hcju/isabelleps huggingface dataset:
simps_24: shows "cospan_shape.comp cospan_shape.BT cospan_shape.TT = cospan_shape.Null"
Inr_in_tailD: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat" assumes "rb_aux_inv (bs, ss, p # ps)"   and "Inr j \<in> set ps" shows "sig_of_pair p \<noteq> term_of_pair (0::'a, j)"
LDD_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and d :: "('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" shows "LDD ars r d = (let (\<beta>, \<alpha>, \<sigma>1, \<sigma>2, \<sigma>3, \<tau>1, \<tau>2, \<tau>3) = d in LDD1 ars r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LDD1 ars r \<alpha> \<beta> \<tau>1 \<tau>2 \<tau>3 \<and> lst \<sigma>3 = lst \<tau>3)"
just_show_all_approximating_bigstep_fun_equalities_with_start_Undecided: fixes s :: "state"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" assumes "s = Undecided \<Longrightarrow> approximating_bigstep_fun \<gamma> p rs1 s = approximating_bigstep_fun \<gamma> p rs2 s" shows "approximating_bigstep_fun \<gamma> p rs1 s = approximating_bigstep_fun \<gamma> p rs2 s"
fun_mult_closed: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F"   and "g \<in> carrier F" shows "f \<otimes>\<^bsub>F\<^esub> g \<in> carrier F"
Sretr_sym: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "sym theta" shows "sym (PL_Indis.Sretr aval tval cval indis theta)"
Constrains_imp_subset: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> A Co A'" shows "reachable F \<inter> A \<subseteq> A'"
graph_pred_it_def: fixes \<alpha> :: "'e \<Rightarrow> ('f, 'g) graph"   and invar :: "'e \<Rightarrow> bool"   and pred_list_it :: "'e \<Rightarrow> 'f \<Rightarrow> (('f \<times> 'g) list \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> ('f \<times> 'g) list \<Rightarrow> ('f \<times> 'g) list) \<Rightarrow> ('f \<times> 'g) list \<Rightarrow> ('f \<times> 'g) list" shows "graph_pred_it \<alpha> invar pred_list_it \<equiv> graph \<alpha> invar \<and> graph_pred_it_axioms \<alpha> invar pred_list_it"
alpha_tConj: fixes tset1 :: "('b, 'c, 'd) Tree set['b]"   and tset2 :: "('b, 'c, 'd) Tree set['b]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
\<Phi>_in_hom: fixes f :: "'t" assumes "S'.arr f" shows "\<Psi> f \<in> S.hom (\<Psi>o (S'.dom f)) (\<Psi>o (S'.cod f))"
SeqFormP_Disj: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and x :: "tm"   and y :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, x, y)"   and "atom k \<sharp> (s1, s2, k1, k2, x, y)" shows "{SeqFormP s1 k1 x, SeqFormP s2 k2 y} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) (Q_Disj x y)))"
borel_measurable_vimage_halfspace_component_lt: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
pre_bfs2_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and succ :: "nat \<Rightarrow> nat list nres"   and src :: "nat"   and dst :: "nat" shows "Graph.pre_bfs2 c succ src dst \<equiv> Graph.init_state src \<bind> (\<lambda>s. while\<^sub>T (\<lambda>(f, PRED, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, PRED, C, N, d). assert (C \<noteq> {}) \<bind> (\<lambda>_. IICF_Set.op_set_pick C \<bind> (\<lambda>v. let C = C - {v} in assert (v \<in> Graph.V c) \<bind> (\<lambda>_. succ v \<bind> (\<lambda>sl. Graph.inner_loop2 dst sl v PRED N \<bind> (\<lambda>(f, PRED, N). if f then return (f, PRED, C, N, d + 1) else assert (case (f, PRED, C, N, d) of (f, PRED, C, N, d) \<Rightarrow> \<not> f \<and> nf_invar' c src dst PRED C N d) \<bind> (\<lambda>_. if C = {} then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d) else return (f, PRED, C, N, d)))))))) s \<bind> (\<lambda>(f, PRED, uu_, uu_, d). if f then return (Some (d, PRED)) else return None))"
collect_basis_in_finite: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> \<E>" shows "finite (Collect (\<E>.basis_in \<E>))"
set_self_img_compr: fixes A :: "'b set" shows "{a |a. a \<in> A} = A"
select_convs_5: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getConflictClause \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getConflictClause"
Fn: shows "{f^n \<bottom> |. n} = range (\<lambda>n. f^n \<bottom>)"
assms: shows "run_one_step d i (s, vs, ves, $Block x51 x52) = (s', vs', res)"
box_equals: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a = b"   and "a = c"   and "b = d" shows "c = d"
finite_dimensional_vector_space_ow_def: fixes U\<^sub>V\<^sub>S :: "'d set"   and plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and minus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale basis \<equiv> vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<and> finite_dimensional_vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis"
iii: shows "continuous_on {0..1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
L_def: shows "L = (\<lambda>i. fst (g i))"
abs_rat: fixes a :: "int"   and b :: "int" shows "\<bar>Rat.Fract a b\<bar> = Rat.Fract \<bar>a\<bar> \<bar>b\<bar>"
wf_flatten_INTERSECT: fixes X :: "'b rexp set"   and n :: "nat" assumes "finite X" shows "local.wf n (INTERSECT (sorted_list_of_set X)) = (\<forall>r\<in>X. local.wf n r)"
check_extension_l_dom_err_def: fixes p :: "nat" shows "check_extension_l_dom_err p = SPEC (\<lambda>_. True)"
B: fixes a :: "'a" shows "(1::'a) r\<rightarrow> a \<le> (1::'a)"
map_vals_of_Vals: fixes vs :: "val list" shows "map_vals_of (map Val vs) = \<lfloor>vs\<rfloor>"
Sbis_trace: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and tr :: "(('test, 'atom) com \<times> 'state) list \<times> 'state" assumes "c \<approx>s d"   and "s \<approx> t"   and "finTrace tr"   and "iconfig tr = (c, s)" shows "\<exists>tr'. finTrace tr' \<and> iconfig tr' = (d, t) \<and> fstate tr \<approx> fstate tr' \<and> lengthFT tr' = lengthFT tr"
elims: fixes x :: "'a interval"   and xa :: "'a interval"   and y :: "'a interval" assumes "x - xa = y" obtains lx :: "'a" and ux :: "'a" and ly :: "'a" and uy :: "'a" where "x = Interval lx ux" and "xa = Interval ly uy" and "y = Interval (lx - uy) (ux - ly)"
aux: fixes ts1 :: "(('a \<times> 'b) \<times> int) list"   and ts2 :: "(('a \<times> 'b) \<times> int) list"   and k :: "'a"   and i :: "'b" assumes "distinct (map fst (ts1 @ ts2))" shows "the_default 0 (case map_of ts1 (k, i) of None \<Rightarrow> map_of ts2 (k, i) | Some x \<Rightarrow> Some x) = the_default 0 (map_of ts1 (k, i)) + the_default 0 (map_of ts2 (k, i))"
reversepath_simps_1: fixes g :: "real \<Rightarrow> 'a" shows "path (reversepath g) = path g"
core_open: fixes T :: "'b set set"   and a :: "'b set" assumes "topology T" shows "core\<^bsub>T\<^esub> a open\<^bsub>T\<^esub>"
dbl_dec_simps_5: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one (numeral.numeral one plus k) = numeral.numeral one plus (Num.BitM k)"
LLL_reconstruction_inner_simps: shows "LLL_reconstruction_inner p l gs f u Degs (Suc j') = (if j' \<notin> Degs then None else if p ^ local.ll \<le> \<bar>lead_coeff local.g\<bar> then None else case div_int_poly f (primitive_part local.g) of None \<Rightarrow> None | Some f' \<Rightarrow> Some (filter (\<lambda>gi. \<not> p.dvdm gi (primitive_part local.g)) gs, lead_coeff f', f', primitive_part local.g))"
mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "c \<squnion> d \<le> a \<squnion> b"
scale_ms_aux_def: fixes c :: "real"   and xs :: "('a \<times> real) msllist" shows "scale_ms_aux c xs = scale_shift_ms_aux (const_expansion c, 0) xs"
sorted_wrt_distinct_set_unique: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "antisymp rel"   and "sorted_wrt rel xs"   and "distinct xs"   and "sorted_wrt rel ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
distinct_37: fixes x4 :: "nat"   and x5 :: "'e" shows "\<not>unicast x4 \<noteq> send x5"
gt_antisym: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "\<not> s >\<^sub>t t"
F_def: fixes ds :: "'b set" shows "F ds = (\<lambda>(XD, XH). (F1 (undual XH), dual (F2 ds (F1 (undual XH)))))"
rtranclp_converseD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "r\<inverse>\<inverse>\<^sup>*\<^sup>* x y" shows "r\<^sup>*\<^sup>* y x"
snd_prod_decode: fixes z :: "nat" shows "pdec2 z = Max_triangle z - pdec1 z"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "char list" and D :: "char list" where "a1 = Class Ca__" and "a2 = Class D" and "P \<turnstile> Ca__ \<preceq>\<^sup>* D"   | C :: "char list" where "a1 = NT" and "a2 = Class C"
left_adjoint_is_ide: fixes f :: "'b" assumes "D.is_left_adjoint f" shows "D.ide f"
walk_last_edge: fixes xs :: "'a list"   and ys :: "'a list" assumes "walk (xs @ ys)"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "last xs \<rightarrow> hd ys"
f_2: fixes i :: "nat" shows "f i \<in> borel_measurable M"
sets_restrict_space: fixes M :: "'c measure"   and \<Omega> :: "'c set" shows "sets (restrict_space M \<Omega>) = sets.restricted_space M \<Omega>"
distinct_7: fixes x2 :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool"   and x31 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and x32 :: "'i"   and x33 :: "'j"   and x34 :: "(('g \<Rightarrow> 'h option) \<Rightarrow> 'h option) list" shows "(x2)\<^sub>\<surd> \<noteq> x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34"
restrict_PiE: fixes f :: "'g \<Rightarrow> 'h"   and I :: "'g set"   and S :: "'g \<Rightarrow> 'h set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
PAC_checker_def: fixes spec :: "int mpoly"   and A :: "nat set \<times> (nat, int mpoly) fmap"   and b :: "status"   and st :: "(int mpoly, nat, nat) pac_step list" shows "PAC_checker spec A b st = WHILE\<^sub>T (\<lambda>((b, A), st). \<not> is_failed b \<and> st \<noteq> []) (\<lambda>(bA, st). ASSERT (st \<noteq> []) \<bind> (\<lambda>_. PAC_checker_step spec bA (lead_coeff st) \<bind> (\<lambda>S. RETURN (S, tl st)))) ((b, A), st) \<bind> (\<lambda>(S, uu_). RETURN S)"
gc_join_mmsaux_alt: shows "gc_join_mmsaux args rel1 aux = join_mmsaux args rel1 (gc_mmsaux aux) \<or> gc_join_mmsaux args rel1 aux = join_mmsaux args rel1 aux"
length_append: fixes xs :: "'a list"   and ys :: "'a list" shows "\<parallel>xs @ ys\<parallel> = \<parallel>xs\<parallel> + \<parallel>ys\<parallel>"
assms_1: shows "lve \<equiv> lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit"
u_simps_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.arr V u"
add_le_same_cancel2: fixes a :: "'h multiset"   and b :: "'h multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "(x < xa) = (int_of_integer x < int_of_integer xa)"
simps_2: fixes v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "Heap.multiset (T v l r) = Heap.multiset l + {#v#} + Heap.multiset r"
bigo_plus_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. (0::'b) \<le> f x"   and "\<forall>x. (0::'b) \<le> g x" shows "O(f + g) = O(f) + O(g)"
put_ask: fixes s :: "'s"   and f :: "'r \<Rightarrow> 'm" shows "put s (ask f) = ask (\<lambda>r. put s (f r))"
s'_nonneg: fixes x :: "real" assumes "0 \<le> v"   and "a \<le> 0" shows "0 \<le> s' x"
WT_gpv_inline1_2: fixes out :: "'call'"   and rpv :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and input :: "'ret'" assumes "Inr (out, rpv, rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s"   and "input \<in> responses_\<I> \<I>' out" shows "\<I>' \<turnstile>g rpv input \<surd>"
conjunct1: fixes P :: "bool"   and Q :: "bool" assumes "P \<and> Q" shows "P"
edom_prognosis: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "edom (prognosis ae as a (\<Gamma>, e, S)) \<subseteq> fv \<Gamma> \<union> fv e \<union> fv S"
deg_minus_eq: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q \<noteq> deg R p" shows "deg R (p \<ominus>\<^bsub>UP R\<^esub> q) = max (deg R p) (deg R q)"
enum_ex_sdim_def: fixes P :: "sdim \<Rightarrow> bool" shows "enum_class.enum_ex P = (P Length \<or> P Mass \<or> P Time \<or> P Current \<or> P Temperature \<or> P Amount \<or> P Intensity)"
out_ocircline_cmat_cvec_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" shows "out_ocircline_cmat_cvec H z = (0 < Re (quad_form z H))"
assms_4: shows "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>r result"
diff_right: fixes a :: "'a"   and b :: "'b"   and b' :: "'b" shows "a ** (b - b') = a ** b - a ** b'"
repv_selectlike_other_converse: fixes x :: "variable"   and y :: "variable"   and \<omega> :: "variable \<Rightarrow> real"   and d :: "real"   and X :: "(variable \<Rightarrow> real) set" assumes "x \<noteq> y" shows "(repv \<omega> x d \<in> X) = (repv \<omega> x d \<in> selectlike X \<omega> {y})"
zero_QDelta_def: shows "0 = QDelta 0 0"
SolveExec_def: fixes solve_exec :: "('a \<times> constraint) list \<Rightarrow> 'a list + (nat, rat) mapping" shows "SolveExec solve_exec \<equiv> (\<forall>cs v. solve_exec cs = Inr v \<longrightarrow> \<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)) \<and> (\<forall>cs I. solve_exec cs = Inl I \<longrightarrow> minimal_unsat_core (set I) cs)"
insertion_Var: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "insertion f (Var i) = f i"
p_eq2: shows "int p = 2 * ((int p - 1) div 2) + 1"
assms_5: shows "(\<delta>', c\<^sub>0', c\<^sub>1') = combine_code (\<delta>\<^sub>L', p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R', p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
RF_circ_edge_forward: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set"   and y :: "'a" assumes "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S"   and "edge \<Gamma> x y" shows "y \<in> RF\<^bsub>\<Gamma>\<^esub> S"
set_sum_alt: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "finite I" shows "sum S I = {sum s I |s. \<forall>i\<in>I. s i \<in> S i}"
SdlHPPR2: shows "SdlHPPB d PP R2"
1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
pdevs_val_uminus_pdevs: fixes e :: "nat \<Rightarrow> real"   and x :: "'b pdevs" shows "pdevs_val e (uminus_pdevs x) = - pdevs_val e x"
that: fixes c :: "'a" assumes "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)" shows "thesis"
nc_iff1: fixes x :: "'a" shows "(x \<le> nc) = (x \<sqinter> 1\<^sub>\<pi> = (0::'a))"
eval_primfun_pos: fixes f :: "primfun \<times> real" shows "\<forall>\<^sub>F x in at_top. 0 < eval_primfun f x"
sendmsg_simps_4: fixes P :: "'e \<Rightarrow> bool"   and ip :: "nat" shows "sendmsg P (\<not>unicast ip) = True"
power_inductl: fixes z :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and i :: "nat" assumes "z \<union> x ; y \<subseteq> y" shows "rel_d.power x i ; z \<subseteq> y"
on_curve_infinity: fixes a :: "'b"   and b :: "'b" shows "on_curve a b Infinity"
add_scale_img: fixes a :: "real"   and b :: "real" assumes "a < b" shows "(\<lambda>x. a + (b - a) * x) ` {0..1} = {a..b}"
scount_eq_0_iff: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(scount P \<omega> = 0) = alw (\<lambda>xs. \<not> P xs) \<omega>"
vector_derivative_at_within_non_trivial_limit: fixes x :: "real"   and s :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "at x within s \<noteq> bot \<and> (f has_vector_derivative f') (at x)" shows "vector_derivative f (at x within s) = f'"
plus_adds_pp_mono: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and u :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "s adds t"   and "term_powerprod.adds_pp pair_of_term u v" shows "term_powerprod.adds_pp pair_of_term (s + u) (term_powerprod.splus pair_of_term term_of_pair t v)"
dist_pos: fixes x :: "fbint"   and y :: "fbint" assumes "x \<noteq> y" shows "0 < local.dist x y"
a: shows "\<forall>x. rep_endofun (abs_endofun x) = x"
arr_char: fixes f :: "'a \<times> 'a" shows "VV.arr f = (B.arr (fst f) \<and> B.arr (snd f) \<and> src (fst f) = trg (snd f))"
eq: fixes s :: "'a" assumes "c \<le> s \<bullet> (1::'a)" shows "fds_nth h N = nat_power N s * eval_fds h s - nat_power N s * remainder s"
simps: fixes s :: "'b word"   and e :: "'b word" shows "interval_of (s, e) = {s..e}"
eventually_at_top_dense: fixes P :: "'b \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n>N. P n)"
labels_not_empty: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp"   and p :: "('c, 'd, 'b, 'e) seqp" assumes "wellformed \<Gamma>" shows "labels \<Gamma> p \<noteq> {}"
ZObis_coind: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "theta \<subseteq> ZOretr (theta \<union> ZObis)" shows "theta \<subseteq> ZObis"
distinct_pair_filter_n: fixes xs :: "('b \<times> 'c) list"   and a :: "'b" assumes "distinct (map fst xs)"   and "a \<notin> set (map fst xs)" shows "map snd (filter (\<lambda>(x, y). x = a) xs) = []"
delete_block_fin_incidence_sys: fixes b :: "'a set" shows "finite_incidence_system \<V> (del_block b)"
Gram_Schmidt_column_k_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and a :: "'c" shows "Gram_Schmidt_column_k_row A k a = (\<chi>b. (if b = mod_type_class.from_nat k then column b A - (\<Sum>x\<in>{column i A |i. i < b}. (column b A \<bullet> x / (x \<bullet> x)) *\<^sub>R x) else column b A) $ a)"
encode1_replicate_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and k :: "nat"   and i :: "nat" assumes "grover_state_sig n f dims R K" shows "partial_state.encode1 (replicate (Suc k) 2) {0..<k} i = i mod 2 ^ k"
permutation: fixes x :: "'a permutation" shows "(\<rightarrow>) x \<in> {f. bij f}"
cases: fixes x :: "'c list" obtains "x = []"   | s :: "'c" and ss :: "'c list" where "x = s # ss"
dist_pos_lt: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "0 < dist x y"
assms: fixes y :: "'a" assumes "y \<in> set ys"   and "P y" shows "False"
simps_4: fixes r :: "'b mrexp2"   and s :: "'b mrexp2"   and f :: "bool"   and n :: "bool" shows "ok2 (Plus2 r s f n) = (ok2 r \<and> ok2 s \<and> (let rs = Plus (strip2 r) (strip2 s) in f = final rs \<and> n = nullable rs))"
distinct_4: fixes x31 :: "com"   and x32 :: "com" shows "x31;; x32 \<noteq> Skip"
set_of_map_map_of: fixes xs :: "('a \<times> 'b) list" shows "set_of_map (map_of xs) = set (AList.clearjunk xs)"
Src_VcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Src (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Src u"
c2: shows "c2 = map (\<lambda>d. (i, \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t)) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))"
scast_eq3: shows "sint (SCAST(32 \<rightarrow> 64) w2) = sint w2"
addSubsetSetI_3: fixes t :: "'b set"   and ts :: "'b set set"   and s :: "'b set" assumes "t \<in> ts" shows "t - s \<in> addSubsetSet s ts"
FSbImorph_igWlsInp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'i, 'j, 'k) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'i option"   and h :: "'i \<Rightarrow> 'l"   and hA :: "'j \<Rightarrow> 'm"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'l, 'm, 'n) model_scheme" assumes "igWlsInp MOD delta inp"   and "FSbImorph h hA MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
Var_apps_eq_Var_apps_conv: fixes m :: "nat"   and rs :: "dB list"   and n :: "nat"   and ss :: "dB list" shows "(dB.Var m \<degree>\<degree> rs = dB.Var n \<degree>\<degree> ss) = (m = n \<and> rs = ss)"
list_of_aux_code_2: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a llist" shows "list_of_aux xs (LCons y ys) = list_of_aux (y # xs) ys"
G_succ_def: shows "G_succ \<equiv> wgraph_by_map.succ lookup"
C: shows "C \<in> Chains (relation_of (\<lesssim>) \<S>)"
extend_def: fixes n :: "nat"   and R :: "('e, 'd) labeled_graph \<times> ('e, 'd) labeled_graph"   and f :: "('d \<times> nat) set" shows "extend n R f \<equiv> f \<union> (let V_new = sorted_list_of_set (vertices (snd R) - vertices (fst R)) in set (zip V_new [n..<n + length V_new]))"
ct_eq_commute: fixes x :: "ct"   and y :: "ct" assumes "ct_eq x y" shows "ct_eq y x"