hcju/isabelleps huggingface dataset:
that: fixes u :: "'a \<Rightarrow> real"   and e :: "real" assumes "s (flow0 x (u x)) = 0"   and "u x = t"   and "\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0"   and "continuous_on (cball x e) u"   and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"   and "0 < e"   and "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)))) o\<^sub>L (Ds (flow0 x t) o\<^sub>L flowderiv x t) o\<^sub>L embed1_blinfun)) (at x)" shows "thesis"
nat_pow_Suc: fixes n :: "nat"   and x :: "nat \<Rightarrow> int" shows "[Suc n] \<cdot> x = [n] \<cdot> x \<oplus> x"
these_image_Some_eq: fixes A :: "'a set" shows "Option.these (Some ` A) = A"
matrix_inv_unique: fixes A :: "(('c, 'd) vec, 'd) vec"   and B :: "(('c, 'd) vec, 'd) vec" assumes "A ** B = mat (1::'c)"   and "B ** A = mat (1::'c)" shows "matrix_inv A = B"
cast_ok_def: fixes P :: "'a prog"   and C :: "char list"   and h :: "heap"   and v :: "val" shows "cast_ok P C h v \<equiv> v = Null \<or> P \<turnstile> cname_of h (the_Addr v) \<preceq>\<^sup>* C"
equal: fixes equal :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
castOp_simps_3: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wlsBinp (fst3 (castOp s X)) (trd3 (castOp s X))"
less_sfirstD: fixes n :: "nat"   and P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "enat n < sfirst P \<omega>" shows "\<not> P (sdrop n \<omega>)"
le_inf_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> y \<cdot> z) = (x \<le> y \<and> x \<le> z)"
cut_greater_Min_all: fixes t :: "'a"   and I :: "'a set" assumes "t < iMin I" shows "I \<down>> t = I"
tdev_slist_sum: fixes xs :: "(nat, 'b) slist" shows "tdev_slist xs = sum (abs \<circ> snd) (set_slist xs)"
euler_genus_rev_swap: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d"   and b :: "'d" assumes "digraph_map G M"   and "arc_to_ends G a = arc_to_ends G b"   and "{a, b} \<subseteq> arcs G" shows "pre_digraph_map.euler_genus G (pre_digraph_map.rev_swap M a b) = pre_digraph_map.euler_genus G M"
adjacent_no_loop: fixes v :: "'v"   and v' :: "'v" assumes "adjacent v v'" shows "v \<noteq> v'"
R2_def: fixes x :: "nat"   and y :: "nat" shows "R2 x y = TI \<bind> (\<lambda>((c1, d1), c2, d2). let e2 = (x + c1) mod q; e1 = (y + (q - c2)) mod q; s1 = (x * e1 + (q - d1)) mod q; s2 = (e2 * c2 + (q - d2)) mod q in return_spmf ((y, c2, d2, e2), s1, s2))"
fst_snd_id_lens: shows "fst\<^sub>L +\<^sub>L snd\<^sub>L = 1\<^sub>L"
K_def: shows "K = (\<lambda>(t, x). distr (exponential (escape_rate x)) borel ((+) t) \<Otimes>\<^sub>M measure_pmf (J x))"
A_def: fixes j :: "nat" shows "A j = {x \<in> S. f x \<le> (real j - 1 / 3) * e}"
if_eval_elim': fixes e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stmt.If e c\<^sub>1 c\<^sub>2, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "(c' = c\<^sub>1 \<and> ev\<^sub>B mem e \<or> c' = c\<^sub>2 \<and> \<not> ev\<^sub>B mem e) \<and> mds' = mds \<and> mem' = mem"
override_on_apply_notin: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a \<notin> A" shows "(f ++\<^bsub>A\<^esub> g) a = f a"
cnf_form_of_cnf: fixes S :: "'b literal list list" shows "cnf (form_of_cnf S) = set (map set S)"
bisimParPresSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> R \<parallel> P \<sim> R \<parallel> Q"
sifum_types_axioms: shows "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"
poincare_map_has_derivative_step: fixes P :: "'a set"   and D :: "'a \<Rightarrow>\<^sub>L 'a"   and x0 :: "'a"   and h :: "real"   and S :: "'a set" assumes "(poincare_map P has_derivative blinfun_apply D) (at (flow0 x0 h))"   and "returns_to P x0"   and "continuous (at x0 within S) (return_time P)"   and "0 \<le> h"   and "h < return_time P x0"   and "closed P"   and "x0 \<in> S" shows "(poincare_map P has_derivative blinfun_apply (D o\<^sub>L Dflow x0 h)) (at x0 within S)"
nonzero_const_is_nonzero: fixes k :: "real" assumes "k \<noteq> 0" shows "Const k \<noteq> 0"
smcf_ArrMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> smc_0\<lparr>Arr\<rparr>"
g: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_network \<Delta> g" shows "flow \<Delta> g"
join_env_restr_UNIV: fixes S1 :: "'c set"   and S2 :: "'c set"   and m :: "'c \<Rightarrow> 'd" assumes "S1 \<union> S2 = UNIV" shows "m f|` S1 \<squnion> m f|` S2 = m"
assms_3: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> theNFTBoth cl dl"   and "I \<in> P n"   and "II = local.shift cl n I"   and "II \<in> UNpart1 cl dl P" shows "phi"
add_diff_inverse: fixes a :: "'k"   and b :: "'k" assumes "\<not> a < b" shows "b + (a - b) = a"
induced_subgraph_refl: shows "induced_subgraph (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
simps_1: fixes s :: "('f, 'g) locks \<times> (('g \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'h) \<times> ('g \<Rightarrow> 'i wait_set_status option) \<times> 'g set"   and t :: "'g"   and T :: "'g" shows "cond_action_ok s t (Join T) = (case thr s T of None \<Rightarrow> True | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> t \<noteq> T \<and> final x \<and> ln = no_wait_locks \<and> wset s T = None)"
card_of_Plus_ordLeq_infinite_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and B :: "'c set" assumes "infinite (Field r)"   and "|A| \<le>o r"   and "|B| \<le>o r"   and "Card_order r" shows "|A <+> B| \<le>o r"
NIC_def: fixes t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" shows "NIC t1 t2 \<equiv> \<exists>u1 u2. Some u1 = NEXT t1 \<and> Some u2 = NEXT t2 \<and> get_S (cpu_reg_val PSR u1) = 0 \<and> get_S (cpu_reg_val PSR u2) = 0 \<and> low_equal u1 u2"
Infinitesimal_SReal_divide: fixes x :: "real star"   and y :: "real star" assumes "x \<in> Infinitesimal"   and "y \<in> \<real>" shows "x / y \<in> Infinitesimal"
counit_simps_5: shows "S.trg (S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g)) = S.trg (S.UP f)"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>P, \<alpha>\<cdot>(\<tau>.Q)) \<in> Rel"
Reverse_def: fixes fsm :: "('b, 'c) fsm_scheme" shows "Reverse fsm = \<lparr>states = states fsm, init = final fsm, final = init fsm, next = \<lambda>st x st'. next fsm st' x st\<rparr>"
runit'_naturality: fixes f :: "'d" assumes "D.arr f" shows "(f \<otimes>\<^sub>D \<I>\<^sub>D) \<cdot>\<^sub>D D.runit' (D.dom f) = D.runit' (D.cod f) \<cdot>\<^sub>D f"
top_right_normalize: fixes a :: "'a" shows "lcm a (1::'a) = normalize a"
tensor_univ_propTr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme"   and Z :: "('j, 'a, 'h) Module_scheme"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "R module Z"   and "bilinear_map f R M N Z" shows "\<exists>g. g \<in> mHom R P Z \<and> compose (M \<times>\<^sub>c N) g (tau R M N P) = f"
and_or_or_s: fixes w1 :: "word32"   and w2 :: "word32"   and w3 :: "word32" assumes "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0" shows "get_S ((w1 AND 4279238655 OR w2) OR w3) = 0"
well_order_on_domain: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "well_order_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
compInt_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs"   and SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c) semDom_scheme" shows "compInt g gA SEM \<equiv> prWlsAll g gA SEM \<and> prCons g gA SEM \<and> prFreshAll g gA SEM \<and> prSwapAll g gA SEM \<and> prSubstAll g gA SEM"
pt_fresh_left: fixes a :: "'b"   and pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
project_extend_eq: fixes C :: "'c set"   and F :: "'a program" shows "project h C (extend h F) = mk_program (Init F, Restrict (project_set h C) ` Acts F, {act. Restrict (project_set h C) act \<in> project_act h ` Restrict C ` project_act h -` AllowedActs F})"
simps_1: fixes a :: "'a"   and p :: "'a point" shows "point_mult a 0 p = Infinity"
goodstein_aux: fixes c :: "nat"   and n :: "nat" shows "goodsteinC (Suc c) (N2H (Suc (Suc c)) (Suc n)) = goodsteinC (c + 2) (N2H (c + 3) (H2N (c + 3) (N2H (c + 2) (n + 1)) - 1))"
ord_FO_\<Gamma>_def: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset" shows "ord_FO_\<Gamma> S = {uu_. \<exists>CAs DA AAs As \<sigma> E. uu_ = old_Infer (mset CAs) DA E \<and> ord_resolve_rename S CAs DA AAs As \<sigma> E}"
coeff_map_poly_hom: fixes p :: "int poly"   and i :: "nat" shows "coeff (of_int_poly p) i = of_int (coeff p i)"
maybe_negate_false: fixes x :: "trilean" shows "(\<not>? x = false) = (x = true)"
subst_mgt_refl: fixes \<theta> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta>"
wf_wcode_fourtimes_case_le: shows "wf wcode_fourtimes_case_le"
cases: fixes a1 :: "'a"   and a2 :: "'a list"   and a3 :: "'a" assumes "bal a1 a2 a3" obtains c :: "'a" and T :: "'a list" and c' :: "'a" where "a1 = c" and "a2 = T" and "a3 = c'" and "trace c T c'" and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'" and "stack c' = stack c"
f: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> f) = (\<exists>k l u. k \<parallel> l \<and> l \<parallel> p \<and> p \<parallel> u \<and> k \<parallel> q \<and> q \<parallel> u)"
dist_riemann_sphere'_def: shows "dist_riemann_sphere' \<equiv> map_fun Rep_riemann_sphere (map_fun Rep_riemann_sphere id) dist_riemann_sphere_r3"
VV_equivalence: fixes v :: "'a"   and p :: "Player" assumes "v \<in> V" shows "(v \<notin> VV p) = (v \<in> VV p**)"
ladder_stepdown_ix: fixes L :: "(nat \<times> nat \<times> nat) list"   and L' :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" assumes "1 < length L"   and "L' = ladder_stepdown L"   and "0 < index"   and "index < length L'" shows "ladder_ix L' index = ladder_ix L (Suc index)"
iT_Mult_card: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "card (I \<otimes> k) = card I"
uses'_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and chooseNext_all :: "('e \<Rightarrow> 'c set) \<Rightarrow> ('e \<times> 'c \<Rightarrow> 'c list option) \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c"   and g :: "'d"   and n :: "'e" shows "CFG_SSA_Transformed_notriv_base.uses' \<alpha>n defs uses phis chooseNext_all g n \<equiv> CFG_SSA_Transformed_notriv_base.substNext \<alpha>n defs uses phis chooseNext_all g ` uses g n"
gcd_non_0_nat: fixes y :: "nat"   and x :: "nat" assumes "y \<noteq> zero_class.zero" shows "gcd x y = gcd y (x mod y)"
mem_samp_primes_gt_2: fixes x :: "nat" assumes "x \<in> set_spmf sample_primes" shows "2 < x"
TOTAL: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "Total r"
subst_same_Var: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and \<phi> :: "'c"   and x :: "'a" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "\<phi> \<in> fmla"   and "x \<in> var" shows "subst \<phi> (Var x) x = \<phi>"
assms_2: shows "component_of_term u < component_of_term v"
injectively_orientable_def: fixes x :: "'a" shows "injectively_orientable x \<equiv> \<exists>y. orientation x y \<and> injective y"
simps_12: shows "iszlfm F = (\<lambda>bs. isatom F \<and> bound0 F)"
partJoin_commute: fixes P :: "'a set set"   and Q :: "'a set set" shows "partJoin P Q = partJoin Q P"
BINDING_eqvt_name_2: fixes pi :: "(name \<times> name) list"   and B :: "ty"   and Y :: "ntrm set" shows "pi \<bullet> BINDINGc B Y = BINDINGc B (pi \<bullet> Y)"
assms_20: shows "vars F0 \<subseteq> Vbl"
image_map_indets_Polys: fixes f :: "'e \<Rightarrow> 'c"   and X :: "'e set" shows "map_indets f ` P[X] = P[f ` X]"
run_asset_iii_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_asset_iii n s \<equiv> SOME x. pred_asset_iii n (run_owner_iii n s) x"
djf_def: fixes f :: "'a \<Rightarrow> fm"   and p :: "'a"   and q :: "fm" shows "djf f p q = (if q = T then T else if q = F then f p else let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q)"
norm_eq_1_iff_in_S: fixes v :: "(real, 2) vec" shows "(norm v = 1) = (proj2_pt v \<in> S)"
triangle_trans: fixes k :: "nat"   and A :: "'a mat"   and k' :: "nat" assumes "local.triangular_to k A"   and "k' < k" shows "local.triangular_to k' A"
top_rule: fixes G :: "('b, 'c) labeled_graph"   and r :: "'b" assumes "graph G" shows "maintained (top_rule r) G = (vertices G \<times> vertices G = getRel r G)"
mono_inf: fixes f :: "'g multiset \<Rightarrow> 'h"   and A :: "'g multiset"   and B :: "'g multiset" assumes "subset_mset.mono f" shows "f (A \<inter># B) \<le> inf (f A) (f B)"
cases: fixes x :: "'h option" obtains "x = None"   | xa :: "'h" where "x = Some xa"
intros_2: fixes c :: "'form"   and \<Gamma> :: "'form fset"   and s :: "'subst"   and i :: "nat"   and fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool" assumes "c |\<notin>| ass_forms"   and "c |\<in>| \<Gamma>"   and "c = subst s (freshen i anyP)" shows "iwf fc (HNode i s []) (\<Gamma> \<turnstile> c)"
squarefree_part_0: shows "squarefree_part 0 = {}"
reads_consistent_buffered_snoc: fixes sb :: "'b memref list"   and a :: "nat"   and v :: "nat"   and pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and volatile :: "bool"   and t :: "nat" assumes "buffered_val sb a = Some v"   and "reads_consistent pending_write \<O> m sb"   and "volatile \<longrightarrow> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "reads_consistent pending_write \<O> m (sb @ [Read\<^sub>s\<^sub>b volatile a t v])"
wf_ruleset_append: fixes \<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" shows "wf_ruleset \<gamma> p (rs1 @ rs2) = (wf_ruleset \<gamma> p rs1 \<and> wf_ruleset \<gamma> p rs2)"
adopt_node_get_shadow_root: fixes parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>adopt_node_locs parent owner_document document_ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
assms: shows "z \<in> set (list_of_oalist_tc xs)"
foldli_map: fixes f1 :: "'g \<Rightarrow> 'f"   and xs :: "'g list"   and c :: "'e \<Rightarrow> bool"   and f2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and \<sigma> :: "'e" shows "foldli (map f1 xs) c f2 \<sigma> = foldli xs c (f2 \<circ> f1) \<sigma>"
awalk_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "awalk u p v \<equiv> u \<in> verts G \<and> set p \<subseteq> arcs G \<and> cas u p v"
Keys_bad_insert_non_LtK: fixes X :: "msg"   and IK :: "msg set"   and Bad :: "agent set" assumes "X \<notin> range LtK" shows "Keys_bad (insert X IK) Bad = Keys_bad IK Bad"
norm_eq_sqrt_inner: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "norm x = sqrt (inner x x)"
mn_correct: fixes rf :: "recf"   and fap :: "abc_inst list"   and far :: "nat"   and fft :: "nat"   and rsx :: "nat"   and xs :: "nat list"   and arity :: "nat"   and B :: "abc_inst list"   and f :: "nat \<Rightarrow> (nat \<times> nat list) \<times> nat \<times> nat"   and x :: "nat"   and anything :: "nat list"   and P :: "(nat \<times> nat list) \<times> nat \<times> nat \<Rightarrow> bool"   and Q :: "(nat \<times> nat list) \<times> nat \<times> nat \<Rightarrow> bool" assumes "rec_ci rf = (fap, far, fft)"   and "0 < rsx"   and "length xs = arity"   and "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"   and "f = (\<lambda>stp. (abc_steps_l (length fap, xs @ x # rsx # anything) (fap @ B) stp, length fap, arity))"   and "P = (\<lambda>((as, lm), ss, arity). as = 0)"   and "Q = (\<lambda>((as, lm), ss, arity). mn_ind_inv (as, lm) (length fap) x rsx anything xs)" shows "\<exists>stp. P (f stp) \<and> Q (f stp)"
properL_proper: fixes cl :: "('a, 'b, 'c) cmd list"   and c :: "('a, 'b, 'c) cmd" assumes "properL cl"   and "c \<in> set cl" shows "proper c"
assms: shows "map_pmf con s = Sum_pmf (8 / 10) Da Db"
distinct_9: fixes x21 :: "POPLmark.type"   and x22 :: "trm"   and x31 :: "POPLmark.type"   and x32 :: "trm" shows "(\<lambda>:x21. x22) \<noteq> (\<lambda><:x31. x32)"
C1: fixes R :: "pi"   and S :: "pi"   and x :: "name" assumes "(R, S) \<in> Rel" shows "(<\<nu>x>R, <\<nu>x>S) \<in> Rel'"
distinct_fst_isin_same_fst: fixes x :: "'a"   and y :: "'b"   and xs :: "('a \<times> 'b) list"   and y' :: "'b" assumes "(x, y) \<in> set xs"   and "(x, y') \<in> set xs"   and "distinct_fst xs" shows "y = y'"
word_ops_msb: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "msb (x || y) = (msb x \<or> msb y) \<and> msb (x && y) = (msb x \<and> msb y) \<and> msb (x xor y) = (msb x \<noteq> msb y) \<and> msb (~~ x) = (\<not> msb x)"
Sup_distl: fixes x :: "'a"   and Y :: "'a set" shows "x \<cdot> \<Squnion> Y = \<Squnion> ((\<cdot>) x ` Y)"
assms_2: shows "\<Psi>' \<simeq> \<Psi>''"
redT_updWs_None_SomeD: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and was :: "('b, 'c) wait_set_action list"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t' :: "'b"   and w :: "'c wait_set_status" assumes "redT_updWs t ws was ws'"   and "ws' t' = \<lfloor>w\<rfloor>"   and "ws t' = None" shows "t = t' \<and> (\<exists>w'. Suspend w' \<in> set was)"
le_zero_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
cases: fixes x :: "bool \<times> tm" obtains k :: "name" and s :: "name" and t :: "tm" and vf :: "bool" where "atom k \<sharp> (s, t)" and "atom s \<sharp> t" and "x = (vf, t)"
interleaves_equal_fst: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and ys' :: "'b list" assumes "xs \<simeq> {ys, zs, P}"   and "xs \<simeq> {ys', zs, P}" shows "ys = ys'"
c: fixes s :: "complex" assumes "s \<in> complex_cone' \<alpha> - {0}" shows "cmod (stirling_integral m s) \<le> c / cmod s ^ m"
map_disc_iff: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e option" shows "(map_option f a = None) = (a = None)"
at_prm_fresh1: fixes c :: "'a"   and pi :: "('a \<times> 'a) list" assumes "at TYPE('a)"   and "c \<sharp> pi" shows "\<forall>(a, b)\<in>set pi. c \<noteq> a \<and> c \<noteq> b"
simps_3: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and t :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><Lam [x].t> = Lam [x].\<theta><t>"
gram_schmidt_wit_def: shows "gram_schmidt_wit = gram_schmidt.main"
monotoneD: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
dimv_ws: shows "\<forall>w\<in>set ws. dim_vec w = n"
iso_adj_map: fixes x :: "'a set"   and y :: "'a set" assumes "maxsimp x"   and "maxsimp y"   and "x \<sim> y"   and "x \<noteq> y" shows "f ` x \<sim> f ` y"
assms_2: shows "heap_upds_ok (\<Gamma>, S)"
monofun_mhomTr1: fixes N :: "('g, 'b, 'h) Module_scheme"   and h :: "'a"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and a :: "'b" assumes "R module N"   and "h \<in> H"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "a \<in> carrier R" shows "monofun R M N f H h (a \<cdot>\<^sub>s h) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f h"
lm029: fixes x :: "'a"   and X :: "'a set" shows "(x \<in> X) = ({x} \<in> finestpart X)"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a \<squnion> b = b)"
float_distinct_4: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (IEEE.is_nan a \<and> IEEE.is_zero a)"
AbstrLevels_S1opt_A12: fixes i :: "AbstrLevelsID" assumes "sS1opt \<in> AbstrLevel i" shows "sA12 \<notin> AbstrLevel i"
ChiStar_Self: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto" shows "(S, S) \<in> ChiStar A"
heads_pos: fixes xs :: "bool list" assumes "True \<in> set xs" shows "0 < heads xs"
porder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.porder less_eq less \<equiv> partial_ordering UNIV less_eq less"
sgn_abs: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "\<bar>sgn x\<bar> = (1::'a)"
zmod_numeral_Bit1: fixes v :: "num"   and w :: "num" shows "numeral (num.Bit1 v) mod numeral (num.Bit0 w) = 2 * (numeral v mod numeral w) + 1"
distinct_1: fixes name :: "name"   and name' :: "name"   and trm' :: "trm" shows "Var name \<noteq> Lam [name'].trm'"
and_negative_int_iff: fixes k :: "int"   and l :: "int" shows "(k AND l < 0) = (k < 0 \<and> l < 0)"
fnn: fixes x :: "'a" assumes "x \<in> A" shows "0 \<le> f x"
set_mode_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_mode_locs shadow_root_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
chain_boundary_of_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}"   and "singular_chain p X c" shows "chain_boundary p c = (if p = 0 \<or> odd p then 0 else frag_extend (\<lambda>f. frag_of (\<lambda>x\<in>standard_simplex (p - 1). a)) c)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_ring_hom hom" shows "map_poly_comm_ring_hom hom"
order_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a = a + b)"
coprime_0_left_iff: fixes a :: "'a" shows "coprime (0::'a) a = is_unit a"
selection_axioms: fixes M :: "'a literal multiset set" shows "selection (S_M S M)"
order_refl: fixes x :: "'loc \<times> 't" shows "x\<le>\<^sub>px"