hcju/isabelleps huggingface dataset:
Root_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" shows "Root F G = (SOME A. A \<in> F \<and> A \<notin> \<Union> (ran G))"
pos: shows "\<forall>\<^sub>F n in sequentially. 0 < f n"
induct: fixes P :: "(name \<times> trm) list \<Rightarrow> name \<Rightarrow> bool"   and a0 :: "(name \<times> trm) list"   and a1 :: "name" assumes "\<And>x. P [] x"   and "\<And>y T \<theta> x. (x \<noteq> y \<Longrightarrow> P \<theta> x) \<Longrightarrow> P ((y, T) # \<theta>) x" shows "P a0 a1"
CH_cop_F_cases: fixes x :: "'a"   and ds :: "'b set"   and fp :: "'a set" assumes "x \<in> CH (cop_F ds fp)" obtains "x \<in> CH fp"   | "x \<in> Rf CH fp"   | "x \<in> CD_on ds (- Rf CH fp) - fp"
polytope_imp_bounded: fixes S :: "'b set" assumes "polytope S" shows "bounded S"
rep_eq: fixes x :: "('c, 'd) mapping_rbt" shows "lookup x = rbt_comp_lookup ccomp (impl_of x)"
prime_dvd_mult_eq_nat: fixes p :: "nat"   and a :: "nat"   and b :: "nat" assumes "prime p" shows "(p dvd a * b) = (p dvd a \<or> p dvd b)"
show_r_zero: fixes x :: "'a vec"   and a :: "'a vec" assumes "x \<in> carrier_vec n"   and "a \<in> carrier_vec n" shows "(x + a = x) = (a = 0\<^sub>v n)"
landau_pair_axioms: shows "landau_pair bigomega smallomega bigomega smallomega bigomega smallomega (\<lambda>x y. y \<le> x)"
ptr_kinds_M_ok: fixes h :: "(_) heap" shows "h \<turnstile> ok node_ptr_kinds_M"
complete_lattice_partial_function_definitions: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
monadic_nfoldli_def: fixes l :: "'f list"   and c :: "'e \<Rightarrow> bool nres"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e nres"   and s :: "'e" shows "monadic_nfoldli l c f s \<equiv> REC\<^sub>T (\<lambda>D (l, s). case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> c s \<bind> (\<lambda>b. if b then f x s \<bind> (\<lambda>s'. D (ls, s')) else RETURN s)) (l, s)"
apath_nonempty_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.apath u p v"   and "p \<noteq> []" shows "u \<noteq> v"
add_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "add_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do operand2 \<leftarrow> gets (get_operand2 op_list); curr_win \<leftarrow> get_curr_win (); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); psr_val \<leftarrow> gets (cpu_reg_val PSR); c_val \<leftarrow> gets (\<lambda>s. get_icc_C psr_val); result \<leftarrow> gets (\<lambda>s. if instr_name = arith_type ADD \<or> instr_name = arith_type ADDcc then rs1_val + operand2 else rs1_val + operand2 + ucast c_val); rd_val \<leftarrow> gets (user_reg_val curr_win rd); new_rd_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then result else rd_val); write_reg new_rd_val curr_win rd; add_instr_sub1 instr_name result rs1_val operand2 od"
2: shows "LeftDerivation x' (derivation_shift D (length u) 0) (drop (length u) w)"
distinct_1: fixes x21 :: "'f"   and x22 :: "'f list" shows "[] \<noteq> x21 # x22"
bl_to_bin_ge0: fixes bs :: "bool list" shows "0 \<le> bl_to_bin bs"
finprod_all1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (0::'c)" shows "sum f A = (0::'c)"
inv_of_domain_ss1: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "continuous_on U f"   and "inj_on f U"   and "f ` U \<subseteq> S"   and "subspace S"   and "openin (top_of_set S) U" shows "openin (top_of_set S) (f ` U)"
dg_prod_2_op_dg_dg_Obj: shows "(op_dg \<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr> = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
Zp_residue_mult_zero_2: fixes a :: "nat \<Rightarrow> int"   and k :: "nat"   and b :: "nat \<Rightarrow> int" assumes "a k = 0" shows "(b \<otimes> a) k = 0"
qFreshAll_simps_2: fixes xs :: "'d"   and x :: "'e"   and delta :: "'h"   and inp :: "'f \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm option"   and binp :: "'g \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qAbs option" shows "qFresh xs x (qOp delta inp binp) = (liftAll (qFresh xs x) inp \<and> liftAll (qFreshAbs xs x) binp)"
refl_onI: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "order.greater_eq (A \<times> A) r"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
const_lt_component_SomeD2: fixes p :: "'t \<Rightarrow>\<^sub>0 'e"   and cmp :: "'k" assumes "const_lt_component p = Some cmp" shows "component_of_term (lt p) = cmp"
brackets_unlift_vars_swap: fixes x :: "nat itrm"   and n :: "nat"   and vs :: "nat list"   and x' :: "nat itrm" assumes "set (opaque x) \<subseteq> {0..<n}"   and "set vs \<subseteq> {0..<n}"   and "itrm_brackets vs x = Some x'" shows "term_brackets vs (unlift_vars n x) = Some (unlift_vars n x')"
oSuc_less_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "(oSuc x < oSuc y) = (x < y)"
pre_gauss_lemma: shows "[a ^ nat ((int p - 1) div 2) = (- 1) ^ card E] (mod int p)"
leg1_in_hom_2: shows "\<guillemotleft>tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 : tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1\<guillemotright>"
ds_dom_llp: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> d y \<cdot> x) = (d x \<sqsubseteq> d y)"
sinks_aux_subset_ipurge_tr: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and I' :: "('d \<times> 'd) set"   and D' :: "'c \<Rightarrow> 'd"   and u' :: "'d"   and xs :: "'c list" shows "sinks_aux I D U (ipurge_tr I' D' u' xs) \<subseteq> sinks_aux I D U xs"
apply_bcontfun_inverse: fixes x :: "'c \<Rightarrow>\<^sub>C 'd" shows "Bcontfun (apply_bcontfun x) = x"
him_mult_left: fixes a :: "real"   and q :: "quat" shows "HIm (quat_of_real a * q) = a *\<^sub>R HIm q"
min_0_1_6: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq (numeral.numeral one plus x) one = one"
fold_insert: fixes A :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c fset set"   and z :: "'c fset set" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>\<phi>. (\<otimes>) (f \<phi>)) z (insert x A) = f x \<otimes> Finite_Set.fold (\<lambda>\<phi>. (\<otimes>) (f \<phi>)) z A"
converse_subset_swap: fixes r :: "('c \<times> 'd) set"   and s :: "('d \<times> 'c) set" shows "(r \<subseteq> s\<inverse>) = (r\<inverse> \<subseteq> s)"
ereal_Inf: fixes A :: "real set" assumes "\<bar>Inf (ereal ` A)\<bar> \<noteq> \<infinity>" shows "ereal (Inf A) = Inf (ereal ` A)"
Map_dom: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" shows "J_C.Map (J_C.dom f) = J_C.Dom f"
def_body: fixes x :: "Set('d)"   and y :: "'d" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "y \<noteq> \<bottom>" shows "\<lfloor>\<lfloor>y \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> \<bottom> \<and> \<lfloor>\<lfloor>y \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> null"
rec_imp_def: shows "rec_imp = CN rec_disj [CN rec_not [recf.Id 2 0], recf.Id 2 1]"
assms_1: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> S"   and "Y \<in> S"   and "X \<noteq> Y" shows "X \<inter> Y = {}"
add_rconjby: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "rconjby x y + rconjby x z = rconjby x (y + z)"
CompFun_HAInitStates_HAStates: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)" shows "InitState SA \<in> HAInitStates A"
below: shows "\<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub>"
simps: fixes a1 :: "cmd"   and a2 :: "cmd" shows "(a1 \<^sub>C\<longlonglongrightarrow>\<^sup>* a2) = ((\<exists>c. a1 = c \<and> a2 = c) \<or> (\<exists>c d e. a1 = c \<and> a2 = e \<and> c \<^sub>C\<longlonglongrightarrow> d \<and> d \<^sub>C\<longlonglongrightarrow>\<^sup>* e))"
assms: shows "P \<longmapsto> \<alpha> \<prec> P'"
hyperpow_two: fixes r :: "'a star" shows "r pow 2 = r * r"
givenBy_imp_eq_Collect: fixes A :: "'d set"   and v :: "'d \<Rightarrow> 'e" assumes "A \<in> givenBy v" shows "\<exists>P. A = {s. P (v s)}"
let\<^sub>1_closed: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "closed (\<Gamma>, Let \<Delta> e, S)"   and "domA \<Delta> \<inter> domA \<Gamma> = {}"   and "domA \<Delta> \<inter> upds S = {}" shows "(\<Gamma>, Let \<Delta> e, S) \<Rightarrow> (\<Delta> @ \<Gamma>, e, S)"
greater_iff_sym_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and b :: "'b"   and a :: "'b" assumes "comparator cmp" shows "(cmp b a = Greater) = (cmp a b = Less)"
par_by_merge_choice_right: fixes P :: "('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and Q :: "('d, 'g) urel"   and R :: "('d, 'g) urel" shows "P \<parallel>\<^bsub>M\<^esub> (Q \<sqinter> R) = P \<parallel>\<^bsub>M\<^esub> Q \<sqinter> P \<parallel>\<^bsub>M\<^esub> R"
monom_mult_smult: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "monom (UP R) (a \<otimes>\<^bsub>R\<^esub> b) n = a \<odot>\<^bsub>UP R\<^esub> monom (UP R) b n"
simps: fixes a1 :: "'v"   and a2 :: "'v"   and a3 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" shows "path a1 a2 a3 = ((\<exists>v. a1 = v \<and> a2 = v \<and> a3 = []) \<or> (\<exists>e v' pth. a1 = edge_begin e \<and> a2 = v' \<and> a3 = e # pth \<and> e \<in> edges \<and> path (edge_end e) v' pth))"
binsert_sorted: fixes t :: "tree"   and x :: "nat" shows "BinaryTree_TacticStyle.sorted t \<longrightarrow> BinaryTree_TacticStyle.sorted (binsert x t)"
carrier_direct_lim_def: shows "carrier_direct_lim \<equiv> Sigma I \<FF> / {(x, y). x \<sim> y}"
assms_2: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "F x \<le> F y"
mult_divides: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R - {\<zero>}"   and "c \<otimes> a divides c \<otimes> b" shows "a divides b"
aHom_def: fixes A :: "('c, 'e) aGroup_scheme"   and B :: "('d, 'f) aGroup_scheme" shows "aHom A B = {f \<in> carrier A \<rightarrow> carrier B. f \<in> extensional (carrier A) \<and> (\<forall>a\<in>carrier A. \<forall>b\<in>carrier A. f (a \<plusminus>\<^bsub>A\<^esub> b) = f a \<plusminus>\<^bsub>B\<^esub> f b)}"
Finite_completion: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto>w A' i \<union> C"   and "\<And>i. i \<in> I \<Longrightarrow> F \<in> A' i Co A' i \<union> C" shows "F \<in> \<Inter> (A ` I) \<longmapsto>w \<Inter> (A' ` I) \<union> C"
less_literal_def: fixes L :: "'b literal"   and M :: "'b literal" shows "(L < M) = (atm_of L < atm_of M \<or> atm_of L \<le> atm_of M \<and> is_neg L < is_neg M)"
length_filter_replace1: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x \<in> set xs"   and "\<not> P x" shows "|filter P (replace x ys xs)| = |filter P xs| + |filter P ys|"
cong1: fixes x :: "'b vec list"   and y :: "'b vec list"   and n :: "nat" assumes "x = y" shows "mat_of_cols n x = mat_of_cols n y"
simps_2: fixes xs :: "'b list" shows "shuffles xs \<epsilon> = {xs}"
has_parametrized_subprob_density_integral: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and x :: "'a" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M" shows "integral\<^sup>N R (f x) \<le> 1"
simps_5: fixes a :: "'a" shows "typeof (Addr a) = None"
alw_sfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "alw (holds P) xs"   and "alw (ev (holds Q)) xs" shows "sfilter (\<lambda>x. P x \<and> Q x) xs = sfilter Q xs"
that: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list" assumes "l = ka + kb - 1"   and "U = {xs, ys}"   and "Form_Body ka kb xs ys (inter_scheme l U)"   and "0 < kb"   and "kb \<le> ka"   and "ka \<le> Suc kb" shows "thesis"
cos_arg_i_mult_zero: fixes y :: "complex" assumes "y \<noteq> 0"   and "Re y = 0" shows "cos (arg y) = 0"
t3: shows "continuous (at_left (\<Squnion> {norm (f x) / norm x |x. True})) ((*) (cmod r))"
extra_odd_nodes: shows "{v, v'} \<subseteq> odd_nodes_set (del_unEdge v w v' G)"
Ops_FvarsT: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and Ops :: "('b \<Rightarrow> 'b) set"   and enc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "TermEncode var trm fmla Var FvarsT substT Fvars subst num Ops enc"   and "f \<in> Ops"   and "t \<in> trm" shows "FvarsT (f t) = FvarsT t"
v_eq0_imp_b_eq_0: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "design point_set block_collection"   and "int (card point_set) = 0" shows "int (size block_collection) = 0"
minus_ow_def: fixes U :: "'b set"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "minus_ow U minus \<equiv> \<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> minus a b \<in> U"
inverse_unique: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.inverse_arrows f g" shows "S.inv f = g"
deriv_bodies: shows "\<forall>p\<in>Procs. \<forall>\<sigma> Z. \<Gamma>,\<Theta>' \<sigma> p\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
NoSuchMethodC_def: shows "NoSuchMethodC \<equiv> (NoSuchMethodError, Object, [], [])"
typeI_cube_line_integral_exists_boundary: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeI_chain i j F two_chain s" shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
weakBisimE_1: 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 \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakStatImp substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq') Q"
rtrancl_path_Range_end: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "rtrancl_path R x xs y"   and "xs \<noteq> []" shows "Rangep R y"
that: fixes U' :: "'a set"   and V :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "open U'"   and "U' \<subseteq> U"   and "x0 \<in> U'"   and "open V"   and "f x0 \<in> V"   and "homeomorphism U' V f g"   and "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative g' y) (at y)"   and "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f' (g y)))"   and "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f' (g y)))" shows "thesis"
eq1: shows "A * mat (dim_vec u) 1 (\<lambda>(i, j). u $ i) = mat (dim_vec (A *\<^sub>v u)) 1 (\<lambda>(i, j). (A *\<^sub>v u) $ i)"
l8_5: fixes A :: "'p"   and B :: "'p" shows "Per A B B"
degree_monom_eq: fixes a :: "'a"   and n :: "nat" assumes "a \<noteq> (0::'a)" shows "degree (Polynomial.monom a n) = n"
menger_trivial: fixes paths :: "'a list set"   and n :: "nat"   and S :: "'a set" assumes "DisjointPaths G v0 v1 paths"   and "card paths = n"   and "Separation G v0 v1 S" shows "n \<le> card S"
retractionE: fixes g :: "'s" assumes "retraction g" obtains f :: "'s" where "ide (g \<cdot> f)"
TT: shows "X closure_of T \<subseteq> X interior_of T"
simps_1: fixes list :: "'c list"   and vals :: "'c \<Rightarrow> 'd option"   and oid :: "'c"   and ref :: "'c option"   and val :: "'d" shows "interp_op (list, vals) (oid, Insert ref val) = (insert_spec list (oid, ref), vals(oid \<mapsto> val))"
InvariantGetReasonIsReasonAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state" shows "let state' = applyUnitPropagate state in InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
mult_assoc: fixes a :: "('a \<times> 'a set) set"   and b :: "('a \<times> 'a set) set"   and c :: "('a \<times> 'a set) set" shows "a \<parallel> b \<parallel> c = a \<parallel> (b \<parallel> c)"
all_countings_set: fixes a :: "nat"   and b :: "nat" shows "all_countings a b = card {V \<in> Pow {0..<a + b}. card V = a}"
HNext_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "HNext rmhist p = ((rmhist!p)$ = (if MemReturn rmCh ires p \<or> RPCFail crCh rmCh rst p then #histB else if MClkReply memCh crCh cst p then #histA else $(rmhist!p)))"
colours_alloc_1: fixes s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and r' :: "'c"   and r :: "'c"   and m :: "'d"   and roots' :: "'c set"   and fl :: "bool" assumes "sys_heap s r' = None" shows "black r (s(mutator m := s (mutator m)\<lparr>roots := roots'\<rparr>, sys := s sys\<lparr>heap := sys_heap s(r' \<mapsto> \<lparr>obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty\<rparr>)\<rparr>)) = (black r s \<or> r' = r \<and> fl = sys_fM s \<and> \<not> grey r' s)"
gcd_fib_mod: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
delete_edges_delete_edge_commute: fixes a1 :: "'b"   and a2 :: "'b"   and G :: "'b list_graph"   and as :: "('b \<times> 'b) list" shows "FiniteListGraph.delete_edges (FiniteListGraph.delete_edge a1 a2 G) as = FiniteListGraph.delete_edge a1 a2 (FiniteListGraph.delete_edges G as)"
sat_models_pow_pow: fixes P :: "'a ltln set" shows "{sat_models (abs_ltln\<^sub>P \<phi>) \<inter> Pow P |\<phi>. prop_atoms \<phi> \<subseteq> P} \<subseteq> Pow (Pow P)"
r_enumdom_nonempty_domain_2: fixes i :: "nat"   and x\<^sub>0 :: "nat"   and x :: "nat" assumes "eval r_phi [i, x\<^sub>0] \<down>" shows "(eval r_phi [i, x] \<down>) = (\<exists>y. eval r_enumdom [i, y] \<down>= x)"
induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "lexordp x1 x2"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x y xs ys. x < y \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>x y xs ys. \<lbrakk>\<not> x < y; \<not> y < x; lexordp xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
norm_Rep: fixes x :: "'b word" shows "take_bit LENGTH('b) (uint x) = uint x"
se_newarg: fixes ei :: "exp"   and es :: "exp list"   and C :: "nat" assumes "ei \<in> set es" shows "ei \<in> isubexprs(New C es)"
induct: fixes xa :: "'a"   and K :: "'a set"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "xa \<in> K.simple_extension K x"   and "P \<zero>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub>"   and "\<And>k1 k2. \<lbrakk>k1 \<in> K.simple_extension K x; P k1; k2 \<in> K\<rbrakk> \<Longrightarrow> P (k1 \<otimes>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> x \<oplus>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> k2)" shows "P xa"
axis_in_Basis_iff: fixes i :: "'b"   and a :: "'a" shows "(axis i a \<in> Basis) = (a \<in> Basis)"
is_volatile_split: fixes r :: "'b memref" shows "is_volatile r = (is_volatile_Read\<^sub>s\<^sub>b r \<or> is_volatile_Write\<^sub>s\<^sub>b r)"
arg_mult_real_positive: fixes k :: "real"   and z :: "complex" assumes "0 < k" shows "arg (cor k * z) = arg z"
plusinf_uset0: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "islin p"   and "\<not> Ifm vs (x # bs) (plusinf p)"   and "Ifm vs (x # bs) p" shows "\<exists>(c, s)\<in>set (uset p). x \<le> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
h2: shows "\<turnstile> \<box>[P \<longrightarrow> Q]_v"
j': shows "atom j' \<sharp> (s, s', i, j, x, y, y')"
rev_eq_Cons_iff: fixes xs :: "'d list"   and y :: "'d"   and ys :: "'d list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
compl_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y = \<bottom>"   and "x \<squnion> y = \<top>"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- x = y"
induct: fixes P :: "'a word \<Rightarrow> 'a simple_rule list \<Rightarrow> 'a wordinterval \<Rightarrow> bool"   and a0 :: "'a word"   and a1 :: "'a simple_rule list"   and a2 :: "'a wordinterval" assumes "\<And>uu_ uv_. P uu_ [] uv_"   and "\<And>s m rs acc_dropped. \<lbrakk>simple_match_ip (src m) s \<Longrightarrow> P s rs acc_dropped; \<not> simple_match_ip (src m) s \<Longrightarrow> P s rs acc_dropped\<rbrakk> \<Longrightarrow> P s (SimpleRule m Accept # rs) acc_dropped"   and "\<And>s m rs acc_dropped. \<lbrakk>simple_match_ip (src m) s \<Longrightarrow> P s rs (wordinterval_union (ipcidr_tuple_to_wordinterval (dst m)) acc_dropped); \<not> simple_match_ip (src m) s \<Longrightarrow> P s rs acc_dropped\<rbrakk> \<Longrightarrow> P s (SimpleRule m Drop # rs) acc_dropped" shows "P a0 a1 a2"
vsingletonD: fixes y :: "V"   and x :: "V" assumes "y \<in>\<^sub>\<circ> set {x}" shows "y = x"
minus_le_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) b = less_eq (uminus b) a"