 :: "jump" shows "Error x3 \<noteq> Jump x2"
False: shows "\<not> (p = 0 \<or> q = 0 \<or> poly p x \<noteq> 0)"
a_kernel_def: fixes G :: "('a, 'b) ring_scheme"   and H :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"
D\<^sub>i\<^sub>v: fixes x :: "i" shows "x \<cdot> (dom x) \<cong> x"
induct: fixes x1 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>b {x1} x2 { x3}"   and "\<And>Pa c Q. \<turnstile>\<^sub>3\<^sub>a {Pa} c { Q} \<Longrightarrow> P Pa c Q"   and "\<And>Pa c Q P' k Q'. \<lbrakk>\<turnstile>\<^sub>3\<^sub>b {Pa} c { Q}; P Pa c Q; \<And>s n. P' (s, n) \<Longrightarrow> Pa (s, k * n); \<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n div k); 0 < k\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
bisim1ThrowNull: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,throw e,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
S2_and_def: fixes B :: "bool \<times> bool"   and s2 :: "bool" shows "S2_and B s2 = (let (a2, b2) = B in S2_14_OT (a2, b2) s2 \<bind> (\<lambda>V. return_spmf (B, V)))"
Not_eqvt: fixes p :: "perm"   and x :: "('b, 'c, 'd) formula" shows "p \<bullet> Formula.Not x = Formula.Not (p \<bullet> x)"
count_of_def: fixes xs :: "('b \<times> nat) list"   and x :: "'b" shows "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
a: shows "steps (start_of ly as + 2 * n, l, r) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stpa = (Suc (start_of ly as) + 2 * n, la, ra) \<and> inv_locate_b (as, lm) (n, la, ra) ires"
map_gpv_id_bind_gpv: fixes f :: "'k \<Rightarrow> 'h"   and gpv :: "('l, 'i, 'j) gpv"   and g :: "'l \<Rightarrow> ('k, 'i, 'j) gpv" shows "map_gpv f id (gpv \<bind> g) = gpv \<bind> map_gpv f id \<circ> g"
iprev_less_mono_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "iprev a I < iprev b I"   and "a \<in> I"   and "b \<in> I" shows "a < b"
seqI: fixes g1 :: "'a"   and f1 :: "'a"   and g2 :: "'a"   and f2 :: "'a" assumes "seq g1 f1 \<and> seq g2 f2" shows "VxV.seq (g1, g2) (f1, f2)"
l: shows "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
row_matrix_matrix_mult: fixes P :: "(('b, 'd) vec, 'e) vec"   and i :: "'e"   and A :: "(('b, 'c) vec, 'd) vec" shows "P $ i v* A = (P ** A) $ i"
Union_natural: fixes f :: "'a \<Rightarrow> 'b" shows "\<mu> \<circ> \<P> (\<P> f) = \<P> f \<circ> \<mu>"
assms_3: shows "q \<in> dgrad_sig_set' (length fs) d"
Exit_not_standard_control_dependent: fixes n :: "'b" assumes "n controls\<^sub>s Exit" shows "False"
abs_cfg_def: shows "abs_cfg \<equiv> quot_type.abs (rel_prod (=) eq_scheduler) Abs_cfg"
simps: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" shows "fixpt M T = (let (b, T2) = iter M T in if b then fixpt M T2 else T2)"
inj_map_strong: fixes x :: "'c pmf"   and xa :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and fa :: "'c \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> set_pmf x; za \<in> set_pmf xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_pmf f x = map_pmf fa xa" shows "x = xa"
assms_1: shows "insert (Var i IN t) (insert (Var i IN u) H) \<turnstile> Fls"
perm_eq_iff: fixes p :: "perm"   and q :: "perm" shows "(p = q) = (\<forall>a. p \<bullet> a = q \<bullet> a)"
0: shows "arr (mkArr (UP ` hfset B) (UP ` hfset C) (\<lambda>x. UP (happ F (DOWN x))))"
Bool: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> boolean_axioms" shows "H \<turnstile> A"