hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'b set" shows "space_as_set (ccspan x) = closure (cspan x)"
contour_integrable_linepath_Reals_iff: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "a \<in> \<real>"   and "b \<in> \<real>"   and "Re a < Re b" shows "(f contour_integrable_on linepath a b) = ((\<lambda>x. f (complex_of_real x)) integrable_on {Re a..Re b})"
unit_simps_5: shows "trg (local.inv \<epsilon>\<^sub>0) = src d\<^sub>0"
i_take_i_drop: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "f \<Up> m \<Down> n = f \<Down> (n + m) \<up> m"
Rep_take_columns: fixes A :: "'b matrix"   and c :: "nat"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (take_columns A c) j i = (if i < c then Rep_matrix A j i else (0::'b))"
dense_access_fp_aux: fixes S :: "'b set"   and U :: "'b set"   and V :: "'b set" assumes "open S"   and "connected S"   and "openin (top_of_set (frontier S)) U"   and "openin (top_of_set (frontier S)) V"   and "V \<noteq> {}"   and "\<not> U \<subseteq> V" obtains g :: "real \<Rightarrow> 'b" where "arc g" and "pathstart g \<in> U" and "pathfinish g \<in> V" and "g ` {0<..<1} \<subseteq> S"
assms_2: shows "h \<circ> f \<le> g \<circ> h"
preserves_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.cod (hom'.map f) = hom'.map (hom'.cod f)"
empt: fixes ks :: "int list"   and l :: "int"   and x :: "int" assumes "dioph_is_node ks l x" shows "\<not> dioph_memb x (dioph_empt ks l)"
ctrm_monom: fixes a :: "'a"   and k :: "nat" assumes "a \<in> carrier R" shows "monom P (monom P a (Suc k) 0) 0 = \<zero>\<^bsub>P\<^esub>"
pos_emb_step_at_disjunct: fixes q :: "dir list"   and p :: "dir list"   and t :: "('c, 'd) tm"   and d1 :: "dir"   and d2 :: "dir" assumes "take (length q) p \<noteq> q"   and "take (length p) q \<noteq> p" shows "position_of t (p @ [d1]) = position_of (emb_step_at q d2 t) (p @ [d1])"
matrix_component_le_onorm: fixes A :: "((real, 'a) vec, 'b) vec"   and i :: "'b"   and j :: "'a" shows "\<bar>A $ i $ j\<bar> \<le> onorm ((*v) A)"
in_multisets_iff: fixes M :: "'b multiset"   and A :: "'b set" shows "(M \<in> multisets A) = (set_mset M \<subseteq> A)"
multiplicity_diff_pow_coprime: fixes p :: "nat"   and x :: "int"   and y :: "int"   and n :: "nat" assumes "prime p"   and "int p dvd x - y"   and "\<not> int p dvd x"   and "\<not> int p dvd y"   and "coprime p n" shows "multiplicity (int p) (x ^ n - y ^ n) = multiplicity (int p) (x - y)"
tSplice_jkbpC: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "{t, t'} \<subseteq> SPR.jkbpC"   and "tObsC t = tObsC t'" shows "t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t' \<in> SPR.jkbpC"
simulation_reaches: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "A.reaches a a'"   and "a \<sim> b" shows "\<exists>b'. B.reaches b b' \<and> a' \<sim> b'"
word_arith_nat_add: fixes a :: "'b word"   and b :: "'b word" shows "a + b = word_of_nat (unat a + unat b)"
assms_3: shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
fdia_one: fixes x :: "'a" shows " |(1::'a)\<rangle> x = d x"
Elem_Lambda_Fun: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and U :: "ZF"   and V :: "ZF" shows "Elem (Lambda A f) (Fun U V) = (A = U \<and> (\<forall>x. Elem x A \<longrightarrow> Elem (f x) V))"
rel_compp_Grp: fixes R1 :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'j \<Rightarrow> bool" shows "Resumption.resumption.rel_resumption R1 R2 = (BNF_Def.Grp {x. results x \<subseteq> {(x, y). R1 x y} \<and> outputs x \<subseteq> {(x, y). R2 x y}} (Resumption.resumption.map_resumption fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. results x \<subseteq> {(x, y). R1 x y} \<and> outputs x \<subseteq> {(x, y). R2 x y}} (Resumption.resumption.map_resumption snd snd)"
A: fixes U :: "'a set"   and ys :: "'b list"   and V :: "'a set" shows "sinks_aux I D U ys \<subseteq> V \<longrightarrow> ipurge_ref_aux I D U ys (ipurge_ref_aux_foldr I D P V xs X) = ipurge_ref_aux_foldr I D P V xs X"
0: shows "ord_Zp y \<le> ord_Zp (x \<oplus> y)"
0: fixes n :: "nat" shows "\<forall>h. h \<in> carrier (UP R) \<and> deg R h \<le> n \<and> h \<in> carrier (UP (R\<lparr>carrier := S\<rparr>)) \<longrightarrow> Cring_Poly.compose R h g = Cring_Poly.compose (R\<lparr>carrier := S\<rparr>) h g"
hyps: shows "(variable \<in> vars formula) = (\<exists>literal. literal el formula \<and> var literal = variable)"
r_dovetail_prim: shows "prim_recfn 2 r_dovetail"
inj_on_Un: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" shows "inj_on f (A \<union> B) = (inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {})"
convert_linearSubstitutionUni: fixes var :: "nat"   and a :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and a' :: "atomUni"   and b :: "real mpoly"   and B :: "real"   and c :: "real mpoly"   and C :: "real" assumes "convert_atom var a (xs' @ x # xs) = Some a'"   and "insertion (nth_default 0 (xs' @ x # xs)) b = B"   and "insertion (nth_default 0 (xs' @ x # xs)) c = C"   and "B \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "length xs' = var" shows "aEval (linear_substitution var (- c) b a) (xs' @ x # xs) = evalUni (linearSubstitutionUni B C a') x"
ldrop_ltl: fixes n :: "enat"   and xs :: "'a llist" shows "ldrop n (ltl xs) = ldrop (eSuc n) xs"
satPB_Un: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi>1 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set"   and \<Phi>2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "satPB \<xi> (\<Phi>1 \<union> \<Phi>2) = (satPB \<xi> \<Phi>1 \<and> satPB \<xi> \<Phi>2)"
R: assumes "C = add_mset s (C - {#s#})"   and "A = B + (C - {#s#})" shows "P"
assms_4: shows "fds_nth f 1 \<noteq> (0::'a)"
inorder_ins: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (ins x t) = ins_list x (Tree2.inorder t)"
has_domain_iff_has_codomain: fixes f :: "'a" shows "(C.domains f \<noteq> {}) = (C.codomains f \<noteq> {})"
"1_2": shows "\<forall>j<k. cs ! j \<noteq> cs ! k"
bal_substitute_separator: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree"   and c :: "'b" assumes "bal (Node (ls @ (a, b) # rs) t)" shows "bal (Node (ls @ (a, c) # rs) t)"
z: fixes j :: "nat"   and x :: "nat"   and r :: "nat"   and i :: "nat" assumes "j < e_length x" shows "eval z [j, r, i, x] = eval r_result1 [e_length x, i, e_take (Suc j) x]"
IH: fixes S :: "variable set"   and s :: "variable" assumes "finite S" shows "distinct (fst (variant_variables n s S))"
iMODb_mod_0_eq_conv: fixes r :: "nat"   and c :: "nat"   and r' :: "nat"   and m' :: "nat"   and c' :: "nat" shows "([ r, mod 0, c ] = [ r', mod m', c' ]) = (r = r' \<and> (m' = 0 \<or> c' = 0))"
topology_finer_closed_id: fixes X :: "'c topology"   and X' :: "'c topology" shows "(\<forall>S. closedin X S \<longrightarrow> closedin X' S) = closed_map X X' id"
B: shows "(xs @ zs @ [z], Z) \<in> failures P"
Hd_head_id: fixes s :: "('c, 'd) tm" assumes "num_args s = 0" shows "Hd (head s) = s"
u: shows "\<Delta> @ \<Gamma> \<turnstile> u : U"
singleton: fixes x :: "'a" shows "prod_mset {#x#} = x"
EWHILEIT_weaken: fixes I :: "'c \<Rightarrow> bool"   and I' :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> ('d + 'c) nres"   and x :: "'c" assumes "\<And>x. I x \<Longrightarrow> I' x" shows "EWHILEIT I' b f x \<le> EWHILEIT I b f x"
Red_F_eq_Red_F_diff_Red_F: fixes N :: "'f set" shows "Red_F N = Red_F (N - Red_F N)"
Inf_greatest: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'b::type set"   and z :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "\<And>x::'b::type. x \<in> (A::'b::type set) \<Longrightarrow> (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (z::'b::type) x" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (z::'b::type) ((Inf::'b::type set \<Rightarrow> 'b::type) (A::'b::type set))"
B_def: fixes y :: "int" shows "B y \<equiv> y \<le> 0"
i_1: shows "2 ^ i < (n - 1) div 2 + 1"
add_mset_remove_trivial: fixes x :: "'c"   and M :: "'c multiset" shows "remove1_mset x (add_mset x M) = M"
empty_LeadsTo: fixes F :: "'c program"   and B :: "'c set" shows "F \<in> {} \<longmapsto>w B"
im_d_gchainTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and l :: "nat" assumes "d_gchain G n f"   and "f l \<in> f ` {i. i \<le> n} - {f 0}" shows "f (LEAST j. f j \<in> f ` {i. i \<le> n} - {f 0}) \<in> f ` {i. i \<le> n} - {f 0}"
max_Var_floatarith_FDERIV_floatariths: fixes xs :: "nat list"   and d :: "floatarith list"   and fa :: "floatarith list" assumes "length xs = length d" shows "max_Var_floatariths (FDERIV_floatariths fa xs d) \<le> max (max_Var_floatariths fa) (max_Var_floatariths d)"
atLeastLessThan_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.order less_eq less"   and "less_eq b a" shows "ord.atLeastLessThan less_eq less a b = {}"
iso_is_retraction: fixes f :: "'a" assumes "local.iso f" shows "retraction f"
face_cycle_set_succ: fixes a :: "'b" shows "face_cycle_set (face_cycle_succ a) = face_cycle_set a"
hIm_hrcis: fixes r :: "real star"   and a :: "real star" shows "hIm (hrcis r a) = r * (*f* sin) a"
eventually_nhds: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "eventually P (nhds a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
invert_subclseq: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "ws_prog G" shows "(D, C) \<notin> (subcls1 G)\<^sup>+"
UNIV_enum: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "UNIV = set enum"
simps: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c" shows "final_thread.not_final_thread final s t = ((\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x) \<or> (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> ln \<noteq> no_wait_locks) \<or> (\<exists>x w ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> wset s t = \<lfloor>w\<rfloor>))"
getEnv_substEnv2: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "ys \<noteq> xs \<or> y \<noteq> x"   and "rho xs x = Some X" shows "(rho &[Y / y]_ys) xs x = Some (X #[Y / y]_ys)"
Dirichlet_L_euler_product_LIMSEQ: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - \<chi>\<^sub>0 p / nat_power p s) else 1) \<longlonglongrightarrow> Dirichlet_L n \<chi>\<^sub>0 s"
sort_oalist_id: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs" shows "sort_oalist xs = xs"
Diff_eq_empty_iff: fixes A :: "'g set"   and B :: "'g set" shows "(A - B = {}) = (A \<subseteq> B)"
beta_upright_add: fixes vb :: "'c zmultiset"   and vc :: "'c zmultiset"   and va :: "'c zmultiset" assumes "upright vb"   and "upright vc"   and "beta_upright va vb" shows "beta_upright va (vb + vc)"
neg: fixes i :: "Integer" assumes "defined i"   and "\<lbrakk>i\<rbrakk> < 0" shows "P i"
mult_zero_left: fixes a :: "'a" shows "(0::'a) * a = (0::'a)"
one_lt_\<omega>: shows "1 < \<omega>\<^sub>z"
mk_pos_leq: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "x \<le> y" shows "mk_pos x \<le> mk_pos y"
rrsmultD1: fixes x :: "('f, 'g) aezfun"   and r :: "('f, 'g) aezfun"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "x \<in> FH" shows "(r \<currency> f) x = f (x * r)"
inj_on_image_Pow: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" assumes "inj_on f A" shows "inj_on ((`) f) (Pow A)"
rep_eq: shows "coeff 0 = (\<lambda>_. 0::'b)"
UnI1: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A" shows "c \<in> A \<union> B"
chart_compose_lemma: fixes c1 :: "('a, 'b) chart"   and c2 :: "('a, 'b) chart" assumes "domain c1 \<inter> domain c2 \<subseteq> carrier"   and "\<And>c. c \<in> charts \<Longrightarrow> k-smooth_compat c1 c"   and "\<And>c. c \<in> charts \<Longrightarrow> k-smooth_compat c2 c" shows "k-smooth_on (apply_chart c1 ` (domain c1 \<inter> domain c2)) (apply_chart c2 \<circ> inv_chart c1)"
dist_riemann_sphere_ge_0: fixes M1 :: "riemann_sphere"   and M2 :: "riemann_sphere" shows "0 \<le> dist_riemann_sphere' M1 M2"
ask_const: fixes m :: "'m" shows "ask (\<lambda>_. m) = m"
mem_CH_Ch: fixes x :: "'a"   and X :: "'a set" shows "(x \<in> CH X) = (x \<in> Ch (Xh x) X)"
one_iff_aux: fixes xs :: "'b com option list"   and s :: "'b" assumes "xs \<noteq> []" shows "(\<forall>ys. ((xs, s) # ys \<in> par_cptn) = (\<exists>clist. length clist = length xs \<and> (xs, s) # ys \<propto> map (\<lambda>i. (fst i, s) # snd i) (zip xs clist) \<and> (\<forall>i<length xs. (xs ! i, s) # clist ! i \<in> cptn))) = (par_cp xs s = {c. \<exists>clist. length clist = length xs \<and> (\<forall>i<length clist. clist ! i \<in> cp (xs ! i) s) \<and> c \<propto> clist})"
zero: shows "{#b \<in># \<B>. b \<notin> bs#} rep p = 0"
transaction_strand_subsets_6: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" shows "set (unlabel (transaction_receive T)) \<subseteq> set (unlabel (transaction_strand T))"
dom_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.dom (\<F>C.mkarr t) = \<F>C.mkarr (Dom t)"
filter_le_avg_def: fixes s :: "'s" shows "filter_le_avg s \<equiv> let a = avg s in iterate s (\<lambda>x s. if x \<le> a then ins x s else s) (local.empty ())"
monoD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and A :: "'b set"   and p :: "nat"   and s :: "'b"   and t :: "'b" assumes "SemilatAlg.mono r step n A"   and "p < n"   and "s \<in> A"   and "s \<sqsubseteq>\<^bsub>r\<^esub> t" shows "set (step p s) {\<sqsubseteq>\<^bsub>r\<^esub>} set (step p t)"
adjoint_pair_antipar_3: fixes f :: "'a"   and g :: "'a" assumes "adjoint_pair f g" shows "src f = trg g"
append_T_imp_tickFree: fixes t :: "'a event list"   and s :: "'a event list"   and P :: "'a process" assumes "t @ s \<in> T P"   and "s \<noteq> []" shows "tickFree t"
sorted_rev_nth_mono: fixes xs :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "linorder.sorted (\<le>lex) (rev xs)"   and "i \<le> j"   and "j < \<^bold>|xs\<^bold>|" shows "xs ! j \<le>lex xs ! i"
fresh_star_Un: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'b" shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
list_of_dlist: fixes x :: "'c set_dlist" shows "list_of_dlist x \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}"
assms: shows "remdups_fwd_acc Acc xs = ys @ x # zs"
mult_mat_vec_carrier: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "A *\<^sub>v v \<in> carrier_vec nr"
less_finite_1_def: fixes x :: "Enum.finite_1"   and y :: "Enum.finite_1" shows "(x < y) = False"
l8_13_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A \<noteq> B"   and "C \<noteq> D"   and "Col X A B"   and "Col X C D"   and "\<exists>U V. Col U A B \<and> Col V C D \<and> U \<noteq> X \<and> V \<noteq> X \<and> Per U X V" shows "X PerpAt A B C D "
section_is_mono: fixes g :: "('a S.E.term, 'a) S.arr" assumes "S.section g" shows "S.mono g"
assms_1: shows "p = {l}\<lbrakk>fa\<rbrakk> qq"
box_phi_a_2: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>"   and y :: "\<nu>" assumes "[\<^bold>\<box>(\<^bold>\<forall>F. \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>\<phi> F) in v]" shows "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) \<^bold>\<rightarrow> \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>y\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
sort_oalist_ko_def: shows "sort_oalist_ko \<equiv> oalist_raw.sort_oalist_raw (\<lambda>x. x)"
non_null_2: fixes i :: "'b" assumes "i \<in> S" shows "0 < e i"
assms_1: shows "chain_reparam_weak_chain one_chain1 one_chain2"
arity_var_ne_infinity_if_\<delta>_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and x :: "'a" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "0 < \<delta>" shows "arity_var x \<noteq> \<infinity>"
setdist_singletons: fixes x :: "'b"   and y :: "'b" shows "setdist {x} {y} = dist x y"
ord: shows "order x p = order x (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
well_valued0_map_map_tree: fixes f :: "'b \<Rightarrow> 'c"   and t :: "'b tree"   and as :: "'b list" assumes "\<forall>x y. f (x + y) = f x + f y"   and "well_shaped t"   and "well_valued0 as t"   and "r t \<le> length as"   and "as \<noteq> []" shows "well_shaped (map_tree f t) \<and> well_valued0 (map f as) (map_tree f t)"
star_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> \<cdot> y\<^sup>\<star>)\<^sup>\<star>"
quote_all_Contra_PfP_ssubst: fixes V :: "name set"   and \<alpha> :: "fm" assumes "V \<subseteq> {}"   and "supp \<alpha> \<subseteq> atom ` {}" shows "quote_all 0 V \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V (make_F {} 0)) IMP PfP (ssubst \<lfloor>Neg \<alpha>\<rfloor>V V (make_F {} 0)) IMP PfP (ssubst \<lfloor>Fls\<rfloor>V V (make_F {} 0))"
map_add_subsumed1: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "f \<subseteq>\<^sub>m g" shows "f ++ g = g"
f1: shows "(cmod (jozsa_algo $$ (0, 0)))\<^sup>2 = (cmod (complex_of_real (\<Sum>k<2 ^ n. (- 1) ^ f k / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))\<^sup>2"
weakenS: fixes s :: "'a"   and s' :: "'a"   and a :: "'c" assumes "S s s' a" shows "S' s s' a"
induct: fixes P :: "'c rexp \<Rightarrow> bool"   and a0 :: "'c rexp" assumes "P Zero"   and "P One"   and "\<And>r s. \<lbrakk>P r; P s\<rbrakk> \<Longrightarrow> P (Plus r s)"   and "P Full"   and "\<And>v. P (Atom v)"   and "\<And>v va. P (Times v va)"   and "\<And>v. P (Star v)"   and "\<And>v. P (rexp.Not v)"   and "\<And>v va. P (rexp.Inter v va)"   and "\<And>v. P (Pr v)" shows "P a0"
continuous_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'd"   and g :: "'a \<Rightarrow> 'e" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. prod (f x) (g x))"
inject_2: fixes x31 :: "('a, 'b, 'c) cmd"   and x32 :: "('a, 'b, 'c) cmd"   and y31 :: "('a, 'b, 'c) cmd"   and y32 :: "('a, 'b, 'c) cmd" shows "(x31 ;; x32 = y31 ;; y32) = (x31 = y31 \<and> x32 = y32)"
clock_repSim: fixes t :: "'s Trace" assumes "t \<in> Clock.jkbpC" shows "sim (clock_simMCt t) ((clock_repMC \<circ> fst) (clock_sim t)) clock_repSim"
uminus_assertion: fixes x :: "'a" assumes "x \<in> assertion" shows "neg_assert x \<in> assertion"
simps_2: fixes bdd :: "bdd" shows "Rmi_g (Suc 0) Trueif bdd = True"
index_smult_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and a :: "'a" assumes "i < dim_row A"   and "j < dim_col A" shows "(a \<cdot>\<^sub>m A) $$ (i, j) = a * A $$ (i, j)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Atm x1 = Atm y1) = (x1 = y1)"
elims: fixes x :: "chan"   and y :: "msg" assumes "implem_sym x = y" obtains A :: "agent" and B :: "agent" and M :: "msg" where "x = Insec Aa__ Ba__ Ma__" and "y = \<langle>InsecTag, Messages.Agent Aa__, Messages.Agent Ba__, Ma__\<rangle>"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Confid Ab__ Bb__ Mb__" and "y = Enc \<langle>ConfidTag, Mb__\<rangle> (shrK Ab__ Bb__)"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Auth Ac__ Bc__ Mc__" and "y = \<langle>Mc__, hmac \<langle>AuthTag, Mc__\<rangle> (shrK Ac__ Bc__)\<rangle>"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Secure A B M" and "y = Enc \<langle>SecureTag, M\<rangle> (shrK A B)"
match_ex_table_pc_length_compE2: fixes P :: "'a prog"   and a :: "String.literal"   and pc :: "nat"   and e :: "'b expr1"   and pc' :: "nat"   and d :: "nat"   and pcd :: "nat \<times> nat" assumes "match_ex_table P a pc (compxE2 e pc' d) = \<lfloor>pcd\<rfloor>" shows "pc' \<le> pc \<and> pc < length (compE2 e) + pc'"
abs_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'a" shows "deg_pp (PP x) = deg_pm x"
midpoint_in_open_segment: fixes a :: "'b"   and b :: "'b" shows "(midpoint a b \<in> {a<--<b}) = (a \<noteq> b)"
perm_1: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list" shows "pi \<bullet> FNil = FNil"
continuous_on_proj_spaceI: fixes S :: "'b proj_space set"   and f :: "'b proj_space \<Rightarrow> 'c" assumes "continuous_on (Proj -` S) (f \<circ> Proj)"   and "open S" shows "continuous_on S f"
conv_compl: fixes x :: "'a" shows "(- x)\<^sup>T = - x\<^sup>T"
pick_mono: fixes n :: "nat"   and S :: "nat set"   and m :: "nat" assumes "n < card S \<or> infinite S"   and "m < n" shows "pick S m < pick S n"
verts_ge_Suc0: shows "[0..<length (g_V G)] \<noteq> [0]"
rel_is_equivalence: shows "equivalence (Sigma I \<FF>2) {(x, y). stk2.rel x y}"
error_free_Norm: fixes s :: "st" shows "error_free (Norm s)"
times_modTr: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a"   and I :: "'a set" assumes "a \<in> carrier R"   and "a' \<in> carrier R"   and "b \<in> carrier R"   and "b' \<in> carrier R"   and "ideal R I"   and "a \<plusminus> -\<^sub>a b \<in> I"   and "a' \<plusminus> -\<^sub>a b' \<in> I" shows "a \<cdot>\<^sub>r a' \<plusminus> -\<^sub>a b \<cdot>\<^sub>r b' \<in> I"
relcomp_UNION_distrib2: fixes r :: "'d \<Rightarrow> ('a \<times> 'c) set"   and I :: "'d set"   and s :: "('c \<times> 'b) set" shows "\<Union> (r ` I) ; s = (\<Union>i\<in>I. r i ; s)"