hcju/isabelleps huggingface dataset:
bal_split_last_2: fixes ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ (sub, sep) # rs) t)" shows "height (Node (ls @ (sub, sep) # rs) t) = height (Node (ls @ rs) t)"
has_branch: fixes T :: "tree" shows "\<exists>b. branch b T"
prj_simps_10: shows "C.arr trg_trg_f.Prj\<^sub>1"
preserves_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "Span.trg (SPN \<mu>) = SPN.map\<^sub>0 (trg \<mu>)"
H: shows "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C D"
CLS_hcomp: fixes f :: "'a"   and g :: "'a" assumes "B.ide f"   and "B.ide g"   and "src f = trg g" shows "\<lbrakk>\<lbrakk>f \<star> g\<rbrakk>\<rbrakk> = MkArr (src g) (trg f) (Comp \<lbrakk>f\<rbrakk>\<^sub>B \<lbrakk>g\<rbrakk>\<^sub>B)"
mor2ZF_Op: fixes C :: "('e, 'd, 'f) LSCategory_scheme" shows "mor2ZF (Op C) = mor2ZF C"
redT_updT_None: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and ta :: "('h, 'f, 'i) new_thread_action"   and t :: "'h" assumes "redT_updT ts ta t = None" shows "ts t = None"
is_non_volatile_Write\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_non_volatile_Write\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
covector_inf_comp_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "vector x" shows "(y \<sqinter> x\<^sup>T) * z = y * (x \<sqinter> z)"
respection_of_barbs_and_closures_5: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_respects_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_respects_binary_pred (Rel\<^sup>*) (\<lambda>a. hasBarb a CWB)"
RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: fixes i :: "nat" shows "RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>r\<^sub>p i,Up\<^sub>r\<^sub>p i\<rbrakk>"
collision_def: fixes time_set :: "real set" shows "collision time_set \<equiv> \<exists>t\<in>time_set. ego.s t = other.s t"
del_invc_invh: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c" assumes "invh t"   and "invc t" shows "invh (del x t) \<and> (color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or> color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
exec_gpv_conv_inline1: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and gpv :: "('a, 'c, 'd) gpv"   and s :: "'b" shows "exec_gpv callee gpv s = map_spmf projl (inline1 (\<lambda>s c. lift_spmf (callee s c)) gpv s)"
left_def: shows "left \<equiv> \<lambda>R. R (jumpF f (at_left b)) 0 \<or> continuous (at_left b) f \<and> R (f b) 0"
N_pos: fixes q :: "real"   and n :: "int" assumes "fbnorm q"   and "n \<noteq> 0" shows "0 < fbnorm.N q n"
dsg1: fixes x :: "'a" shows "ans_d x \<cdot> x = x"
det_four_block_mat_lower_left_zero_col: fixes A1 :: "'b mat"   and A2 :: "'b mat"   and n :: "nat"   and A3 :: "'b mat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat 1 1"   and "A2 \<in> carrier_mat 1 n"   and "A3 = 0\<^sub>m n 1"   and "A4 \<in> carrier_mat n n" shows "Determinant.det (four_block_mat A1 A2 A3 A4) = Determinant.det A1 * Determinant.det A4"
opti_iterate: fixes y :: "'a"   and x :: "'a" shows "d y +  |(x \<cdot> ad y)\<^sup>\<star>\<rangle>  |x\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
additive_wp_Abort: shows "Transformers.additive (wp Abort)"
pmdl_image_monic: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl (monic ` B) = pmdl B"
fstI: fixes x :: "'g \<times> 'h"   and y :: "'g"   and z :: "'h" assumes "x = (y, z)" shows "fst x = y"
lc_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.lc (local.punit.monom_mult c t p) = c * punit.lc p"
NEmpt: fixes z :: "NominalSequents.var"   and y :: "NominalSequents.var"   and x :: "NominalSequents.var"   and ys :: "NominalSequents.var list" shows "[z;y](x # ys) = (if x = y then z # [z;y]ys else x # [z;y]ys)"
ide_char: fixes \<mu> :: "'a arrow_of_spans_data" shows "V.ide \<mu> = (arrow_of_spans (\<cdot>) \<mu> \<and> C.ide (Chn \<mu>))"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Can \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Can t \<and> Can u \<and> Can v \<and> Src t = Trg u \<and> Src u = Trg v)"
1: shows "\<guillemotleft>\<r>[f] : f \<star> src f \<Rightarrow> f\<guillemotright>"
outs'_gpv_Out: fixes generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat"   and gpv :: "('d, 'e, 'f) gpv"   and x :: "'e" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "x \<in> generat_outs generat" shows "x \<in> outs'_gpv gpv"
interpretation_subst_exists': fixes X :: "'c set" shows "\<exists>\<theta>. subst_domain \<theta> = X \<and> ground (subst_range \<theta>)"
emeasure_eq_measure: fixes A :: "real set" shows "emeasure (exponential l) A = ennreal (prob A)"
reward_until_simps_1: fixes s :: "'s"   and X :: "'s set"   and \<omega> :: "'s stream" assumes "s \<in> X" shows "reward_until X s \<omega> = 0"
cut_le_Un: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<union> B) \<down>\<le> t = A \<down>\<le> t \<union> B \<down>\<le> t"
trancl_to_fmrel: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e"   and ym :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "(fmrel f)\<^sup>+\<^sup>+ xm ym" shows "fmrel f\<^sup>+\<^sup>+ xm ym"
beh1_def: fixes s1 :: "'f \<Rightarrow> ('d, 'f, 'g) F1"   and s2 :: "'g \<Rightarrow> ('e, 'f, 'g) F2"   and a :: "'f" shows "beh1 s1 s2 a = (\<Union>n. Lev1 s1 s2 n a, Lab1 s1 s2 a)"
succs: fixes g' :: "'a"   and succs :: "'a \<Rightarrow> 'a list"   and g :: "'a"   and g'' :: "'a" assumes "g' \<in> set (succs g)"   and "g' [succs]\<rightarrow>* g''" shows "g [succs]\<rightarrow>* g''"
0: shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
Rep_proj2: fixes x :: "proj2" shows "Rep_proj2 x \<in> non_zero_vectors // proportionality"
sep_disj_addI3: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj x (plus y z)"
case_eq_if: fixes f :: "(('f, 'g) T\<^sub>m, 'g T\<^sub>h, 'h, 'i) F\<^sub>m \<Rightarrow> 'e"   and G\<^sub>m :: "('f, 'g, 'h, 'i) G\<^sub>m" shows "(case G\<^sub>m of G\<^sub>m x \<Rightarrow> f x) = f (the_G\<^sub>m G\<^sub>m)"
gen_Union: fixes I :: "'b set"   and P :: "'b set set" assumes "I \<in> P" shows "gen P I = \<Union> {J \<in> P. J \<subseteq> gen P I}"
rationalizes_on_f_range_on_V_axiom_on_1: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set"   and r :: "('b \<times> 'b) set" assumes "rationalizes_on A f r" shows "f_range_on A f"
S12l: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
simps_2: fixes X :: "(('fun, 'atom) Term.term \<times> nat) list"   and F :: "(('fun, ('fun, 'atom) Term.term \<times> nat) Term.term \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term) list"   and F' :: "(('fun, ('fun, 'atom) Term.term \<times> nat) Term.term \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term) list" shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = (F' = [] \<and> (\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a) \<or> (\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> pair ` set F' \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X)))"
gla_1: fixes x :: "'a"   and y :: "'a" assumes "ad x \<cdot> y = (zero_class.zero::'a)" shows "ad x \<le> ad y"
tendsto_tan: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "cos a \<noteq> (0::'a)" shows "((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
no_ofail_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option" shows "no_ofail P f \<equiv> \<forall>s. P s \<longrightarrow> f s \<noteq> None"
ascii_char_of: fixes c :: "nat" assumes "c < 128" shows "\<not> digit7 (char_of c)"
no_selfloop: fixes L :: "(nat \<times> int \<times> nat) list" assumes "fromlist L" shows "\<forall>u w v. (u, w, v) \<in> set L \<longrightarrow> u \<noteq> v"
matched_same_level_CFG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains as :: "'a list" where "CFG.same_level_path' sourcenode targetnode kind valid_edge get_return_edges (parent_node n) as (parent_node n')"
diff_fun_scaleR: fixes a :: "'a \<Rightarrow> real"   and b :: "'a \<Rightarrow> 'c" assumes "diff_fun k (charts_submanifold W) a"   and "diff_fun k (charts_submanifold W) b" shows "diff_fun k (charts_submanifold W) (\<lambda>x. a x *\<^sub>R b x)"
sortedE: fixes j :: "nat" assumes "j \<le> i" shows "length (xs ! i) \<le> length (xs ! j)"
f_2: shows "integrable (lebesgue_on S) (\<lambda>x. (f x)\<^sup>2)"
true_lit_simps_1: fixes I :: "'b set"   and A :: "'b" shows "(I \<Turnstile>l Pos A) = (A \<in> I)"
ptc: shows "pt TYPE('y \<Rightarrow> 'a) TYPE('x)"
in1: shows "map of_int_hom.vec_hom fs ! i - gs2.gso i - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j \<in> gs.span (gs1.gso ` {0..<i})"
XE1: fixes u :: "'a" assumes "u \<in> N1" shows "(x, u) \<in> (edges T1)\<^sup>*"
atoms_deriv_subset: fixes x :: "'a"   and r :: "'a rexp" shows "atoms (deriv x r) \<subseteq> atoms r"
N: fixes k :: "nat"   and n :: "nat" assumes "N k \<le> n" shows "dist (rep_metric_completion (X k) (N k)) (rep_metric_completion (X k) n) < 1 / real (Suc k)"
cbsub_1: shows "cbox (- b) b \<subseteq> box (- c) c"
real_add_less_0_iff: fixes x :: "real"   and y :: "real" shows "(x + y < 0) = (y < - x)"
prj_simps_5: shows "C.dom trg_f.prj\<^sub>0 = trg.leg0 \<down>\<down> f.leg1"
abs_eq: fixes x :: "dflt_size word" shows "uint16_of_uint (Abs_uint x) = Abs_uint16 (ucast x)"
take0: shows "take 0 = (\<lambda>xs. \<epsilon>)"
decrpoly_normh: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "polybound0 p" shows "isnpolyh (decrpoly p) (n0 - 1)"
collapse: fixes complex :: "complex" shows "Complex (Re complex) (Im complex) = complex"
sub_unit_closed: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "submonoid N M composition unit" shows "unit \<in> N"
wave_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g"   and "wave (quotient_web \<Gamma> f) g" shows "wave \<Gamma> (plus_current f g)"
alw_implies_until: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream"   and \<psi> :: "'b stream \<Rightarrow> bool" assumes "alw \<phi> \<omega>" shows "(\<phi> until \<psi>) \<omega>"
complete_imp_closed: fixes S :: "'c set" assumes "complete S" shows "closed S"
mono_Sup: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "mono f" shows "\<Squnion> (f ` A) \<le> f (\<Squnion> A)"
braun_merge: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "braun (merge l r)"
indRelSTEQ_refl: fixes TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "refl TRel" shows "refl (indRelSTEQ SRel TRel)"
2: shows "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"
functor_is_transformation: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" assumes "functor A B F" shows "natural_transformation A B F F F"
joinI_right: fixes b :: "'b"   and a :: "'b"   and A :: "('b \<times> 'b) set" assumes "(b, a) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<down>"
hd_prog_prog_configs: fixes p :: "'a"   and sb :: "'a memref list" shows "hd_prog p sb = p \<or> hd_prog p sb \<in> prog_configs sb"
pw_econc_iff_3: fixes E :: "('c \<times> 'e) set"   and R :: "('d \<times> 'f) set"   and m :: "('e + 'f) nres" shows "nofail (\<Down>\<^sub>E E R m) = nofail m"
wt_external_call: fixes P :: "'addr J_prog"   and h :: "'heap"   and va :: "'addr extCallRet"   and T :: "ty"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr" assumes "conf_extRet P h va T"   and "P,E,h \<turnstile> e : T" shows "\<exists>T'. P,E,h \<turnstile> extRet2J e va : T' \<and> P \<turnstile> T' \<le> T"
snd_divide_out: fixes p :: "'a"   and x :: "'a" shows "snd (divide_out p x) = multiplicity p x"
exetyp_arity_of_def: fixes exesignature :: "exesignature" shows "exetyp_arity_of exesignature = (case exesignature of ExeSignature x1 x2 x3 \<Rightarrow> x2)"
vfsequence_vcons_nin_vrange: fixes x :: "V" assumes "vdistinct (xs #\<^sub>\<circ> x)" shows "x \<notin>\<^sub>\<circ> \<R>\<^sub>\<circ> xs"
invertible_mat: fixes c :: "'d" assumes "c \<noteq> (0::'d)" shows "invertible (mat c)"
strong_del_block_alt_def: fixes b :: "'a set" shows "str_del_block b = {#bl - b. bl \<in># removeAll_mset b \<B>#}"
simps: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" shows "Lfilled a1 a2 a3 a4 = ((\<exists>vs lholed es' es. a1 = 0 \<and> a2 = lholed \<and> a3 = es \<and> a4 = vs @ es @ es' \<and> const_list vs \<and> lholed = LBase vs es') \<or> (\<exists>vs lholed n es' l es'' k es lfilledk. a1 = k + 1 \<and> a2 = lholed \<and> a3 = es \<and> a4 = vs @ [Label n es' lfilledk] @ es'' \<and> const_list vs \<and> lholed = LRec vs n es' l es'' \<and> Lfilled k l es lfilledk))"
circline_intersection_def: fixes H1 :: "circline"   and H2 :: "circline" shows "circline_intersection H1 H2 = {z. on_circline H1 z \<and> on_circline H2 z}"
inter_comm: fixes l :: "Lines"   and m :: "Lines" shows "Pascal_Property.inter l m = Pascal_Property.inter m l"
composition_series_triv_group: shows "(carrier G = {\<one>\<^bsub>G\<^esub>}) = (\<GG> = [{\<one>\<^bsub>G\<^esub>}])"
lnth_lmap: fixes n :: "nat"   and xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "enat n < llength xs" shows "lmap f xs $ n = f (xs $ n)"
monomial_single_power: fixes x :: "'b"   and k :: "nat"   and c :: "'c"   and n :: "nat" shows "monomial c (monomial k x) ^ n = monomial (c ^ n) (monomial (k * n) x)"
affine_parallel_subspace: fixes S :: "'b set" assumes "affine S"   and "S \<noteq> {}" shows "\<exists>!L. subspace L \<and> affine_parallel S L"
m_closed: fixes R :: "('e, 'f) ring_scheme"   and N :: "'g set"   and M :: "('e, 'g, 'h) module_scheme"   and v :: "'g"   and w :: "'g" assumes "LinearCombinations.submodule R N M"   and "v \<in> N"   and "w \<in> N" shows "v \<oplus>\<^bsub>M\<^esub> w \<in> N"
assms: shows "FI m ps (qs * q) (p * up) uq f"
sum_atLeastAtMost_code: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" shows "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a. (+) (f a)) a b (0::'b)"
subst_to_var_is_var: fixes t :: "('c, 'e) Term.term"   and s :: "'e \<Rightarrow> ('c, 'd) Term.term"   and v :: "'d" assumes "t \<cdot> s = Var v" shows "\<exists>w. t = Var w"
constr_rep_3: fixes x2 :: "ty"   and x1 :: "ty" shows "Rep_ty (x2\<rightarrow>x1) = Arrow_Rep (Rep_ty x2) (Rep_ty x1)"
rfb_set: fixes R :: "('c \<times> 'd) set"   and Y :: "'d set" shows "fb\<^sub>\<R> R Y = {x. \<forall>y. s2p R (x, y) \<longrightarrow> s2p Y y}"
pp_surjective_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.surjective (\<ominus> \<ominus> f) = (\<forall>j. \<exists>i. f (i, j) \<noteq> bot)"
seq_comp_sequential: fixes P :: "'c option process"   and Q :: "'c option process" assumes "sequential P"   and "sequential Q" shows "sequential (P ; Q)"
ide_right: shows "ide r\<^sub>0\<^sup>*"
simps_5: fixes x :: "floatarith"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> {interpret_floatarith a vs..interpret_floatarith b vs})"
pell_valuation_power_nat: fixes z :: "nat \<times> nat"   and n :: "nat" shows "pell_valuation (case pell_power z n of (a, b) \<Rightarrow> (int a, int b)) = pell_valuation (case z of (x, y) \<Rightarrow> (int x, int y)) ^ n"
path: shows "P,C \<turnstile> Cs \<sqsubseteq> Cs' @\<^sub>p Cs"
distinct_8: fixes x51 :: "expr"   and x52 :: "cmd" shows "while (x51) x52 \<noteq> Skip"
sum_union_disjoint': fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "A \<union> B = C" shows "sum g C = sum g A + sum g B"
sup_compl_top_left1: fixes x :: "'a"   and y :: "'a" shows "- x + (x + y) = 1"
simps_3: fixes s :: "dB"   and k :: "nat" shows "lift (Abs s) k = Abs (lift s (k + 1))"
currentLevel_def: fixes t :: "('b \<times> bool) list" shows "currentLevel t = length (markedElements t)"
coeff_pcompose_monom: fixes j :: "nat"   and n :: "nat"   and f :: "'b poly"   and i :: "nat" assumes "j < n" shows "coeff (f \<circ>\<^sub>p monom (1::'b) n) (n * i + j) = (if j = 0 then coeff f i else (0::'b))"
lnth_Suc_LCons: fixes x :: "'c"   and xs :: "'c llist"   and n :: "nat" shows "lnth (LCons x xs) (Suc n) = lnth xs n"
ln_fact_conv_sum_upto: fixes n :: "nat" shows "ln (fact n) = sum_upto (\<lambda>x. ln (real x)) (real n)"
not_le: fixes x :: "'g"   and y :: "'g" shows "(\<not> x \<le> y) = (y < x)"
phiArg_def: fixes g :: "'g"   and v :: "'var \<times> 'node \<times> Def"   and v' :: "'var \<times> 'node \<times> Def" shows "braun_ssa.phiArg g v v' \<equiv> \<exists>vs. braun_ssa.phi g v = Some vs \<and> v' \<in> set vs"
\<F>_suffix: fixes \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" shows "\<F> \<phi> (suffix i w) \<subseteq> \<F> \<phi> w"
linear_compose_sum: fixes S :: "'b set"   and f :: "'b \<Rightarrow> ('c, 'd) vec \<Rightarrow> ('c, 'e) vec" assumes "\<forall>a\<in>S. Vector_Spaces.linear (*s) (*s) (f a)" shows "Vector_Spaces.linear (*s) (*s) (\<lambda>x. \<Sum>a\<in>S. f a x)"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>trg\<^sub>D f : trg\<^sub>D a \<Rightarrow>\<^sub>D trg\<^sub>D b\<guillemotright>"
pol_expr_unique: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>"   and "pol_coeff S c"   and "p = polyn_expr R X (fst c) c"   and "snd c (fst c) \<noteq> \<zero>\<^bsub>S\<^esub>"   and "pol_coeff S d"   and "p = polyn_expr R X (fst d) d"   and "snd d (fst d) \<noteq> \<zero>\<^bsub>S\<^esub>" shows "fst c = fst d \<and> (\<forall>j\<le>fst c. snd c j = snd d j)"
mult_not_zero: fixes a :: "'b hrel"   and b :: "'b hrel" assumes "\<not> a ;; b = false" shows "\<not> a = false \<and> \<not> b = false"
trivial_ker_imp_inj: assumes "kernel G H h = {\<one>}" shows "inj_on h (carrier G)"
hoare_r_def: fixes p :: "'a upred"   and Q :: "'a hrel"   and r :: "'a upred" shows "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u = ((\<lceil>p\<rceil>\<^sub>< \<Rightarrow> \<lceil>r\<rceil>\<^sub>>) \<sqsubseteq> Q)"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "seq \<nu> \<mu>"   and "seq \<tau> \<sigma>" shows "\<nu> \<cdot> \<mu> \<star> \<tau> \<cdot> \<sigma> = (\<nu> \<star> \<tau>) \<cdot> (\<mu> \<star> \<sigma>)"
intro: fixes P :: "'a \<Rightarrow> bool"   and lookup :: "'b \<Rightarrow> ('a, 'c option) state"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> ('a, unit) state"   and cnt :: "'b \<Rightarrow> bool"   and nxt :: "'b \<Rightarrow> 'b"   and sizef :: "'b \<Rightarrow> nat"   and empty :: "'a" assumes "dp_consistency_iterator P lookup update cnt nxt sizef"   and "dp_consistency_empty lookup update P empty" shows "dp_consistency_iterator_empty P lookup update cnt nxt sizef empty"
Vfrom_mono: fixes i :: "V"   and a :: "V"   and b :: "V"   and j :: "V" assumes "Ord i"   and "a \<le> b"   and "i \<le> j" shows "Vfrom a i \<le> Vfrom b j"
older_seniors_card_le: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n) < card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n)) = (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n \<subset> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n)"
cr_view_Grp: shows "cr_view = Grp UNIV to_view"
distinct_5: fixes x11 :: "'a"   and x12 :: "'b"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x11 \<leftarrow> x12 \<noteq> x41 ;; x42"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C FG.map FG.cmp I\<^sub>C.map I\<^sub>C.cmp \<epsilon>\<^sub>0 \<epsilon>\<^sub>1"
Inf_between_def: fixes N :: "'f set"   and M :: "'f set" shows "Inf_between N M = Inf_from (N \<union> M) - Inf_from (N - M)"
in_arcs_single_per_vert: fixes v :: "'a"   and u :: "'a"   and e1 :: "'b"   and e2 :: "'b" assumes "v \<in> verts G"   and "u \<in> verts G"   and "e1 \<in> in_arcs G v"   and "e2 \<in> in_arcs G v"   and "tail G e1 = u"   and "tail G e2 = u" shows "e1 = e2"
shift_commute: fixes e :: "nat \<Rightarrow> 'a"   and i :: "nat"   and U :: "'a"   and T :: "'a" shows "e\<langle>i:U\<rangle>\<langle>0:T\<rangle> = e\<langle>0:T\<rangle>\<langle>Suc i:U\<rangle>"
summable_offset: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "summable (\<lambda>n. f (n + k))" shows "summable f"
ospec_alt: fixes m :: "'b option"   and P :: "'b \<Rightarrow> bool" shows "ospec m P = (case m of None \<Rightarrow> False | Some x \<Rightarrow> P x)"
homf_def: fixes A :: "'o" shows "Hom(A,_) \<equiv> \<lparr>om = restrict (hom AA A) (Ob), am = \<lambda>f\<in>Ar. \<lparr>set_dom = Hom A (Dom f), set_func = restrict ((\<bullet>) f) (Hom A (Dom f)), set_cod = Hom A (Cod f)\<rparr>\<rparr>"
real_exp_eq: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and w :: "bool stream" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "real_cond_exp N (stoch_proc_filt N geom_proc borel n) der w = expl_cond_expect N (proj_stoch_proc geom_proc n) der w"
hyps_1: shows "((\<sigma>, k), \<sigma>', k') \<in> cp"
1: shows "e * - (1::'a) * e\<^sup>T \<le> bot"
mod_mult_cong: fixes a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "a mod c = a' mod c"   and "b mod c = b' mod c" shows "a * b mod c = a' * b' mod c"
assms_2: fixes n :: "nat" shows "almost_full_on (adds) {x. d x \<le> n}"
assms_3: shows "m3 \<in> carrier_mat d2 d2"
algebraic_int_sqrt: fixes x :: "real" assumes "algebraic_int x" shows "algebraic_int (sqrt x)"