hcju/isabelleps huggingface dataset:
slice_kinds_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and S :: "'b set"   and as :: "'a list" assumes "BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice" shows "BackwardSlice.slice_kinds sourcenode targetnode kind valid_edge backward_slice S as \<equiv> map (BackwardSlice.slice_kind sourcenode targetnode kind valid_edge backward_slice S) as"
AssertionStatEq_def: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<simeq> \<Psi>' \<equiv> \<Psi> \<hookrightarrow> \<Psi>' \<and> \<Psi>' \<hookrightarrow> \<Psi>"
Cs_def: shows "Cs \<equiv> image_mset f_i Ds"
acyclic_6_1_injectively_orientable: fixes x :: "'a" shows "acyclic_6 x = (acyclic_1 x \<and> injectively_orientable x)"
goodBinp_def: fixes binp :: "'c \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option" shows "goodBinp binp \<equiv> liftAll goodAbs binp \<and> |{i. binp i \<noteq> None}| <o |UNIV|"
countable_dense_exists: shows "\<exists>D. countable D \<and> (\<forall>X. p X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d\<in>D. d \<in> X))"
b_holds_only_current: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and tk :: "nat" assumes "B.reachable (c, n, ts)"   and "t < N"   and "locks_ticket (ts t) tk" shows "tk = c"
good_item_simps_22: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xa :: "'var"   and xsa :: "'varSort" assumes "good X"   and "goodEnv rhoa" shows "goodEnv (rhoa [xa \<leftarrow> X]_xsa)"
word_and_le1: fixes y :: "'b word"   and a :: "'b word" shows "y && a \<le> a"
semiring_norm_80: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m < num.Bit1 n) = (m < n)"
fixp_spmf_parametric: fixes F :: "('a \<Rightarrow> 'b spmf) \<Rightarrow> 'a \<Rightarrow> 'b spmf"   and G :: "('c \<Rightarrow> 'd spmf) \<Rightarrow> 'c \<Rightarrow> 'd spmf"   and A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. spmf.mono_body (\<lambda>f. F f x)"   and "\<And>x. spmf.mono_body (\<lambda>f. G f x)"   and "((A ===> rel_spmf R) ===> A ===> rel_spmf R) F G" shows "(A ===> rel_spmf R) (spmf.fixp_fun F) (spmf.fixp_fun G)"
comp_red_monic_basis_is_auto_reduced: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "is_auto_reduced (set (comp_red_monic_basis xs))"
minim_Under_under: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> Field r" shows "local.Under A = local.under (minim A)"
length_Lev2: fixes xa :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and x :: "'e" assumes "xa \<in> Lev2 s1 s2 n x" shows "length xa = n"
disjoint_Nv2_inter_Delta2_inter_E1: shows "N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {}"
mismatchCases: fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual"   and F :: "name \<Rightarrow> name \<Rightarrow> bool" assumes "[a\<noteq>b]P \<longmapsto> Rs"   and "\<lbrakk>P \<longmapsto> Rs; a \<noteq> b\<rbrakk> \<Longrightarrow> F a b" shows "F a b"
IH: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "B \<in> set (tr\<^sub>p\<^sub>c S D)" shows "proj n B \<in> set (tr\<^sub>p\<^sub>c (proj n S) (proj n D))"
e2ennreal_infty: shows "e2ennreal \<infinity> = \<top>"
complete_sc_in_Runs: 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" assumes "sc_completion s vs"   and "\<And>vs ln. ta_seq_consist P vs (llist_of (convert_RA ln))" shows "mthr.Runs s (complete_sc s vs)"
rHom_tOp: fixes A :: "('e, 'f) Ring_scheme"   and R :: "('g, 'h) Ring_scheme"   and x :: "'e"   and y :: "'e"   and f :: "'e \<Rightarrow> 'g" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "f \<in> rHom A R" shows "f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y"
dom_ran_disj_comp: fixes R :: "('a \<times> 'a) set" assumes "Domain R \<inter> Range R = {}" shows "R O R = {}"
assms_2: shows "cos_aform_err p X0 = Some Y"
cnj_minus: fixes x :: "octo" shows "Octonions.cnj (- x) = - Octonions.cnj x"
func_plus: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f + g = (\<lambda>x. f x + g x)"
assms: fixes i :: "'a"   and j :: "'b" assumes "i \<in> CBasis"   and "j \<in> CBasis" shows "((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
oalist_inv_raw_tl: fixes xs :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (tl xs)"
P_def: shows "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0))"
6: shows "\<Union> {if xa \<in> Fvars \<phi> - {x} \<union> f t x then FvarsT ta else {} |ta xa. (ta, xa) \<in> set txs} = \<Union> {f ta xa |ta xa. (ta, xa) \<in> set txs}"
subgroup_self: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "subgroup (carrier G) G"
invariant: fixes a :: "'a"   and b :: "'a" assumes "Q a"   and "a \<rightarrow> b" shows "Q b"
vs_type_all_def: shows "vs_type_all \<equiv> \<lambda>uu uua. (case uua of (ad, al) \<Rightarrow> \<lambda>P. {v. \<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T}) uu"
distinct_3: fixes x3 :: "tnam" shows "Object' \<noteq> TName x3"
simps_5: fixes r :: "nat \<times> char list list" shows "typeof (Ref r) = None"
prv_imp_cnjR: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (cnj \<phi> \<chi>) \<chi>)"
star_inductl_eq: fixes z :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "sup_prod z (times_prod' x y) = y" shows "less_eq_prod (times_prod' (prod.qstar x) z) y"
rel_gpv''_mono: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R' :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "A \<le> A'"   and "C \<le> C'"   and "R' \<le> R" shows "rel_gpv'' A C R \<le> rel_gpv'' A' C' R'"
espclosure_DA_delta_is_steps: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and w :: "'d list"   and Q :: "'c set" shows "(eps A)\<^sup>* `` DA.delta (nae2da A) w Q = NAe.steps A w `` Q"
mono_D_syn: fixes S :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "D (S \<lbrakk>A\<rbrakk> Q) = D (Q \<lbrakk>A\<rbrakk> S)"
R27_bc_1: shows "pmf (sds R27) b = 0"
fmrel_to_subtuple: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xm :: "'c \<rightharpoonup>\<^sub>f 'a"   and ym :: "'c \<rightharpoonup>\<^sub>f 'b" assumes "fmrel r xm ym" shows "subtuple r xm ym"
monom_mult_zero_left: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and t :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'e" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair (0::'e) t p = 0"
Cod_F: shows "(\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Cod\<rparr> \<lparr>f, f', f''\<rparr>\<^sub>\<bullet> = B"
uv: fixes x :: "complex" assumes "x \<in> closed_segment a b" shows "v < u \<bullet> x"
borel_measurable_sum: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
cases: fixes a :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" assumes "local.wf a" obtains t :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" where "a = t" and "snd (tree.root t) \<in> R" and "effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t)" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> local.wf t'"
arrI: fixes f1 :: "'j"   and f2 :: "'a" assumes "J.arr f1"   and "A.arr f2" shows "JxA.arr (f1, f2)"
endpoint_in_S_1: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S" shows "endpoint_in_S a b \<in> S"
closed_halfspace_component_le: fixes i :: "'b"   and a :: "real" shows "closed {x. x \<bullet> i \<le> a}"
Snd_def: fixes q :: "ZF" shows "Snd q \<equiv> SOME y. \<exists>x. q = Opair x y"
comp_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "g \<in> B \<rightarrow> C"   and "inj_on g B" shows "inj_on (compose A g f) A"
sdrop_all: fixes xs :: "[:'a:]"   and i :: "Integer" assumes "le\<cdot>(slength\<cdot>xs)\<cdot>i = TT" shows "sdrop\<cdot>i\<cdot>xs = [::]"
octo_of_real_times_left_commute: fixes r :: "real"   and x :: "octo"   and q :: "octo" shows "octo_of_real r * (x * q) = x * (octo_of_real r * q)"
no_free_return: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Return], Undecided\<rangle> \<Rightarrow> t"   and "matches \<gamma> m p" shows "False"
gval_foldr_gOr_false: fixes l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "(gval (fold gOr l g) s = false) = (\<forall>g'\<in>set (g # l). gval g' s = false)"
subst_def: shows "subst \<equiv> \<lambda>x y s. lam_rec (\<lambda>x. if x = y then s else VAR x) (\<lambda>t\<^sub>1 t\<^sub>2. APP) (\<lambda>x t. LAM x) NUM (\<lambda>t\<^sub>1 t\<^sub>2. DIFF) (\<lambda>t\<^sub>1 t\<^sub>2. PLUS) TRUE FALSE (\<lambda>t1 t2 t3. IF) (\<lambda>x t. FIX x) (\<lambda>t. ZET) (\<lambda>t1 t2. EQI) x"
assms_3: fixes X :: "'a set" assumes "X \<in> \<F>" shows "openin T X"
check_del_l_def: fixes spec :: "'a"   and A :: "'b"   and p :: "nat" shows "check_del_l spec A p = RETURN CSUCCESS"
ground_on_def: fixes V :: "'b set"   and \<sigma> :: "('b \<times> 'b trm) list" shows "ground_on V \<sigma> = (\<forall>x\<in>V. ground_term (Var x \<lhd> \<sigma>))"
group_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and S :: "'a set" assumes "Group.group G" shows "Group.group (subgroup_generated G S)"
insertI1: fixes a :: "'b"   and B :: "'b set" shows "a \<in> a \<triangleright> B"
simps_3: fixes v :: "bool" shows "unbox_num (DBool v) = None"
NF_E: fixes a :: "'b"   and A :: "('b \<times> 'b) set"   and b :: "'b" assumes "a \<in> NF A" obtains "(a, b) \<notin> A"
po_on_imp_irreflp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "po_on P A" shows "irreflp_on P A"
set_return_pmf: fixes x :: "'c" shows "set_pmf (return_pmf x) = {x}"
simps_1: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a"   and v :: "(real, 'a) vec" shows "frechet I (Var x) v = (\<lambda>v'. v' \<bullet> axis x 1)"
exhaust: fixes y :: "'a option" obtains "y = None"   | x2 :: "'a" where "y = pure x2"
weakenpre: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"   and "\<forall>s. P s \<le> P' s" shows "\<turnstile>\<^sub>2\<^sub>' {P'} c {Q}"
assms_5: fixes i :: "'a" assumes "i \<in> A" shows "\<exists>x\<in>B. f i \<le>\<^sub>o\<^sub>w g x"
consequence_relation_axioms: shows "consequence_relation Bot (\<Turnstile>)"
covector_inf_comp_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "vector x" shows "y * (x \<sqinter> z) = (y \<sqinter> x\<^sup>T) * (x \<sqinter> z)"
walk_edges_decomp: fixes xs :: "'a list"   and x :: "'a"   and xs' :: "'a list" shows "H.edges_of_walk (xs @ x # xs') = H.edges_of_walk (xs @ [x]) \<union> H.edges_of_walk (x # xs')"
istopology_def: fixes L :: "'c set \<Rightarrow> bool" shows "istopology L \<equiv> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>\<K>. (\<forall>K\<in>\<K>. L K) \<longrightarrow> L (\<Union> \<K>))"
integral_empty: fixes f :: "'b \<Rightarrow> 'a" shows "integral {} f = (0::'a)"
set_iterator_product_correct: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_a :: "'e set"   and it_b :: "'e \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_b :: "'e \<Rightarrow> 'f set" assumes "set_iterator it_a S_a"   and "\<And>a. a \<in> S_a \<Longrightarrow> set_iterator (it_b a) (S_b a)" shows "set_iterator (set_iterator_product it_a it_b) (Sigma S_a S_b)"
inj_on_iff_eq_card: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
algebraic_mult_cancel_left: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "x \<noteq> (0::'b)" shows "algebraic (x * y) = algebraic y"
formulaEntailsLiteralSubset: fixes formula :: "Literal list list"   and literal :: "Literal"   and formula' :: "Literal list list" assumes "formulaEntailsLiteral formula literal"   and "\<forall>c. c el formula \<longrightarrow> c el formula'" shows "formulaEntailsLiteral formula' literal"
iso_digraph_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<exists>\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
InfinitesimalI: fixes x :: "'a star" assumes "\<And>r. \<lbrakk>r \<in> \<real>; 0 < r\<rbrakk> \<Longrightarrow> hnorm x < r" shows "x \<in> Infinitesimal"
collision_excluded: fixes T :: "real set"   and S :: "real set" assumes "\<And>t. t \<in> T \<Longrightarrow> ego.s t \<noteq> other.s t" shows "collision S = collision (S - T)"
C0_is: shows "prems_of \<iota> ! j0 = (C0, active)"
less_path_\<mu>: shows "ereal (awalk_cost f p) < (INF p\<in>{p. apath u p v}. ereal (awalk_cost f p))"
left_neutral: fixes a :: "'b hrel" shows "false \<sqinter> a = a"
assms_1: shows "AE x in M. f x \<le> ennreal c"
wfp_onE_min: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and Q :: "'b set" assumes "wfp_on r A"   and "x \<in> Q"   and "Q \<subseteq> A" obtains z :: "'b" where "z \<in> Q" and "\<And>y. r y z \<Longrightarrow> y \<notin> Q"
zero_one_infty_not_equal_3: shows "0\<^sub>h \<noteq> 1\<^sub>h"
CUT_SNa: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and x :: "name"   and N :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>"   and "x:N \<in> \<parallel>(B)\<parallel>" shows "SNa (Cut <a>.M x.N)"
mem_not_sym: fixes x :: "V"   and y :: "V" shows "\<not> (x \<in>\<^sub>\<circ> y \<and> y \<in>\<^sub>\<circ> x)"
term_variants_pred_fv_eq: fixes P :: "'a \<Rightarrow> 'a list"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "term_variants_pred P s t" shows "fv s = fv t"
vimage_snd: fixes A :: "'c set" shows "snd -` A = UNIV \<times> A"
ga_step_alt: shows "ga_step = rel_of_enex (ga_en, ga_ex)"
assms: shows "l\<^sub>1\<^sup>c = l\<^sub>2\<^sup>c"
ces_append: fixes v1 :: "'c \<times> nat"   and es1 :: "('c \<times> nat) edge list"   and es2 :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.ces v1 (es1 @ es2) v2 subs = (\<exists>v. SubRel.ces v1 es1 v subs \<and> SubRel.ces v es2 v2 subs)"
onechain_boundary1: shows "boundary_chain (one_chain1 - ps1)"
h': shows "h2 \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"
Red1AAcc: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and v :: "'addr val"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and xs :: "'addr val list" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "heap_read h a (ACell (nat (sint i))) v" shows "uf,P,t \<turnstile>1 \<langle>addr a\<lfloor>Val (Intg i)\<rceil>,(h, xs)\<rangle> -\<lbrace>ReadMem a (ACell (nat (sint i))) v\<rbrace>\<rightarrow> \<langle>Val v,(h, xs)\<rangle>"
image_mset_Diff: fixes B :: "'c multiset"   and A :: "'c multiset"   and f :: "'c \<Rightarrow> 'd" assumes "B \<subseteq># A" shows "image_mset f (A - B) = image_mset f A - image_mset f B"
suffix_comp: fixes f :: "'c \<Rightarrow> 'b"   and k :: "nat"   and w :: "nat \<Rightarrow> 'c" shows "(\<lambda>i. f (suffix k w i)) = suffix k (f \<circ> w)"
continuous_on_compact_bound: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact A"   and "continuous_on A f" obtains B :: "real" where "0 \<le> B" and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> B"
preserves_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" shows "G (g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f) = G g \<cdot>\<^sub>B G f"
set_gen2: fixes A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" shows "set (gen2 A B m n) = {(x, y). length x = m \<and> length y = n \<and> (\<forall>i<m. x ! i \<le> A) \<and> (\<forall>j<n. y ! j \<le> B)}"
s_cond_def_2: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> delayed_safe_distance.other.t_stop" shows "delayed_safe_distance.other.s t = delayed_safe_distance.other.p t"
red_simps_4: fixes t :: "'c local.term" assumes "Ide t" shows "Cod (t\<^bold>\<down>) = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
boundl_def: fixes x :: "nat list"   and y :: "nat list" shows "boundl x y = (\<forall>i<m. x ! i \<le> max_x y i)"
assms_2: shows "\<exists>bs. nfa_reach A (nfa_startnode A) bs q1 \<and> list_all (is_alph n) bs"
bij_enumerate: fixes S :: "nat set" assumes "infinite S" shows "bij_betw (wellorder_class.enumerate S) UNIV S"
collapse_1: fixes sum :: "'d + 'e" assumes "isl sum" shows "Inl (projl sum) = sum"
elementary_unions_intervals: fixes f :: "'a set set" assumes "finite f"   and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a b" obtains p :: "'a set set" where "p division_of \<Union> f"
lappend_eq_lappend_conv: fixes xs :: "'b llist"   and us :: "'b llist"   and ys :: "'b llist"   and vs :: "'b llist" assumes "llength xs = llength us" shows "(lappend xs ys = lappend us vs) = (xs = us \<and> (lfinite xs \<longrightarrow> ys = vs))"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "(\<Sum>d | d dvd n. \<chi> d) = (\<Prod>p\<in>prime_factors n. \<Sum>d | d dvd p ^ multiplicity p n. \<chi> d)"
ISTEP: fixes s :: "'a" assumes "RETURN s \<le> I"   and "b s" shows "f s \<le> I"
distinct_2: fixes x2 :: "'c" shows "Some x2 \<noteq> None"
lemma_R_for_Lim_effective: shows "\<exists>g\<in>\<R>. \<forall>i. \<phi> (the (g i)) \<in> \<R> \<and> (\<forall>U \<psi>. learn_lim \<psi> U (\<phi> i) \<longrightarrow> learn_lim \<psi> U (\<phi> (the (g i))))"
simps_13: fixes var :: "nat"   and i :: "nat"   and uy :: "atom fm" shows "grab_atoms var (ExN i uy) = None"
galois_def: fixes l :: "'a \<Rightarrow> 'a"   and u :: "'a \<Rightarrow> 'a" shows "galois l u \<equiv> \<forall>x y. (l x \<le> y) = (x \<le> u y)"
emeasure_subadditive_finite: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "A ` I \<subseteq> sets M" shows "emeasure M (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
lookup_update': fixes k :: "'e"   and v :: "'d"   and m :: "('e, 'd) mapping"   and k' :: "'e" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
irreducible_imp_prime_elem_gcd: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x" shows "comm_semiring_1.prime_elem times one zero x"
Morera_local_triangle: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "\<And>z. z \<in> S \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and> (\<forall>a b c. convex hull {a, b, c} \<subseteq> t \<longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)" shows "f analytic_on S"
zero_local_ring_at_def: shows "pi.zero_local_ring_at \<equiv> local.zero_rel"
star_class_defs_5: shows "uminus \<equiv> *f* uminus"
GodDefsAreEquivalent: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w) = (\<forall>xa. \<P> xa w = xa x w))\<rfloor>"
bet2_le2__le_R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and P :: "'a"   and b :: "'a"   and A :: "'a"   and Q :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet a P b"   and "Bet A Q B"   and "Tarski_neutral_dimensionless.Le Bet Cong P a Q A"   and "Tarski_neutral_dimensionless.Le Bet Cong P b Q B"   and "B = Q" shows "Tarski_neutral_dimensionless.Le Bet Cong a b A B"
fresh_bool: fixes a :: "'a"   and b :: "bool" shows "a \<sharp> b"
assms_2: shows "(I, J) = getIJ Q_pos Q_neg V"
o_id: fixes f :: "'a \<Rightarrow> 'b" shows "f \<circ> id = f"
limit_in_range: fixes r :: "nat \<Rightarrow> 'a" shows "limit r \<subseteq> range r"
IN: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>"   and "edge \<Delta> x y" shows "antiparallel_edges.edg \<Delta> (antiparallel_edges.Edge x y) (antiparallel_edges.Vertex y)"
r_maxlt: fixes e :: "nat" shows "eval r_maxlt [e] \<down>= (GREATEST y. triangle y \<le> e)"
SUCC_def: fixes x :: "tm" shows "SUCC x \<equiv> Eats x x"
blinfun_of_matrix_works: fixes f :: "'b \<Rightarrow>\<^sub>L 'c" shows "blinfun_of_matrix (\<lambda>i j. blinfun_apply f j \<bullet> i) = f"
Main_no_call_target: 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 a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and f :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>Main\<^esub>f" shows "False"
simps_8: fixes f1 :: "'d"   and f2 :: "'d"   and f3 :: "'e \<Rightarrow> 'e ifex \<Rightarrow> 'e ifex \<Rightarrow> 'd" shows "(case Trueif of Trueif \<Rightarrow> f1 | Falseif \<Rightarrow> f2 | IF x xa xb \<Rightarrow> f3 x xa xb) = f1"
assms_2: shows "pA \<in> io_targets A qA w"