hcju/isabelleps huggingface dataset:
compl_structD4: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and h :: "('t \<Rightarrow>\<^sub>0 'e) \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and b :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "h \<in> set (fst (compl gs bs (ps -- sps) sps data))"   and "b \<in> set gs \<union> set bs"   and "fst b \<noteq> 0" shows "\<not> lt (fst b) adds\<^sub>t lt (fst h)"
altc_env_def: fixes C :: "'c cset"   and f :: "'c \<Rightarrow> ('d, 'm) envT" shows "local.altc_env C f = EnvT (\<lambda>r. altc C (\<lambda>c. run_env (f c) r))"
range_cblinfun_code_def: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "range_cblinfun_code A = A *\<^sub>S \<top>"
bhv2cnf_cnf2bhv: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n" shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>(\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n) = n"
paths_tl_notnil: fixes xs :: "'a list" assumes "xs \<in> Q" shows "tl xs \<noteq> []"
that: fixes l :: "'a" assumes "l \<in> s"   and "f \<longlonglongrightarrow> l" shows "thesis"
has_disc_negated_has_disc: fixes disc :: "'c \<Rightarrow> bool"   and neg :: "bool"   and m :: "'c match_expr" assumes "has_disc_negated disc neg m" shows "has_disc disc m"
R_is_R1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is R healthy" shows "P is R1 healthy"
sint_up_scast: fixes w :: "'b word" assumes "is_up SCAST('b \<rightarrow> 'c)" shows "sint (SCAST('b \<rightarrow> 'c) w) = sint w"
assms_2: shows "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> \<Psi>  t"
rel_map_1: fixes S1b :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S2b :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i1 :: "'f \<Rightarrow> 'a"   and i2 :: "'g \<Rightarrow> 'c"   and x :: "('e, 'f, 'g) sp\<^sub>\<mu>"   and y :: "('e, 'b, 'd) sp\<^sub>\<mu>" shows "rel_sp\<^sub>\<mu> S1b S2b (map_sp\<^sub>\<mu> i1 i2 x) y = rel_sp\<^sub>\<mu> (\<lambda>x. S1b (i1 x)) (\<lambda>x. S2b (i2 x)) x y"
exec_step_ind_Store: fixes n :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI (Store n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (tl stk, loc[n := hd stk], C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
scale_sum_right: fixes a :: "real"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "a *\<^sub>R sum f A = (\<Sum>x\<in>A. a *\<^sub>R f x)"
vcomp_def: fixes r :: "V"   and s :: "V" shows "r \<circ>\<^sub>\<circ> s = set {\<langle>a, c\<rangle> |a c. \<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> s \<and> \<langle>b, c\<rangle> \<in>\<^sub>\<circ> r}"
CastThrow: fixes 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"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
set_tag_name_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_tag_name_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
adjoint_one: fixes n :: "nat" shows "adjoint (1\<^sub>m n) = 1\<^sub>m n"
try_swap_inv: fixes p :: "'person"   and c\<^sub>1 :: "'color"   and c\<^sub>2 :: "'color"   and w :: "'person \<Rightarrow> 'color" shows "local.try_swap p c\<^sub>1 c\<^sub>2 (local.try_swap p c\<^sub>1 c\<^sub>2 w) = w"
equivalent_transitive: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set"   and S3 :: "'at Literal set set" assumes "equivalent S1 S2"   and "equivalent S2 S3" shows "equivalent S1 S3"
wt_lbv_def: fixes P :: "jvm_prog"   and C :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and et :: "ex_table"   and cert :: "(ty list \<times> ty err list) option err list"   and ins :: "bytecode" shows "wt_lbv P C b Ts T\<^sub>r mxs mxl\<^sub>0 et cert ins \<equiv> check_cert P mxs ((case b of Static \<Rightarrow> 0 | NonStatic \<Rightarrow> 1) + length Ts + mxl\<^sub>0) (length ins) cert \<and> 0 < length ins \<and> (let start = \<lfloor>([], (case b of Static \<Rightarrow> [] | NonStatic \<Rightarrow> [OK (Class C)]) @ map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor>; result = lbvjvm P mxs ((case b of Static \<Rightarrow> 0 | NonStatic \<Rightarrow> 1) + length Ts + mxl\<^sub>0) T\<^sub>r et cert ins (OK start) in result \<noteq> Err)"
simps_7: fixes v :: "'c" shows "nnf (\<^bold>\<not> (Atom v)) = \<^bold>\<not> (Atom v)"
mem_fTo_sub_carrier: fixes X :: "'a set" assumes "X \<in> carrier fTo D" shows "X \<subseteq> carrier D"
below_CoCallsI: fixes G :: "CoCalls"   and G' :: "CoCalls" assumes "\<And>x y. x--y\<in>G \<Longrightarrow> x--y\<in>G'" shows "G \<sqsubseteq> G'"
bind_is_matrix_vector_mult: fixes x :: "'a pmf"   and A :: "'a st_mat" shows "x \<bind> transition_of_st_mat A = pmf_of_st_vec (A *st st_vec_of_pmf x)"
nop_vrange: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop A n f" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
le_less_linear: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
zero_bcontfun_def: shows "0 \<equiv> Bcontfun (\<lambda>_. 0::'b)"
embed_ge_0: fixes G :: "'c \<Rightarrow> bool"   and s :: "'c" shows "0 \<le> \<guillemotleft> G \<guillemotright> s"
split_less_Suc: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>x<Suc k. P x) = (P k \<and> (\<forall>x<k. P x))"
hyps_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "n = length xs"   and "distinct xs" shows "(ys \<in> set (permutations xs)) = (distinct ys \<and> set xs = set ys)"
nn: shows "nsphere n closure_of {x. x k < - 1 / 2} \<subseteq> nsphere n interior_of {x. x k \<le> 0}"
hyps: shows "<list @ [Suc ab]> = <list @ [ab]> @ [Oc]"
assms_3: shows "other.s \<delta> \<le> u_max"
inner_verts_rev_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "inner_verts (rev_path p) = rev (inner_verts p)"
simps_6: fixes v :: "'b"   and va :: "'b"   and vb :: "('b, 'c) Combinators"   and vc :: "('b, 'c) Combinators list" shows "removeShadowRules1_alternative_rev (DenyAllFromTo v va # vb # vc) = DenyAllFromTo v va # removeShadowRules1_alternative_rev (vb # vc)"
assms_1: shows "path_image g1 \<subseteq> s"
real_root_power_cancel: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x" shows "root n (x ^ n) = x"
fin_L: shows "finite (acc_on (- H) `` {s})"
Mapping_lookup_filter_keys: fixes k :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) mapping" assumes "k \<in> Mapping.keys (Mapping.filter f m)" shows "Mapping.lookup (Mapping.filter f m) k = Mapping.lookup m k"
card_PiE: fixes S :: "'b set"   and T :: "'b \<Rightarrow> 'c set" assumes "finite S" shows "card (Pi\<^sub>E S T) = (\<Prod>i\<in>S. card (T i))"
ex_minimal_witness: fixes P :: "nat \<Rightarrow> bool" assumes "\<exists>k. P k" obtains K :: "nat" where "P K" and "\<And>k. k < K \<Longrightarrow> \<not> P k"
homeomorphic_space_prod_topology_sing2: fixes a :: "'d"   and X :: "'d topology"   and Y :: "'e topology" assumes "a \<in> topspace X" shows "Y homeomorphic_space prod_topology (subtopology X {a}) Y"
sup_supF: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "\<exists>x. x \<in> c"   and "Subspace.subspace F E" shows "Subspace.subspace F H"
surj_vimage_empty: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'b set" assumes "surj f" shows "(f -` A = {}) = (A = {})"
Merge_hash: fixes A :: "transaction\<^sub>m set"   and x :: "transaction\<^sub>m"   and a :: "transaction\<^sub>m" assumes "transaction.Merge A = Some x"   and "a \<in> A" shows "hash_transaction a = hash_transaction x"
acyclic_heap_def: fixes h :: "(_) heap" shows "local.CD.a_acyclic_heap h = acyclic (parent_child_rel h)"
card_of_Well_order: fixes A :: "'f set" shows "Well_order |A|"
precedesAntisymmetry: fixes a :: "'a"   and l :: "'a list"   and b :: "'a" assumes "a \<in> set l"   and "b \<in> set l"   and "precedes a b l"   and "precedes b a l" shows "a = b"
Inf_Assertion_def: fixes A :: "'a Assertion set" shows "\<Sqinter>A = - \<Squnion>(uminus ` A)"
projection_split_arbitrary_element: fixes \<tau> :: "'b list"   and X :: "'b set"   and \<alpha> :: "'b list"   and x :: "'b"   and \<beta> :: "'b list" assumes "\<tau> \<upharpoonleft> X = (\<alpha> @ [x] @ \<beta>) \<upharpoonleft> X"   and "x \<in> X" shows "\<exists>\<alpha>' \<beta>'. \<tau> = \<alpha>' @ [x] @ \<beta>' \<and> \<alpha>' \<upharpoonleft> X = \<alpha> \<upharpoonleft> X \<and> \<beta>' \<upharpoonleft> X = \<beta> \<upharpoonleft> X"
Bignorabimus_step: fixes S :: "variable set set"   and \<alpha> :: "game" assumes "\<And>M. M \<in> S \<Longrightarrow> M \<in> Bignorabimus \<alpha>" shows "\<Union> S \<in> Bignorabimus \<alpha>"
assms_2: fixes c :: "'a Abs_State.st option acom" shows "strip (f c) = strip c"
mut_initial_state_def: fixes s :: "('a, 'b, 'c, 'd) Model.local_state" shows "mut_initial_state s = (ghost_hs_phase s = hp_IdleMarkSweep \<and> ghost_honorary_grey s = {} \<and> ghost_honorary_root s = {} \<and> W s = {})"
inf_greatest: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" assumes "x \<subseteq># y"   and "x \<subseteq># z" shows "x \<subseteq># y \<inter># z"
order_le_less_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y < z" shows "x < z"
hom_0: fixes x :: "'a poly" assumes "poly_lift x = 0" shows "x = 0"
simps_12: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (And p q) = (Ifm bbs bs p \<and> Ifm bbs bs q)"
chine_hcomp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "('b, 'c) arrow_of_spans_data_scheme"   and \<mu> :: "('b, 'd) arrow_of_spans_data_scheme" assumes "span_bicategory C prj0 prj1" shows "span_bicategory.chine_hcomp C prj0 prj1 \<nu> \<mu> \<equiv> elementary_category_with_pullbacks.tuple C prj0 prj1 (C (Chn \<nu>) (prj1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)))) (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)) (C (Chn \<mu>) (prj0 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))))"
density_context_insert: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and F :: "(nat \<Rightarrow> val) \<Rightarrow> val measure"   and t :: "pdf_type"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "density_context V V' \<Gamma> \<delta>"   and "has_parametrized_subprob_density (state_measure (V \<union> V') \<Gamma>) F (stock_measure t) f" shows "density_context (shift_var_set V) (Suc ` V') (case_nat t \<Gamma>) (insert_dens V V' f \<delta>)"
extf_snoc_if_\<delta>_eq_\<epsilon>: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and xs :: "('s, 'v) tm list"   and x :: "('s, 'v) tm" assumes "\<delta> = \<epsilon>" shows "extf f gt (xs @ [x]) xs"
twiddleVal_Loc: fixes l1 :: "Location"   and l2 :: "Location"   and \<beta> :: "(Location \<times> Location) set" assumes "(l1, l2) \<in> \<beta>" shows "(\<beta>, RVal (Loc l1), RVal (Loc l2)) \<in> twiddleVal"
in_homE: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
cyclic_group_axioms: shows "Finite_And_Cyclic_Groups.cyclic_group (H\<lparr>carrier := generate H {b}\<rparr>) b"
insert_mod_def: fixes x :: "'b"   and m :: "'m" shows "insert_mod x m = (let k = key x; ys = map_of m k in if \<exists>y\<in>set ys. subsumed x y then m else up k (x # ys) m)"
simplicial_subdivision_of: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" shows "simplicial_subdivision p (frag_of f) = (if p = 0 then frag_of f else simplicial_cone (p - 1) (\<lambda>i. (\<Sum>j\<le>p. simplicial_vertex j f i) / real (Suc p)) (simplicial_subdivision (p - 1) (chain_boundary p (frag_of f))))"
OTNSepaux: fixes a :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list" assumes "onlyTwoNets (a \<oplus> y) \<and> OnlyTwoNets z \<longrightarrow> OnlyTwoNets (separate (a \<oplus> y # z))"   and "\<not> member DenyAll a"   and "\<not> member DenyAll y"   and "noDenyAll z"   and "onlyTwoNets a"   and "OnlyTwoNets (y # z)"   and "first_bothNet a = first_bothNet y" shows "OnlyTwoNets (separate (a \<oplus> y # z))"
transition_idx_def: fixes s :: "nat \<Rightarrow> 'b"   and T :: "nat \<Rightarrow> ('b \<times> 'b) set"   and A :: "nat set" shows "transition_idx s T A = (LEAST k. \<exists>i j. A = {i, j} \<and> i < j \<and> (s j, s i) \<in> T k)"
lhd_lSup: fixes A :: "'c llist set" assumes "\<exists>x\<in>A. \<not> lnull x" shows "lhd (lSup A) = (THE x. x \<in> lhd ` (A \<inter> {xs. \<not> lnull xs}))"
isLub_unique: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isLub R S x"   and "isLub R S y" shows "x = y"
odd_distinct: fixes p :: "nat \<Rightarrow> nat" assumes "\<And>i. p i \<noteq> 0 \<Longrightarrow> odd i" shows "odd_of_distinct (distinct_of_odd p) = p"
has_le_has: fixes n :: "nat"   and S :: "'b set"   and n' :: "nat" assumes "has n S"   and "n' \<le> n" shows "has n' S"
fun_rel_mono: fixes R1 :: "('a \<times> 'b) set"   and R2 :: "('a \<times> 'b) set"   and R3 :: "('c \<times> 'f) set"   and R4 :: "('c \<times> 'f) set" assumes "R1 \<subseteq> R2"   and "R3 \<subseteq> R4" shows "R2 \<rightarrow> R3 \<subseteq> R1 \<rightarrow> R4"
simps_3: fixes x :: "char list"   and D :: "sdiagram" shows "com_cell (Exists_sdia x D) = com_sdia D"
runit_in_hom_1: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<r>[f] : src f \<rightarrow> trg f\<guillemotright>"
mat_annir: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" shows "f \<otimes> \<delta> = \<delta>"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> parts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>K X. \<lbrakk>Crypt K X \<in> parts H; P (Crypt K X)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y" shows "P x"
check_ex_unique: fixes eX :: "'b withERR" assumes "eX \<noteq> ERR" shows "\<exists>!X. eX = OK X"
hypreal_sqrt_mult_distrib2: fixes x :: "real star"   and y :: "real star" assumes "0 \<le> x"   and "0 \<le> y" shows "(*f* sqrt) (x * y) = (*f* sqrt) x * (*f* sqrt) y"
finite_acyclic_wf_converse: fixes r :: "('b \<times> 'b) set" assumes "finite r"   and "acyclic r" shows "wf (r\<inverse>)"
set_eq: shows "{i. 1 \<le> i \<and> i \<le> n} = (\<Union>m\<in>{m. m \<le> n \<and> odd m}. {i. \<exists>k. i = 2 ^ k * m \<and> i \<le> n})"
inf_idem: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
node_ptr_inclusion: fixes node_ptr :: "(_) node_ptr"   and node_ptrs :: "(_) node_ptr set" shows "(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> 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_ptrs) = (node_ptr \<in> node_ptrs)"
left_impl_one: fixes a :: "'a" shows "a l\<rightarrow> a = (1::'a)"
mkAutoEC_def: shows "mkAutoEC \<equiv> \<lambda>a. \<lparr>pInit = \<lambda>obs. {t \<in> jkbpC. jviewInit a obs = jview a t}, pTrans = \<lambda>obs ps. {uu_. \<exists>t t'. uu_ = t \<and> t \<in> jkbpC \<and> t' \<in> ps \<and> jview a t = jviewIncr a obs (jview a t')}, pAct = \<lambda>ps. jAction MC (SOME t. t \<in> ps) a\<rparr>"
that: fixes p :: "'a set set" assumes "p division_of {}" shows "thesis"
simps_2: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and a :: "'b"   and ts :: "'b FOL_Fitting.term list" shows "evalt e f (App a ts) = f a (evalts e f ts)"
borel_singleton: fixes A :: "'a set"   and x :: "'a" assumes "A \<in> sets borel" shows "insert x A \<in> sets borel"
elims: fixes x :: "('a \<times> 'a) list"   and xa :: "'a"   and y :: "'a" assumes "inverse_permutation_of_list x xa = y" obtains xb :: "'a" where "x = []" and "xa = xb__" and "y = xb__"   | ya :: "'a" and x' :: "'a" and xs :: "('a \<times> 'a) list" and xb :: "'a" where "x = (ya, x') # xs" and "xa = xb" and "y = (if xb = x' then ya else inverse_permutation_of_list xs xb)"
bisim: shows "weak_reduction_bisimulation (indRelRTPO TRel) (STCal Source Target)"
assms: shows "max_Var_floatariths ode_e \<le> length xs"
fmdrop_set_fmdrop: fixes S :: "'c set"   and b :: "'c"   and m :: "('c, 'd) fmap" shows "fmdrop_set S (fmdrop b m) = fmdrop_set (insert b S) m"
bigo_const_iff: fixes c1 :: "'c"   and F :: "'b filter"   and c2 :: "'c" shows "((\<lambda>_. c1) \<in> O[F](\<lambda>_. c2)) = (F = bot \<or> c1 = (0::'c) \<or> c2 \<noteq> (0::'c))"
arr_cod_iff_arr: fixes f :: "'a" shows "R.arr (R.cod f) = R.arr f"
conc_fun_def: fixes R :: "('c \<times> 'd) set"   and m :: "'d nres" shows "\<Down> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> RES (R\<inverse> `` X)"
vbrelation_vcomp_vid_on_left: fixes r :: "V"   and A :: "V" assumes "vbrelation r"   and "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A" shows "vid_on A \<circ>\<^sub>\<circ> r = r"
SDG_edge_valid_SDG_node_1: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and n' :: "'node SDG_node" assumes "SDG_edge n Vopt popt n'" shows "valid_SDG_node n"
prefix_of_antimono: fixes \<pi> :: "'a prefix"   and \<pi>' :: "'a prefix"   and s :: "'a trace" assumes "\<pi> \<le> \<pi>'"   and "prefix_of \<pi>' s" shows "prefix_of \<pi> s"
SS_unique: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "bprv (all yy (all yy' (imp (cnj (SS \<langle>\<phi>\<rangle> (Var yy)) (SS \<langle>\<phi>\<rangle> (Var yy'))) (eql (Var yy) (Var yy')))))"
asym: fixes a :: "'b word"   and b :: "'b word" assumes "a <s b"   and "b <s a" shows "False"
Rep_mini_alg_unique_cases: fixes y :: "mini_alg" assumes "y \<in> {r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r)}" obtains x :: "mini_alg_unique" where "y = Rep_mini_alg_unique x"
inner_loop2_refine: fixes succl :: "'c list"   and succ :: "'c set"   and dst :: "'c"   and u :: "'d"   and PRED :: "'c \<Rightarrow> 'd option"   and N :: "'c set" assumes "(succl, succ) \<in> \<langle>Id\<rangle>list_set_rel" shows "inner_loop2 dst succl u PRED N \<le> \<Down> Id (inner_loop dst succ u PRED N)"
lessThan_def: fixes u :: "'b word" shows "signed.lessThan u \<equiv> {x. x <s u}"
list_of_oalist_filter: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and xs :: "'x" shows "list_of_oalist (local.filter P xs) = filter_raw P (list_of_oalist xs)"
proj_append_lfinite: fixes t :: "cnf llist"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t' :: "cnf llist" assumes "lfinite t" shows "dynamic_component.proj tCMP active c (t @\<^sub>l t') = dynamic_component.proj tCMP active c t @\<^sub>l dynamic_component.proj tCMP active c t'"
append_def: fixes xq :: "'a lazy_sequence"   and yq :: "'a lazy_sequence" shows "Lazy_Sequence.append xq yq = lazy_sequence_of_list (list_of_lazy_sequence xq @ list_of_lazy_sequence yq)"
Sec_PriK_def: shows "Sec_PriK \<equiv> SOME Sec_PriK. \<exists>Tok_PriK. inj Sec_PriK \<and> inj Tok_PriK \<and> Rev_PriK \<notin> range Sec_PriK \<and> Rev_PriK \<notin> range Tok_PriK \<and> range Sec_PriK \<inter> range Tok_PriK = {}"
check_eqv_def: shows "check_eqv \<equiv> WS1S.check_eqv"
exhaust: fixes y :: "'b + 'c" obtains x1 :: "'b" where "y = Inl x1"   | x2 :: "'c" where "y = Inr x2"
tdev': fixes x :: "'b pdevs"   and p :: "nat" shows "tdev x \<le> tdev' p x"
Ccod: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
lifting_wf_def_ass: fixes P :: "'addr J_prog" assumes "wf_J_prog P" shows "lifting_wf final_expr (\<lambda>t ((e, l), h) ta ((e', l'), h'). extTA2J P,P,t \<turnstile> \<langle>e,(h, l)\<rangle> -ta\<rightarrow> \<langle>e',(h', l')\<rangle>) (\<lambda>t (e, x) m. \<D> e \<lfloor>dom x\<rfloor>)"
deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and a :: "action"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p"   and "a = action.Drop \<or> a = Reject" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
rep_nat_term_pp_def: fixes t :: "('a, 'b) pp" shows "rep_nat_term t = (rep_nat_pp t, 0)"
card_isolated_verts_dv: shows "card isolated_verts = Suc (card H.isolated_verts)"
sorted_sort: fixes xs :: "'d Word.word list" shows "signed.sorted (signed.sort xs)"
\<pi>_in: shows "\<pi> \<in> lins2 S G"
noDAID: fixes p :: "('a, 'b) Combinators list" assumes "noDenyAll p" shows "noDenyAll (insertDenies p)"
mk_poly_sub_carrier: fixes A :: "'b poly mat"   and nr :: "nat"   and nc :: "nat"   and l :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mk_poly_sub A l j \<in> carrier_mat nr nc"
vsv_axioms: shows "vsv (\<AA>\<lparr>Cod\<rparr>)"
inject_2: fixes x2 :: "('b, 'c, 'd, 'e) Tree"   and y2 :: "('b, 'c, 'd, 'e) Tree" shows "(tNot x2 = tNot y2) = (x2 = y2)"