hcju/isabelleps huggingface dataset:
toplevel_summands_final: fixes s :: "'c rexp" shows "final s = (\<exists>r\<in>toplevel_summands s. final r)"
2: shows "punit.lc (rep_list q) \<noteq> (0::'b)"
in_lim: shows "X \<subseteq> Liminf_llist Xs"
paths_le_is_filter: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "paths_le k P \<subseteq> P"
subst_atm_list_nth: fixes i :: "nat"   and As :: "'a list"   and \<tau> :: "'s" assumes "i < length As" shows "(As \<cdot>al \<tau>) ! i = As ! i \<cdot>a \<tau>"
ct_def: fixes s :: "'s" shows "ct s = (SOME D. D \<in> K s \<and> (s \<in> S1 \<longrightarrow> lfp F_inf s = integral\<^sup>N (measure_pmf D) (lfp F_inf)))"
S1Infinite: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<not> S1 rmhist p \<leadsto> S6 rmhist p"   and "sigma \<Turnstile> \<box>\<diamond>S6 rmhist p \<longrightarrow> \<box>\<diamond>S1 rmhist p" shows "sigma \<Turnstile> \<box>\<diamond>S1 rmhist p"
enum_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(t.enum i \<le> t.enum j) = (i \<le> j)"
prefix_sum: shows "(\<Sum>i = 0..<i. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) = (\<Sum>ia = 0..<i. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0)"
assms_2: shows "\<^bold>|p\<^bold>| \<le> \<^bold>|u\<^bold>|"
rtrancl_induct3: fixes r :: "'d \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and ax :: "'d"   and ay :: "'e"   and az :: "'f"   and bx :: "'d"   and "by" :: "'e"   and bz :: "'f"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay, az) (bx, by, bz)"   and "P ax ay az"   and "\<And>a aa b ab ac ba. \<lbrakk>r\<^sup>*\<^sup>* (ax, ay, az) (a, aa, b); r (a, aa, b) (ab, ac, ba); P a aa b\<rbrakk> \<Longrightarrow> P ab ac ba" shows "P bx by bz"
Not\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "p \<bullet> Not\<^sub>\<alpha> t\<^sub>\<alpha> = Not\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
prime_field: fixes \<G> :: "'a cyclic_group"   and a :: "nat" assumes "np_base \<G>"   and "a < order \<G>"   and "a \<noteq> 0" shows "coprime a (order \<G>)"
assms_1: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" assumes "P c0"   and "next_propagate' c0 c1 loc t" shows "P c1"
solvable: fixes eq :: "'a"   and xs :: "'b list" assumes "solvable\<^sub>0 eq" shows "\<exists>x. I\<^sub>a eq (x # xs)"
preserves_hom: fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>R f : R a \<Rightarrow> R b\<guillemotright>"
id_funpow_id: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and n :: "nat" assumes "f x = x" shows "(f ^^ n) x = x"
B: fixes x :: "complex" assumes "x \<in> closed_segment a b" shows "cmod (f x) \<le> B"
delete_def: fixes x :: "'c"   and t :: "('c \<times> 'd) tree" shows "Splay_Map.delete x t = (if t = \<langle>\<rangle> then \<langle>\<rangle> else case Splay_Map.splay x t of \<langle>l, a, r\<rangle> \<Rightarrow> if x = fst a then if l = \<langle>\<rangle> then r else case splay_max l of \<langle>l', m, r'\<rangle> \<Rightarrow> \<langle>l', m, r\<rangle> else \<langle>l, a, r\<rangle>)"
compatible_iff_seq: fixes g :: "'b"   and f :: "'b" shows "(\<phi>.B.domains g \<inter> \<phi>.B.codomains f \<noteq> {}) = \<phi>.B.seq g f"
Inf_is_action: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "(\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` A}) \<in> {p. is_CSP_process p}"
cases: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "('b, 'd) dds"   and a2 :: "('c, 'e) dds" assumes "rel_dds A B a1 a2" obtains S :: "('b, 'd) dds" and S' :: "('c, 'e) dds" where "a1 = S" and "a2 = S'" and "(A ===> rel_prod B (rel_dds A B)) (run S) (run S')"
assms_4: shows "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) (\<ominus> b)"
distinct_17: fixes x31 :: "nat"   and x32 :: "nat"   and x33 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x34 :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and x35 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x36 :: "nat set"   and x37 :: "nat set"   and x38 :: "nat set"   and x39 :: "nat set"   and x51 :: "nat set"   and x52 :: "nat set"   and x53 :: "nat set"   and x54 :: "nat set" shows "RMW x31 x32 x33 x34 x35 x36 x37 x38 x39 \<noteq> Ghost x51 x52 x53 x54"
upper_usphere: shows "subtopology (nsphere n) (upper n) = usphere n"
that_1: shows "x - e i \<in>\<^sub>r a ! i"
split_at_conv_split_at_it_prod: fixes ts :: "'b list"   and ds :: "'b list"   and n :: "nat"   and xs :: "'b list"   and ts' :: "'b list"   and ds' :: "'b list" assumes "(ts, ds) = split_at n xs"   and "(ts', ds') = split_at_it n xs" shows "(ts, ds) = (ts', ds')"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<iota> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and w' :: "'c"   and \<theta>' :: "'c" assumes "tabulation_data V H \<a> \<iota> src trg r \<rho> f g"   and "uw\<theta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta>"   and "uw\<theta> V H \<a> \<iota> src trg r \<rho> f g u w' \<theta>'" shows "uw\<theta>w'\<theta>' V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>'"
project_act_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and act :: "('f \<times> 'f) set" shows "project_act h act = {(x, x'). \<exists>y y'. (h (x, y), h (x', y')) \<in> act}"
shiftl_int_def: fixes x :: "int"   and n :: "nat" shows "x << n = x * 2 ^ n"
cSup_atMost: fixes x :: "'a multiset" shows "\<Squnion> subset_mset.atMost x = x"
inf_idem: fixes x :: "'a" shows "x \<cdot> x = x"
matched_Append_intra_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "matched n ns n''"   and "n'' i-ns'\<rightarrow>\<^sub>d* n'" shows "matched n (ns @ ns') n'"
canonize_def: fixes A :: "('b \<Rightarrow> bool) set"   and s :: "'b" shows "canonize A s \<equiv> SOME t. t ~A~ s"
fg: shows "src\<^sub>C f = trg\<^sub>C g"
sff_list: shows "f = Polynomial.smult a (\<Prod>(a, i)\<leftarrow>fs. a ^ Suc i)"
simps_24: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm"   and x2 :: "'b" shows "PPair x51 x52 \<noteq> PVar x2"
hyps: assumes "is_tangle_diagram (zs \<circ> y)" shows "is_tangle_diagram y"
padd_residue_eq: fixes m :: "int"   and a :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" assumes "residues m" shows "cring.padd (residue_ring m) a p\<^sub>1 p\<^sub>2 = mpadd m a p\<^sub>1 p\<^sub>2"
lhd_llist_of_tllist: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "lhd (llist_of_tllist xs) = thd xs"
z_2: shows "z \<in> closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g) \<union> path_image g \<union> closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1]) \<union> closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1]) \<union> closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3])"
finprod_def: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'e \<Rightarrow> 'c"   and A :: "'e set" shows "finprod G f A = (if finite A then foldD (carrier G) ((\<otimes>\<^bsub>G\<^esub>) \<circ> f) \<one>\<^bsub>G\<^esub> A else \<one>\<^bsub>G\<^esub>)"
attach_compose: fixes conv1 :: "('a, 'b, 'e, 'f) converter"   and conv2 :: "('e, 'f, 'c, 'd) converter"   and res :: "('c, 'd) resource" shows "conv1 \<odot> conv2 \<rhd> res = conv1 \<rhd> conv2 \<rhd> res"
OrdSum_def: shows "OrdSum = {x. (\<exists>a. x = (a, 1::'c)) \<or> (\<exists>b. x = (1::'b, b))}"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr = delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t (cast shadow_root_ptr)"
bind_distrib_sup1: fixes M :: "'d nres"   and N :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres" shows "sup M N \<bind> f = sup (M \<bind> f) (N \<bind> f)"
sorted_lin_data_prev': shows "sorted (map fst (linearize data_prev'))"
split_mult_neg_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Rings.class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b" shows "less_eq (times a b) zero"
inverse_mult: fixes p :: "quat"   and q :: "quat" shows "inverse (p * q) = inverse q * inverse p"
assms: shows "\<S>\<bullet>\<C> \<turnstile> $* b_es : tf"
simps_2: fixes t :: "'a treeFI" shows "stl (konigPath t) = konigPath (SOME tr. tr \<in> set (sub t) \<and> infiniteTr tr)"
assms_1: fixes v :: "'a" assumes "v = Time_Monad.val n" shows "f v = g v"
prime_pr_reverse: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and m :: "nat" assumes "primerec (Pr n f g) m" obtains "m = Suc n" and "primerec f n" and "primerec g (Suc (Suc n))"
InvariantCFalseAfterUnitPropagate: fixes conflictFlag :: "bool"   and M :: "(Literal \<times> bool) list"   and C :: "Literal list"   and M' :: "(Literal \<times> bool) list"   and uLiteral :: "Literal" assumes "InvariantCFalse conflictFlag M C"   and "M' = M @ [(uLiteral, False)]" shows "InvariantCFalse conflictFlag M' C"
simps: fixes a :: "'form"   and t :: "('form, 'rule, 'subst, 'a) itnode rose_tree"   and "is" :: "nat list" shows "(a \<in> hyps_along t is) = (\<exists>is' i h. a = subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h)) \<and> prefix (is' @ [i]) is \<and> i < length (inPorts' (iNodeOf (tree_at t is'))) \<and> hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i))"
set_cases: fixes e :: "'d"   and a :: "'d ifex" assumes "e \<in> set_ifex a" obtains z2 :: "'d ifex" and z3 :: "'d ifex" where "a = IF e z2__ z3__"   | z1 :: "'d" and z2 :: "'d ifex" and z3 :: "'d ifex" where "a = IF z1__ z2a__ z3a__" and "e \<in> set_ifex z2a__"   | z1 :: "'d" and z2 :: "'d ifex" and z3 :: "'d ifex" where "a = IF z1 z2 z3" and "e \<in> set_ifex z3"
degree_def: fixes v :: "'c"   and g :: "('c, 'd) graph" shows "degree v g \<equiv> card {e \<in> edges g. fst e = v}"
list_update_length: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and y :: "'a" shows "(xs @ x # ys)[length xs := y] = xs @ y # ys"
homeomorphic_path_connectedness: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "path_connected S = path_connected T"
y_def: shows "y = quot_of_ratfps (x - x')"
set_child_nodes_split: fixes P :: "((_) heap, exception, unit) prog \<Rightarrow> bool"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog)) list"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "P (invoke (a_set_child_nodes_tups @ xs) ptr children) = ((known_ptr ptr \<longrightarrow> P (set_child_nodes ptr children)) \<and> (\<not> known_ptr ptr \<longrightarrow> P (invoke xs ptr children)))"
path_card_nil: fixes Q :: "'a set" assumes "Q \<in> \<P>" shows "card Q = 0"
c_traces_failures: fixes xps :: "('e \<times> 'f) list"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" assumes "xps \<in> traces (c_process step out s\<^sub>0)" shows "(xps, {(x, p). p \<noteq> out (foldl step s\<^sub>0 (map fst xps)) x}) \<in> c_failures step out s\<^sub>0"
assms: shows "a{b}.P \<longmapsto> \<tau> \<prec> P'"
f': fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>y\<in>S. norm (y - a) < d \<longrightarrow> norm (f y - f a - f' (y - a)) \<le> e * norm (y - a)"
flip_flip: fixes tup :: "'c \<times> 'd" shows "flip (flip tup) = tup"
sub_complement: fixes x :: "'a"   and y :: "'a" shows "- x = - (- - x \<squnion> - y) \<squnion> - (- - x \<squnion> - - y)"
le_refl: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x"
simps_16: fixes x1 :: "'a"   and x5 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<l>\<^bold>[x5\<^bold>]"
nk: shows "degree g - J + (degree f - J) = degree f - J + (degree g - J)"
prod_pOp_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_pOp I A \<in> carr_prodag I A \<rightarrow> carr_prodag I A \<rightarrow> carr_prodag I A"
step: fixes x :: "'b node \<times> 'b node set" assumes "((\<forall>nm\<in>incoming n'. nm < name n') \<and> name n' \<le> name (fst x) \<and> (incoming n' \<inter> incoming (fst x) \<noteq> {} \<longrightarrow> new n' \<subseteq> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd)" shows "f x \<le> SPEC (\<lambda>r. name n' \<le> fst r \<and> (\<forall>nd\<in>snd r. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd))"
h2: shows "(\<exists>y. \<forall>x<y. aEval (Eq p) (L[var := x])) = (\<exists>y. \<forall>x<y. (\<Sum>i\<le>n. k i x * x ^ i) = 0)"
REP_simps_2: fixes A :: "('a, 'a set) arr" assumes "arr A" shows "src (REP A) = local.Dom A"
max_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
prems_2: shows "is_longest_prefix_routing (rr # rtbl)"
groupF_set: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "\<forall>X\<in>set (groupF f xs). \<forall>x\<in>set X. x \<in> set xs"
prems_3: shows "(Fun.swap a b id \<circ> p) a = a"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "(x < xa) = (Rep_pos x < Rep_pos xa)"
i1: shows "univ_list ! i = a \<and> i < length univ_list"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "sublists (x # xs) = sublists xs @ map ((#) x) (prefixes xs)"
parts_image: fixes f :: "'a \<Rightarrow> msg"   and A :: "'a set" shows "parts (f ` A) = (\<Union>x\<in>A. parts {f x})"
factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "q.factorization_m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> q.eq_m f (Polynomial.smult c (\<Prod>\<^sub># fs)) \<and> (\<forall>f\<in>#fs. q.irreducible\<^sub>d_m f \<and> monic (q.Mp f))"
redT_preserves: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "ts_ok P (thr s) (shr s)" shows "ts_ok P (thr s') (shr s')"
assert_bound'_def: fixes dir :: "('i, 'a) Direction"   and i :: "'i"   and x :: "nat"   and c :: "'a"   and s :: "('i, 'a) state" shows "assert_bound' dir i x c s \<equiv> if \<unrhd>\<^sub>u\<^sub>b (Direction.lt dir) c (Direction.UB dir s x) then s else let s' = update\<B>\<I> (Direction.UBI_upd dir) i x c s in if \<lhd>\<^sub>l\<^sub>b (Direction.lt dir) c (Direction.LB dir s x) then set_unsat [i, Direction.LI dir s x] s' else if x \<notin> lvars (\<T> s') \<and> Direction.lt dir c (\<langle>\<V> s\<rangle> x) then update x c s' else s'"
call_return: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list"   and rrs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)"   and "matches \<gamma> m' p"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m (Call chain) # rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
lub_spmf_empty: shows "lub_spmf {} = return_pmf None"
set_sel: fixes a :: "'d Monomorphic_Monad.id" shows "extract a \<in> set_id a"
alpha_qSwap_preserves_qGood1: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"   and "qGood X" shows "qGood X'"
pd_path: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob y x" obtains \<pi> :: "nat \<Rightarrow> 'a" and n :: "nat" and k :: "nat" where "IFC_def.is_path prob \<pi>" and "\<pi> 0 = x" and "\<pi> n = IFC_def.return prob" and "\<pi> k = y" and "k \<le> n"
IH: assumes "\<nu> REP \<nu>_" shows "[\<theta>\<^sub>1_]<>\<nu> \<equiv>\<^sub>P r1_"
extend_leadsTo: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "Extend h" shows "(extend h F \<in> extend_set h A \<longmapsto> extend_set h B) = (F \<in> A \<longmapsto> B)"
subset_\<J>k: fixes u :: "nat"   and v :: "nat"   and k :: "nat" assumes "u \<le> v" shows "\<J> k u \<subseteq> \<J> k v"
mat_to_polys_to_mat: fixes ts :: "'c list"   and A :: "'d mat" assumes "distinct ts"   and "length ts = dim_col A" shows "polys_to_mat ts (mat_to_polys ts A) = A"
t: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
ipoly_poly_div_nonzero: fixes p :: "int poly"   and q :: "int poly"   and x :: "'a"   and y :: "'a" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "ipoly p x = (0::'a)"   and "ipoly q y = (0::'a)"   and "poly p 0 \<noteq> 0 \<or> poly q 0 \<noteq> 0" shows "poly_div p q \<noteq> 0"
less_imp_le: fixes y :: "'c"   and x :: "'c" assumes "y < x" shows "y \<le> x"
nat_to_sch_at_2: shows "nat_to_sch 2 = Base_suc"
hd_last_singletonI: fixes xs :: "'c list" assumes "xs \<noteq> []"   and "hd xs = last xs"   and "distinct xs" shows "xs = [hd xs]"
simps_1: fixes w :: "'b llist"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "lfinite w" shows "lfold f w = fold f (list_of w)"
rtrancl3p_step: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and bs :: "'c list"   and a' :: "'b"   and b :: "'c"   and a'' :: "'b" assumes "rtrancl3p r a bs a'"   and "r a' b a''" shows "rtrancl3p r a (bs @ [b]) a''"
append_take_drop_id: fixes n :: "nat"   and xs :: "'d list" shows "xs \<down> n @ xs \<up> n = xs"
not_eq_extremum: fixes a :: "'b" shows "(a \<noteq> bot) = (bot < a)"
clop_cl_op: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.clattice_with_clop less_eq less cl_op Inf Sup inf sup bot top" shows "clop cl_op"
conjugate_zero: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate zero = zero"
prems_5: shows "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"
that: fixes s :: "'b" assumes "s \<in> M"   and "s \<le> t"   and "\<forall>u\<in>M. \<not> u < s" shows "thesis"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'i \<Rightarrow> 'f"   and f3 :: "'j \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> 'f"   and f4 :: "('g, 'h, 'j, 'k) resumption \<Rightarrow> 'f"   and resumption_corec :: "('g, 'h, 'j, 'k, 'i) resumption_corec" shows "P (case resumption_corec of Terminal_corec x \<Rightarrow> f1 x | Linear_corec x xa \<Rightarrow> f2 x xa | Branch_corec x xa \<Rightarrow> f3 x xa | Resumption_corec x \<Rightarrow> f4 x) = ((\<forall>x1. resumption_corec = Terminal_corec x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. resumption_corec = Linear_corec x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. resumption_corec = Branch_corec x31 x32 \<longrightarrow> P (f3 x31 x32)) \<and> (\<forall>x4. resumption_corec = Resumption_corec x4 \<longrightarrow> P (f4 x4)))"
Confidentiality_Tgs_bis: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Key servK \<notin> analz (knows Spy evs)"
i: fixes s :: "'c \<Rightarrow> ('a, 'b, 'd, 'e) com list \<times> 'b option \<times> 'e" assumes "initial_state sys s" shows "P \<lparr>GST = s, HST = []\<rparr>"
parent_child_rel_def: shows "parent_child_rel = (\<lambda>h. {(parent, child). parent |\<in>| object_ptr_kinds h \<and> child \<in> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` set |h \<turnstile> get_child_nodes parent|\<^sub>r})"
binomial_Suc_n: fixes n :: "nat" shows "Suc n choose n = Suc n"
PER: shows "part_equiv {(x, y) |x y. nodes_connected H x y}"
fmdom'_restrict_set_precise: fixes A :: "'d set"   and m :: "('d, 'e) fmap" shows "fmdom' (fmrestrict_set A m) = fmdom' m \<inter> A"
r2: shows "r2 \<in> set (snd (splitAt r1 vs))"
Gromov_extension_quasi_isometry_boundary_to_boundary: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c Gromov_completion" assumes "lambda C -quasi_isometry f"   and "x \<in> Gromov_boundary" shows "Gromov_extension f x \<in> Gromov_boundary"
nth_tl: fixes n :: "nat"   and xs :: "'c list" assumes "n < length (tl xs)" shows "tl xs ! n = xs ! Suc n"
finite_divisors_nat: fixes m :: "nat" assumes "0 < m" shows "finite {d. d dvd m}"
simps_1: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" shows "find_closest_pair (c\<^sub>0, c\<^sub>1) [] = (c\<^sub>0, c\<^sub>1)"
image_eqI: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and A :: "'d set" assumes "b = f x"   and "s2p A x" shows "s2p (\<P> f A) b"
sumSS'_2: shows "sum cc' S = 1 - c' a"
simps_9: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a" shows "(v ::= va {vb} \<le> {vc} WHILE vd DO ve {vf}) = False"