hcju/isabelleps huggingface dataset:
is_density_exprD_nonneg: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and t :: "pdf_type"   and e :: "cexpr" assumes "is_density_expr (vs, vs', \<Gamma>, \<delta>) t e" shows "nonneg_cexpr (shift_var_set (set vs')) (case_nat t \<Gamma>) e"
vcomp_vid_on_idem: fixes A :: "V" shows "vid_on A \<circ>\<^sub>\<circ> vid_on A = vid_on A"
a_def: shows "a \<equiv> SOME x. x \<in> A"
Dirichlet_L_minus_partial_sum_bound: fixes s :: "complex"   and x :: "real" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "0 < Re s"   and "0 < x" shows "cmod (sum_upto (\<lambda>n. \<chi> n * of_nat n powr - s) x - Dirichlet_L n \<chi> s) \<le> real (totient n) * (2 + cmod s / Re s) / x powr Re s"
distinct_5: fixes x4 :: "nat" shows "INIT \<noteq> REL x4"
hull_Un_left: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" shows "P hull (S \<union> T) = P hull (P hull S \<union> T)"
assms_6: shows "\<forall>n. \<forall>asset\<in>support_set pf. random_variable borel (prices Mkt asset n)"
sexp_rec_def: fixes M :: "('d, unit) Old_Datatype.node set"   and c :: "'d \<Rightarrow> 'c"   and d :: "nat \<Rightarrow> 'c"   and e :: "('d, unit) Old_Datatype.node set \<Rightarrow> ('d, unit) Old_Datatype.node set \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "sexp_rec M c d e = wfrec pred_sexp (\<lambda>g. sexp_case c d (\<lambda>N1 N2. e N1 N2 (g N1) (g N2))) M"
put_get: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'a" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
simps_1: fixes p :: "perm"   and x :: "'b \<times> 'b fs_set \<times> 'a" shows "p \<bullet> AC x = AC (p \<bullet> x)"
cut_and_update_def: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" shows "cut_and_update s vs = (\<forall>ttas s' t x ta x' m'. s -\<triangleright>ttas\<rightarrow>* s' \<longrightarrow> ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) \<longrightarrow> thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<longrightarrow> actions_ok s' t ta \<longrightarrow> (\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> ta_seq_consist P (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>) \<and> eq_upto_seq_inconsist P \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))))"
remove_child_is_component_unsafe: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) 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> remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component 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 child)|\<^sub>r)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
frechet_derivative_worksI: fixes f :: "'b \<Rightarrow> 'c"   and net :: "'b filter" assumes "f differentiable net" shows "(f has_derivative frechet_derivative f net) net"
assms_1: shows "M \<subseteq> Pow \<Omega>"
prob_compl: fixes A :: "real set" assumes "A \<in> events" shows "prob (space M - A) = 1 - prob A"
assms_2: shows "(x, y) \<in> r\<^sup>\<down>"
vector_7_1: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 1 = x1"
true_clss_singleton: fixes I :: "'b set"   and C :: "'b literal multiset" shows "(I \<TTurnstile>s {C}) = (I \<TTurnstile> C)"
gcd_dvd_antisym: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times (gcd a b) (gcd c d)"   and "dvd.dvd times (gcd c d) (gcd a b)" shows "gcd a b = gcd c d"
p_nonpos: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and t :: "real"   and s :: "'d"   and s' :: "'d" assumes "transition_rates R"   and "t \<le> 0" shows "transition_rates.p R s s' t = of_bool (s = s')"
sup_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = Collect ((\<lambda>x. x \<in> A) \<squnion> (\<lambda>x. x \<in> B))"
normal_density_def: fixes \<mu> :: "real"   and \<sigma> :: "real"   and x :: "real" shows "normal_density \<mu> \<sigma> x = 1 / sqrt (2 * pi * \<sigma>\<^sup>2) * exp (- (x - \<mu>)\<^sup>2 / (2 * \<sigma>\<^sup>2))"
lst_of_sp_is_vert: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es :: "'d edge list"   and v2 :: "'d" assumes "Graph.subpath g v1 es v2" shows "v2 \<in> Graph.vertices g"
part_equiv_trans_sym_2: fixes R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "part_equiv R"   and "(a, b) \<in> R"   and "(a, c) \<in> R" shows "(b, c) \<in> R"
diagvector_const: fixes k :: "'a" shows "diagvector (\<lambda>x. k) = mat k"
cmp_simps'_4: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.dom (\<Phi> (\<mu>, \<nu>)) = F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>)"
that_2: shows "((=) ===> A ===> A) (*\<^sub>R) (*\<^sub>R)"
SubstTermP_Ind: fixes v :: "tm"   and w :: "tm"   and t :: "tm" shows "{VarP v, IndP w, TermP t} \<turnstile> SubstTermP v t w w"
i_expand_i_take_mult_Suc: fixes i :: "nat"   and k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "i < k" shows "f \<odot> k \<Down> (n * k + Suc i) = f \<Down> n \<odot> k @ f n # NoMsg\<^bsup>i\<^esup>"
print_sets_def: fixes s :: "bool"   and X :: "'c set" shows "print_sets s X = sets_of_coll (X ::: clw_rel appr_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN ()) (\<lambda>X. RETURN (print_set s X)) (\<lambda>_ _. RETURN ()))"
csubspace_set_plus: fixes A :: "'d set"   and B :: "'d set" assumes "csubspace A"   and "csubspace B" shows "csubspace (A + B)"
exec_move_exec_1: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and body :: "'a expr1"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and xcp' :: "'a option"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and D :: "String.literal"   and frs :: "'a frame list" assumes "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t body h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D" shows "JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write (compP2 P) t (xcp, h, (stk, loc, C, M, pc) # frs) ta (xcp', h', (stk', loc', C, M, pc') # frs)"
fls_subdegree_fls_X_pow: fixes n :: "nat" shows "fls_subdegree (fls_X ^ n) = int n"
inf_map: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and \<iota> :: "'a inference" assumes "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"   and "\<iota> \<in> Inf_F"   and "\<G>_I \<iota> \<noteq> None" shows "the (\<G>_I \<iota>) \<subseteq> Red_I_G (\<G>_F (concl_of \<iota>))"
A2: shows "\<forall>y. f b = Intg y \<longrightarrow> y \<noteq> x"
cong_mult_self_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times a b) zero a"
max_tstp_intro: fixes tstp :: "nat + nat"   and ts :: "nat"   and tp :: "nat"   and tstp' :: "nat + nat" assumes "tstp_lt tstp ts tp"   and "tstp_lt tstp' ts tp"   and "isl tstp = isl tstp'" shows "tstp_lt (max_tstp tstp tstp') ts tp"
uminus_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> f = (\<lambda>e. - f e)"
abs_eq: fixes x :: "conc_interface" shows "asn (abs_interface x) = conc_asn x"
pure_conj_sep_conj: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P h \<and> Q h"   and "pure P \<or> pure Q" shows "(P \<and>* Q) h"
qItem_simps_6: fixes xsa :: "'e"   and xa :: "'f"   and ya :: "'f"   and deltaa :: "'g"   and qinp :: "'c \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qTerm option"   and qbinp :: "'d \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qAbs option" shows "qOp deltaa qinp qbinp #[[xa \<and> ya]]_xsa = qOp deltaa (qSwapInp xsa xa ya qinp) (qSwapBinp xsa xa ya qbinp)"
TESL_interp_neutral2: fixes \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> @ [] \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
polyneg_norm: fixes p :: "poly" assumes "isnpoly p" shows "isnpoly (~\<^sub>p p)"
rel_S_neg_distr: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and uu :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L2' :: "'s \<Rightarrow> 't \<Rightarrow> bool" assumes "rel_S_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_S (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_S L1 L2 Co1 Co2 Contra1 Contra2 OO rel_S L1' L2' Co1' Co2' Contra1' Contra2'"
trace_decrease_mul_adj: fixes \<rho> :: "complex mat"   and x :: "complex mat" assumes "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d"   and "x \<in> carrier_mat d d"   and "adjoint x * x \<le>\<^sub>L 1\<^sub>m d" shows "trace (x * \<rho> * adjoint x) \<le> trace \<rho>"
Compl_NegPf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in isTrue (B.neg (PPf n \<langle>\<phi>\<rangle>)) \<longrightarrow> bprv (B.neg (PPf n \<langle>\<phi>\<rangle>))"
f_def: fixes env_e_s :: "v sem_env \<times> exp0 \<times> 'a state" shows "f env_e_s = (case env_e_s of (env, e, s) \<Rightarrow> evaluate True env s e)"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', R, S) \<in> Rel"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>R, \<lparr>\<nu>*xvec\<rparr>S) \<in> Rel"
o_inv_distrib: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "bij f"   and "bij g" shows "inv (f \<circ> g) = inv g \<circ> inv f"
closedin_standard_simplex: fixes p :: "nat" shows "closedin (powertop_real UNIV) (standard_simplex p)"
f: shows "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
intros_5: fixes s :: "'a"   and b :: "'a set"   and x :: "'a set"   and y :: "'a \<Rightarrow> nat"   and c :: "'a com" assumes "s \<notin> b" shows "Sem (While b x y c) s s"
big_small_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> O[F](g1)"   and "f2 \<in> o[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
path_connected_punctured_convex: fixes S :: "'b set"   and a :: "'b" assumes "convex S"   and "aff_dim S \<noteq> 1" shows "path_connected (S - {a})"
prob_space: fixes M :: "'c pmf" shows "measure_pmf.prob M (space (measure_pmf M)) = 1"
proofTreeTerminal: fixes A :: "(nat \<times> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list) set"   and n :: "nat"   and delta :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "proofTree A"   and "(n, delta) \<in> A"   and "terminal subs delta" shows "SATAxiom (sequent delta)"
snd_mult_sv: fixes k :: "complex"   and v :: "complex \<times> complex" shows "snd (k *\<^sub>s\<^sub>v v) = k * snd v"
assms_3: shows "a \<bullet> u = b \<bullet> v"
borel_measurable_continuous_on_restrict: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on A f" shows "f \<in> borel_measurable (restrict_space borel A)"
tranclp_imp_exists_finite_chain_list: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>xs. chain R (llist_of (x # xs @ [y]))"
boundOutputPar2Dest: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' (Q \<parallel> R)"   and "xvec \<sharp>* Q"   and "yvec \<sharp>* Q" obtains T :: "('d, 'e, 'f) psi" where "P = Q \<parallel> T" and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' R"
weak_eq: fixes x :: "'a"   and y :: "'a" assumes "comp x y = Eq" shows "x = y"
cmp_simps_1: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.arr (S.cmp\<^sub>U\<^sub>P (f, g))"
qs_cost_average_conv_rqs_cost: fixes A :: "'a set"   and B :: "'a set"   and R :: "('a \<times> 'a) set" assumes "finite A"   and "linorder_on B R"   and "A \<subseteq> B" shows "map_pmf (qs_cost R) (pmf_of_set (permutations_of_set A)) = rqs_cost (card A)"
added_ids_def: fixes es :: "('d \<times> ('d, 'b) operation) event list"   and p :: "'b" shows "added_ids es p \<equiv> List.map_filter (\<lambda>x. case x of Broadcast x \<Rightarrow> Map.empty x | Deliver (i, Create j e) \<Rightarrow> if e = p then Some j else None | Deliver (i, Delete set x) \<Rightarrow> Map.empty x | Deliver (i, Append id x) \<Rightarrow> Map.empty x | Deliver (i, Expunge e mo j) \<Rightarrow> if e = p then Some j else None | Deliver (i, Store aa id1 x) \<Rightarrow> Map.empty x) es"
memoized_correct: fixes os :: "'t iarray"   and a :: "'s"   and b :: "'s"   and c :: "nat" shows "viterbi_ix_rec os a b c = fst (run_state (viterbi_ix\<^sub>m' os a b c) Mapping.empty)"
E_extendsS: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>b {P} c { F}" shows "\<turnstile>\<^sub>3\<^sub>' {P} c { F}"
simps: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list"   and ox :: "'a"   and ys :: "('b \<times> 'c) list \<times> 'a" shows "oalist_raw.oalist_eq_raw rep_key_order (xs, ox) ys = (xs = oalist_raw.sort_oalist_aux rep_key_order ox ys)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b stream" shows "sset (smap f v) = f ` sset v"
1: shows "trace (ioa.asig A) e_A' = trace (ioa.asig A) e_B'"
reach_trans: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set" assumes "reach T \<subseteq> I" shows "{reach T} TS.trans T {> I}"
setdist_def: fixes s :: "'c set"   and t :: "'c set" shows "setdist s t \<equiv> if s = {} \<or> t = {} then 0 else Inf {dist x y |x y. x \<in> s \<and> y \<in> t}"
oLimit_mem_INTER: fixes A :: "nat \<Rightarrow> ordinal set"   and f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. normal_set (A n)"   and "\<forall>n. A (Suc n) \<subseteq> A n"   and "\<forall>n. f n \<in> A n"   and "mono f" shows "oLimit f \<in> \<Inter> (range A)"
exetheory_full_exhaust: fixes \<Theta> :: "exetheory" obtains const_type :: "(String.literal \<times> typ) list" and typ_arity :: "(String.literal \<times> nat) list" and sorts :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" and axioms :: "Core.term list" where "\<Theta> = ExeTheory (ExeSignature const_type typ_arity sorts) axioms"
simps_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "not_shadows_vconsts (Vrecabs css name \<Gamma>) = (pred_fmap id (fmmap not_shadows_vconsts \<Gamma>) \<and> pred_fmap (list_all (\<lambda>(pat, t). fdisjnt all_consts (frees pat) \<and> \<not> shadows_consts t)) css \<and> name |\<in>| fmdom css \<and> True)"
imp_graph_vars: fixes l :: "lit"   and l' :: "lit"   and cnf :: "lit uprod set" assumes "(l, l') \<in> imp_graph cnf" shows "var l \<in> vars_of_cnf cnf"
llast_conv_lnth: fixes xs :: "'b llist"   and n :: "nat" assumes "llength xs = eSuc (enat n)" shows "llast xs = xs ?! n"
nth_append: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "(xs \<cdot> ys) ! n = (if n < \<^bold>|xs\<^bold>| then xs ! n else ys ! (n - \<^bold>|xs\<^bold>|))"
H_loop: fixes p :: "'a"   and r :: "'a"   and x :: "'a" assumes "H (t p \<cdot> t r) x p" shows "H p (while r do x od) (t p \<cdot> !r)"
Range_Diff_subset: fixes A :: "('b \<times> 'a) set"   and B :: "('b \<times> 'a) set" shows "Range A - Range B \<subseteq> Range (A - B)"
simps_2: fixes r :: "('a \<times> bool) language" shows "\<dd> (Times_Plus r) = (\<lambda>a. Times_Plus (Plus (\<dd> r (a, True)) (\<dd> r (a, False))))"
assms_11: shows "B = \<Union> (roots ` X (<))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
v0_conforms: assumes "v0 \<in> VV p" shows "\<sigma> v0 = w0"
mu_no_change_large_row: fixes ii :: "nat"   and j :: "nat" assumes "i < ii"   and "ii < m"   and "j \<noteq> i"   and "j \<noteq> i - 1" shows "gs2.\<mu> ii j = fs.gs.\<mu> ii j"
SPROPS_1: shows "dsqr = {r \<in> \<delta>. qw \<in> set (rhsq r) \<and> the (rcm r) \<le> 1}"
f4: shows "(r'' \<otimes> r, s'' \<otimes> s) \<in> carrier rel"
crit_pair_cbelow_same: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m" shows "crit_pair_cbelow_on d m F p p"
elims_3: fixes x :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "\<not> FixVars.qGoodAbs x" obtains xs :: "'e" and x :: "'d" and X :: "('f, 'g, 'e, 'd, 'h) qTerm" where "xa__ = qAbs xs x X" and "\<not> FixVars.qGood X"
assoc: fixes a :: "int poly fract"   and b :: "int poly fract"   and c :: "int poly fract" shows "rat_poly_times (rat_poly_times a b) c = rat_poly_times a (rat_poly_times b c)"
discI_2: fixes tm :: "('a, 'b) tm"   and x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" assumes "tm = App x21 x22" shows "is_App tm"
False_imp_not_eventually: fixes P :: "'a \<Rightarrow> bool"   and net :: "'a filter" assumes "\<forall>x. \<not> P x"   and "net \<noteq> bot" shows "\<not> eventually P net"
ide_char: fixes f :: "Arr" shows "ide f = (f = AA \<or> f = BB \<or> f = TT)"