hcju/isabelleps huggingface dataset:
func_lemma2: fixes I :: "('e, 'f, 'd) interp"   and args :: "'d \<Rightarrow> ('e, 'd) trm"   and \<nu> :: "(real, 'd) vec"   and f :: "'e" assumes "\<forall>x i. (Functions I i has_derivative (THE f'. \<forall>x. (Functions I i has_derivative f' x) (at x)) x) (at x) \<and> continuous_on UNIV (\<lambda>x. Blinfun ((THE f'. \<forall>x. (Functions I i has_derivative f' x) (at x)) x))"   and "\<And>\<theta>. \<theta> \<in> range args \<Longrightarrow> (sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)" shows "((\<lambda>v. Functions I f (\<chi>i. sterm_sem I (args i) v)) has_derivative (\<lambda>v'. (THE f'. \<forall>x. (Functions I f has_derivative f' x) (at x)) (\<chi>i. sterm_sem I (args i) \<nu>) (\<chi>i. frechet I (args i) \<nu> v'))) (at \<nu>)"
Units_not_left_zero_divisor: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> Units Zp"   and "b \<in> carrier Zp"   and "b \<otimes> a = \<zero>" shows "b = \<zero>"
Jordan_inside_outside: fixes c :: "real \<Rightarrow> complex" assumes "simple_path c"   and "pathfinish c = pathstart c" shows "inside (path_image c) \<noteq> {} \<and> open (inside (path_image c)) \<and> connected (inside (path_image c)) \<and> outside (path_image c) \<noteq> {} \<and> open (outside (path_image c)) \<and> connected (outside (path_image c)) \<and> bounded (inside (path_image c)) \<and> \<not> bounded (outside (path_image c)) \<and> inside (path_image c) \<inter> outside (path_image c) = {} \<and> inside (path_image c) \<union> outside (path_image c) = - path_image c \<and> frontier (inside (path_image c)) = path_image c \<and> frontier (outside (path_image c)) = path_image c"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "longest_common_prefix x xa = y" obtains x :: "'b" and xs :: "'b list" and ya :: "'b" and ys :: "'b list" where "xa__ = x # xs" and "xa = ya # ys" and "y = (if x = ya then x # longest_common_prefix xs ys else [])"   | uv_ :: "'b list" where "xa__ = []" and "xa = uv_" and "y = []"   | uu_ :: "'b list" where "xa__ = uu_" and "xa = []" and "y = []"
process_prop_6_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_6 P \<equiv> \<forall>xs X. xs \<in> snd P \<longrightarrow> (xs, X) \<in> fst P"
strictly_preserves_T: fixes f :: "'b"   and g :: "'b" assumes "F.D.arr f"   and "F.D.arr g" shows "G (F.D.tensor f g) = G.D.tensor (G f) (G g)"
AE_iff_emeasure_eq_1: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "prob_space M"   and "Measurable.pred M P" shows "almost_everywhere M P = (emeasure M {x \<in> space M. P x} = 1)"
space_rT: fixes \<omega> :: "'s stream"   and x :: "'s" shows "(\<omega> \<in> space (rT x)) = enabled x \<omega>"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "c.ptensor_mat m1 m2 = tensor_mat c.dims0 c.vars1' m1 m2"
is_partial_magma_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "horizontal_composition V H src trg" shows "partial_magma H"
effectE: fixes c :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect c h h' r" obtains "r = fst (the (execute c h))" and "h' = snd (the (execute c h))" and "success c h"
size_4: fixes x21 :: "'c"   and x22 :: "'c list" shows "|x21 # x22| = |x22| + Suc 0"
upto_Nil2: fixes i :: "int"   and j :: "int" shows "([] = [i..j]) = (j < i)"
insertI_1: fixes s :: "real"   and a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> bool" assumes "s \<in> {a..<b}"   and "s = a \<or> P s"   and "\<forall>t\<in>{s<..<b}. P t"   and "finite_Psegments P a s" shows "finite_Psegments P a b"
context_equiv_tyenv_eq: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem :: "'Var \<Rightarrow> 'Val"   and mem' :: "'Var \<Rightarrow> 'Val"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option" assumes "mem =\<^bsub>\<Gamma>\<^esub> mem'"   and "\<Gamma> =:\<^sub>P \<Gamma>'"   and "pred P mem" shows "mem =\<^bsub>\<Gamma>'\<^esub> mem'"
that: fixes h :: "real \<Rightarrow> 'a" assumes "path h"   and "pathstart h = pathstart g"   and "path_image h \<subseteq> path_image g"   and "path_image h - {pathfinish h} \<subseteq> interior S"   and "pathfinish h \<in> frontier S" shows "thesis"
pairDifference: fixes x :: "'c"   and X :: "'d"   and Y :: "'d" shows "{(x, X)} - {(x, Y)} = {x} \<times> ({X} - {Y})"
sums_complex_iff: fixes f :: "nat \<Rightarrow> complex"   and x :: "complex" shows "f sums x = ((\<lambda>x. Re (f x)) sums Re x \<and> (\<lambda>x. Im (f x)) sums Im x)"
simps_2: fixes uv :: "'a"   and \<sigma> :: "'c" shows "pol uv (Guard \<sigma>) = Fext"
order_trans_rules_28: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b < c" shows "a < c"
Abs1_eq_iff_3: fixes a :: "'c"   and x :: "'b"   and b :: "'c"   and y :: "'b" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = (a \<leftrightarrow> b) \<bullet> y \<and> atom a \<sharp> y)"
box_ne_empty_1: fixes a :: "'c"   and b :: "'c" shows "(cbox a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
in_set_insert: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
simple_match_any: fixes p :: "('a, 'b) simple_packet_scheme" shows "simple_matches simple_match_any p"
simps_12: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Sqrt f) env = compute_tm_by_comp prec ord I a (Sqrt (Var 0)) (approx_tm prec ord I a f env) (\<lambda>x. 0 < lower x)"
simps_4: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "upds_list (Arg x # S) = upds_list S"
tuple_simps_2: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.dom (hfsetcat.tuple f g) = local.dom f"
preserves_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "src\<^sub>C (H \<mu>) = H.map\<^sub>0 (src\<^sub>B \<mu>)"
hyps: fixes A :: "'a tensor" assumes "A \<in> set As" shows "dims A = ds"
finish_simps_6: fixes u :: "'c"   and s :: "('c, 'd) state_scheme" shows "tree_edges (param_DFS_defs.finish u s) = tree_edges s"
1: shows "\<parallel>f\<parallel>\<^sup>2 = (\<Sum>i = 0..<n. F i * F i)"
Un_iff: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
f2: shows "rk {a, c, A, C, R, \<beta>} \<le> 3"
preserves_iso: fixes f :: "'b \<times> 'b" assumes "B.VV.iso f" shows "B.iso (fst f \<star>\<^sub>B snd f)"
bounded_iff: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "(Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
mUnionM_def: shows "mUnionM \<equiv> fold_mset munionM (return {#})"
rotate_3: assumes "round_robin_reschedule t0 (queue @ [t']) n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>" shows "\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t ta"
SCF_R4: fixes xs :: "'c option list"   and X :: "'c option set"   and P :: "'c option process"   and Q :: "'c option process"   and Y :: "'c option set" assumes "(xs, X) \<in> seq_comp_failures P Q"   and "(xs, Y) \<in> seq_comp_failures P Q" shows "(xs, X \<union> Y) \<in> seq_comp_failures P Q"
cone_subsetD_2: fixes h1 :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U1 :: "'x set"   and h2 :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U2 :: "'x set" assumes "cone (h1, U1) \<subseteq> cone (h2, U2)"   and "h1 \<noteq> 0" shows "U1 \<subseteq> U2"
order_linear: fixes x :: "'a"   and a :: "'a" shows "order x [:- a, 1::'a:] = (if x = a then 1 else 0)"
insert_remove: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and x :: "'d" assumes "finite A" shows "sum_0 g (insert x A) = g x + sum_0 g (A - {x})"
scaleRat_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat 1 x = x"
path_split: fixes v :: "'d"   and v' :: "'d"   and pth1 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list"   and e :: "('d \<times> 'c) \<times> 'd \<times> 'b"   and pth2 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" shows "path v v' (pth1 @ [e] @ pth2) = (path v (edge_end e) (pth1 @ [e]) \<and> path (edge_end e) v' pth2)"
iso_comp: fixes Z :: "'b set set"   and g :: "'a \<Rightarrow> 'b" assumes "ChamberComplexIsomorphism X Z g" shows "ChamberComplexIsomorphism X Z (g \<circ> f)"
fd_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.domain_semiringl plus times one zero domain_op less_eq less" shows "domain_semiringl.fd times domain_op x y = domain_op (times x y)"
cobounded1: fixes a :: "'a word"   and b :: "'a word" shows "a \<le>s signed.max a b"
sup_inf_distrib1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
deny_all_fun_def: fixes f :: "'c \<Rightarrow> 'd" shows "D\<^sub>f f \<equiv> D\<^sub>p (Some \<circ> f)"
distinctEqvt: fixes p :: "(name \<times> name) list"   and xs :: "'a list" shows "p \<bullet> distinct xs = distinct (p \<bullet> xs)"
widen_array_array: fixes P :: "'b prog"   and A :: "ty"   and B :: "ty" assumes "P \<turnstile> A \<le> B" shows "P \<turnstile> A\<lfloor>\<rceil> \<le> B\<lfloor>\<rceil>"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (b < a \<or> a = b)"
fg: fixes z :: "complex" assumes "z \<in> ball 0 r" shows "f (g z) = z"
reachable_induce_ss: fixes S :: "'a set"   and u :: "'a"   and v :: "'a"   and T :: "'a set" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"   and "S \<subseteq> T" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> T\<^esub> v"
pendulum_inv: fixes r :: "real"   and G :: "(real, 2) vec \<Rightarrow> bool" shows "(\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2) \<le> |x\<acute>=\<lambda>t s. fball (- s $ 1) s & G on \<lambda>s. {t. 0 \<le> t} UNIV @ 0] (\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2)"
eq: shows "word_of_int (- (sint a div - sint b)) * b + word_of_int (sint a mod - sint b) = a"
const_least: fixes f :: "'b \<Rightarrow> 'a" assumes "Field r \<noteq> {}"   and "f \<in> FINFUNC" shows "(const, f) \<in> oexp"
overlap_extensionality: fixes x :: "i"   and y :: "i" shows "(x = y) = (\<forall>z. O z x = O z y)"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "vc C Q" shows "\<turnstile>\<^sub>2 {pre C Q} strip C {Q}"
main_is_initial: fixes fg :: "('c, 'd, 'e, 'f, 'g) flowgraph_rec_scheme" shows "initialproc fg (main fg)"
DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow> t"
finalStateNonApplicable: fixes state :: "State"   and decisionVars :: "nat set" shows "isFinalState state decisionVars = (\<not> applicableDecide state decisionVars \<and> \<not> applicableUnitPropagate state \<and> \<not> applicableBackjump state \<and> \<not> applicableLearn state)"
intra_kind_def: fixes et :: "('g, 'h, 'i, 'j) edge_kind" shows "intra_kind et \<equiv> (\<exists>f. et = \<Up>f) \<or> (\<exists>Q. et = (Q)\<^sub>\<surd>)"
Struct_axioms: shows "Struct wtFsym wtPsym arOf resOf intT intF intP"
multi_drop_mem_not_eq: fixes c :: "'b"   and B :: "'b multiset" assumes "c \<in># B" shows "remove1_mset c B \<noteq> B"
bl_shiftl: fixes w :: "'a word"   and n :: "nat" shows "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False"
mult: fixes x :: "'a"   and a :: "real" assumes "x \<in> V" shows "f (a \<cdot> x) = a * f x"
merge_on_subset: fixes B :: "'c set" assumes "B \<subseteq> UNIV" shows "merge_on B rhb bob mb"
arr_dom_iff_arr: fixes f :: "'a1" shows "C1.arr (C1.dom f) = C1.arr f"
m_comm: fixes x :: "int vec"   and y :: "int vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + y = y + x"
sumPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>[Rel] Q"   and "Id \<subseteq> Rel'"   and "Rel \<subseteq> Rel'" shows "P \<oplus> R \<leadsto>[Rel'] Q \<oplus> R"
att_or_empty: fixes s' :: "'b set"   and s'' :: "'b set" assumes "\<turnstile>[] \<oplus>\<^sub>\<or>\<^bsup>(s', s'')\<^esup>" shows "s' \<subseteq> s''"
soconf_new: fixes P :: "'b prog"   and h :: "heap"   and C :: "char list"   and obj :: "sfields"   and a :: "nat"   and x :: "obj" assumes "P,h,C \<turnstile>\<^sub>s obj \<surd>"   and "h a = None" shows "P,h(a \<mapsto> x),C \<turnstile>\<^sub>s obj \<surd>"
thrb: shows "row i (\<chi>k. if k = i then 0 else row k A) = 0"
vreal_one_closed: shows "1\<^sub>\<real> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
Program: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'sb \<times> 'dirty \<times> 'owns \<times> 'rels) list"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'sb"   and \<D> :: "'dirty"   and \<O> :: "'owns"   and \<R> :: "'rels"   and p' :: "'p"   and is' :: "instr list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'shared" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')" shows "(ts, m, \<S>) \<Rightarrow> (ts[i := (p', is @ is', \<theta>, record p p' is' sb, \<D>, \<O>, \<R>)], m, \<S>)"
monoI: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
empty_map_add: fixes m :: "'b \<Rightarrow> 'c option" shows "Map.empty ++ m = m"
simps_1: fixes f :: "'b \<Rightarrow> 'a" shows "ListSum [] f = (0::'a)"
simps_2: fixes r :: "'b memref"   and rs :: "'b memref list" shows "write_sops (r # rs) = (case r of Write\<^sub>s\<^sub>b volatile a sop v x xa xb xc \<Rightarrow> insert sop (write_sops rs) | _ \<Rightarrow> write_sops rs)"
eval_uminus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (uminus F) = (\<lambda>x. - eval F x)"
flrestriction_vunion: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<union>\<^sub>\<circ> s) \<restriction>\<^sup>l\<^sub>\<bullet> A = r \<restriction>\<^sup>l\<^sub>\<bullet> A \<union>\<^sub>\<circ> s \<restriction>\<^sup>l\<^sub>\<bullet> A"
add_self_div_2: fixes m :: "nat" shows "(m + m) div 2 = m"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(sup b c \<le> a) = (b \<le> a \<and> c \<le> a)"
eval_evalUni: fixes var :: "nat"   and F :: "atom fm"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and F' :: "atomUni fmUni" assumes "convert_fm var F (xs' @ x # xs) = Some F'"   and "length xs' = var" shows "eval F (xs' @ x # xs) = evalUni F' x"
SD_refl': fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le" shows "SD le p p = (p \<in> lotteries_on carrier)"
UnE: fixes c :: "'l"   and A :: "'l set"   and B :: "'l set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
assms_1: shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
coms_loop_inv: fixes G :: "diagram"   and c :: "command" assumes "coms_com (Loop_dia G) c" obtains ca :: "command" where "c = Loop ca" and "coms_dia G ca"
ex1_implies_ex: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "\<exists>x. P x"
is_proper_suffix_length_cmp: fixes a :: "'a list"   and b :: "'a list" assumes "is_proper_suffix a b" shows "length a < length b"
int_div_less_self: fixes x :: "int"   and k :: "int" assumes "0 < x"   and "1 < k" shows "x div k < x"
mult_im_sheaf_def: shows "mult_im_sheaf \<equiv> \<lambda>V. mult_str (f \<^sup>\<inverse> X V)"
inv_equality: fixes y :: "'a"   and x :: "'a" assumes "y \<otimes> x = \<one>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv x = y"
simps: fixes a :: "('a, 'b) trm" shows "dsafe a = ((\<exists>i. a = Var i) \<or> (\<exists>r. a = Const r) \<or> (\<exists>args i. a = $f i args \<and> (\<forall>x. dsafe (args x))) \<or> (\<exists>\<theta>\<^sub>1 \<theta>\<^sub>2. a = Plus \<theta>\<^sub>1 \<theta>\<^sub>2 \<and> dsafe \<theta>\<^sub>1 \<and> dsafe \<theta>\<^sub>2) \<or> (\<exists>\<theta>\<^sub>1 \<theta>\<^sub>2. a = Times \<theta>\<^sub>1 \<theta>\<^sub>2 \<and> dsafe \<theta>\<^sub>1 \<and> dsafe \<theta>\<^sub>2) \<or> (\<exists>\<theta>. a = Differential \<theta> \<and> dfree \<theta>) \<or> (\<exists>i. a = $' i))"
Ugps_isomorphic: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme" assumes "Ugp G"   and "Ugp E" shows "G \<cong> E"
bre_cond_imp_br_cond: fixes Qi :: "'d set" shows "bre_cond Qi \<subseteq> br_cond"
elims_2: fixes x :: "bool \<times> (bool \<times> 'b) rexp" assumes "final_a x" obtains b :: "bool" and r :: "(bool \<times> 'b) rexp" where "x = (b, r)" and "final r \<or> b \<and> nullable r"
upto_split2: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "j \<le> k" shows "[i..k] = [i..j] @ [j + 1..k]"
extf_trans_from_irrefl: fixes A :: "('s, 'v) tm set"   and zs :: "('s, 'v) tm list"   and ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "finite A"   and "zs \<in> lists A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>x\<in>A. \<not> gt x x"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "extf f gt zs ys"   and "extf f gt ys xs" shows "extf f gt zs xs"
assms_2: shows "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"
simps_3: fixes p :: "dir list"   and dir :: "dir"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at (dir.Left # p) dir (App t s) = App (emb_step_at p dir t) s"
assms_4: shows "is_a_intersec P A B' A' B"
Q_def: shows "Q = Pascal_Property.inter (line A B) (line B' C')"
pair_spmf_return_spmf2: fixes p :: "'a spmf"   and y :: "'b" shows "pair_spmf p (return_spmf y) = map_spmf (\<lambda>x. (x, y)) p"
simps_1: shows "fin_string_upto 0 = [fin_string_empty]"
closed: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and b :: "'nid list" assumes "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>"   and "b \<in> pin (\<sigma>\<^bsub>nid\<^esub>t n)" shows "\<exists>nid'. \<parallel>nid'\<parallel>\<^bsub>t n\<^esub> \<and> bc (\<sigma>\<^bsub>nid'\<^esub>t n) = b"
floor_frac_add_preservation: fixes d :: "real"   and a :: "real" assumes "0 < d"   and "d < 1 - frac a" shows "\<lfloor>a\<rfloor> = \<lfloor>a + d\<rfloor>"
return_pure: fixes x :: "'z"   and h :: "'x" shows "pure (return x) h"
exec_bind_SE_success: fixes A :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and b :: "'d"   and \<sigma>' :: "'e"   and M :: "'d \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "A \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M) = (\<sigma>' \<Turnstile> M b)"
c_sources_ipurge: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xs :: "'f list" shows "c_sources I D u (c_ipurge I D u xs) = c_sources I D u xs"
cts_wp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "bd_cts (wp body)" shows "bd_cts (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
walk_decomp_1: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and ys :: "'a list" assumes "Digraph G"   and "Digraph.walk G (xs @ ys)" shows "Digraph.walk G xs"
emb_step_fun: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b fun t"
update_convs_6: fixes local' :: "t list \<Rightarrow> t list"   and types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "local_update local' \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local' local, label = label, return = return, \<dots> = more\<rparr>"
Sem_eval_cases_6: fixes s :: "Var \<Rightarrow> Val"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , Call \<rightarrow>\<^sub>n  t " obtains na :: "nat" where "n = Suc na" and " s , body \<rightarrow>\<^sub>na  t "
IH: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "P t u"   and "feasible t"   and "\<And>Q. unitary Q \<Longrightarrow> unitary (u Q)" shows "P (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wlp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
dvdg'_2: shows "gcd (a div gcd a b) (b div gcd a b) dvd b div gcd a b"
Min_less_iff: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Min A <s x) = (\<exists>a\<in>A. a <s x)"
semiring_norm_159: shows "(0::'a) \<noteq> (1::'a)"
fin: shows "finite {U \<in> \<A>. U \<inter> V \<noteq> {}}"
b_def': shows "b = \<phi> i\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"
dres_Sup_cases: fixes S :: "'b dres set" obtains "S \<subseteq> {dSUCCEED}" and "Sup S = dSUCCEED"   | "dFAIL \<in> S" and "Sup S = dFAIL"   | a :: "'b" and b :: "'b" where "a__ \<noteq> b" and "dRETURN a__ \<in> S" and "dRETURN b \<in> S" and "dFAIL \<notin> S" and "Sup S = dFAIL"   | a :: "'b" where "S \<subseteq> {dSUCCEED, dRETURN a}" and "dRETURN a \<in> S" and "Sup S = dRETURN a"