hcju/isabelleps huggingface dataset:
WHILEIT_unfold: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm"   and x :: "'a" shows "WHILEIT I b f x = (if I x then if b x then bind (f x) (WHILEIT I b f) else return x else top)"
secondh: fixes x :: "real" assumes "\<forall>y<x. a * y\<^sup>2 + b * y + c < 0"   and "\<not> a < 0"   and "\<not> 0 < b" shows "b = 0"
heap_is_wellformed_children_disc_nodes: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> Shadow_DOM.get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
assms: fixes a :: "'a" shows "relations M a \<subseteq> worlds M \<times> worlds M"
indaut_bij_btw_halfchsys_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "bij_betw ((`) (OpposedThinChamberComplexFoldings.induced_automorphism X f g)) (ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X) (f \<turnstile> ChamberComplex.chamber_system X)"
Func_option_def: fixes A :: "'c set"   and B :: "'d set" shows "Func_option A B \<equiv> {f. (\<forall>a. (f a \<noteq> None) = (a \<in> A)) \<and> (\<forall>a\<in>A. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
in_lists_iff_set: fixes xs :: "'b list"   and A :: "'b set" shows "(xs \<in> lists A) = (set xs \<subseteq> A)"
vifunion_vifunion_flip: fixes B :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>B. \<Union>\<^sub>\<circ> (VLambda A (f a) `\<^sub>\<circ> A)) `\<^sub>\<circ> B) = \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> ((\<lambda>aa\<in>\<^sub>\<circ>B. f aa a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A)"
h4: shows "t mod cycleLength (nC k) mem schedule (nC k)"
F: shows "0 < snd (mediant (X \<otimes> UR \<otimes> Y))"
scomp_def: fixes f :: "'a \<Rightarrow> 'c \<times> 'd"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b" shows "scomp f g = (\<lambda>x. case f x of (x, xa) \<Rightarrow> g x xa)"
reindex: fixes h :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'a" assumes "inj_on h A" shows "F.F g (h ` A) = F.F (g \<circ> h) A"
swap_cs_Trans_RecvMarker: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev d"   and "distributed_system.next channel trans send recv d ev' e"   and "isTrans ev"   and "isRecvMarker ev'"   and "distributed_system.next channel trans send recv c ev' d'"   and "distributed_system.next channel trans send recv d' ev e'" shows "cs e i = cs e' i"
nsum_eq: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A"   and "\<forall>j\<le>n. f j = g j" shows "\<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A g n"
neg_literal_notin_imp_true_cls: fixes A :: "'b"   and C :: "'b literal multiset"   and I :: "'b set" assumes "Neg A \<in># C"   and "A \<notin> I" shows "I \<Turnstile> C"
subst_dnf_dnf: fixes \<phi> :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "min_set (subst_dnf (dnf \<phi>) m) = min_dnf (subst \<phi> m)"
OclOr_bot2: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "X \<tau> \<noteq> true \<tau>" shows "(X or UML_Types.bot_class.bot) \<tau> = UML_Types.bot_class.bot \<tau>"
prime_elem_normalize_iff: fixes x :: "'a" shows "prime_elem (normalize x) = prime_elem x"
jumpNestingOk_eval_no_jumpE: fixes Env :: "env"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and j :: "jump"   and T :: "tys"   and w :: "val"   and upd :: "val \<Rightarrow> state \<Rightarrow> state"   and s :: "state"   and j' :: "jump"   and val :: "val" assumes "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "jumpNestingOk {} t"   and "fst s0 \<noteq> Some (Jump j)"   and "Env\<turnstile>t\<Colon>T"   and "wf_prog (prg Env)" obtains "fst s1 \<noteq> Some (Jump j)" and "\<lbrakk>normal s1; v = In2 (w, upd); fst s \<noteq> Some (Jump j')\<rbrakk> \<Longrightarrow> fst (upd val s) \<noteq> Some (Jump j')"
iteratei_correct': fixes hm :: "(uint32, ('a, 'b) assoc_list) RBT.rbt" assumes "HashMap_Impl.invar hm" shows "map_iterator (HashMap_Impl.iteratei hm) (ahm_\<alpha> (hm_\<alpha>' hm))"
gt_imp_vars: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s" shows "vars s \<subseteq> vars t"
insert_def: fixes a :: "'a"   and B :: "'a set" shows "Set.insert a B = {x. x = a} \<union> B"
rhs_no_zero_tableau_start: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" shows "0 \<notin> rhs ` set (Tableau (preprocess' cs start))"
lipschitz_incr: fixes s :: "real"   and h :: "real" assumes "s \<in> {t0..t1}"   and "h \<in> {0..t1 - s}" shows "L-lipschitz_on (cball (x s) \<bar>r\<bar>) (incr h s)"
scc_of_def: fixes E :: "('b \<times> 'b) set"   and v :: "'b" shows "scc_of E v = (THE scc. is_scc E scc \<and> v \<in> scc)"
e_type_const_list_cons: fixes vs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts1 :: "t list"   and ts2 :: "t list" assumes "const_list vs"   and "\<S>\<bullet>\<C> \<turnstile> vs : [] _> ts1 @ ts2" shows "\<exists>vs1 vs2. \<S>\<bullet>\<C> \<turnstile> vs1 : [] _> ts1 \<and> \<S>\<bullet>\<C> \<turnstile> vs2 : ts1 _> ts1 @ ts2 \<and> vs = vs1 @ vs2 \<and> const_list vs1 \<and> const_list vs2"
put_put: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'a"   and u :: "'a" assumes "mwb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
similar_mat_refl: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat A A"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> 'c"   and blindable\<^sub>h :: "'d blindable\<^sub>h" shows "P (case blindable\<^sub>h of Content x \<Rightarrow> f1 x | Garbage x \<Rightarrow> f2 x) = ((\<forall>x1. blindable\<^sub>h = Content x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. blindable\<^sub>h = Garbage x2 \<longrightarrow> P (f2 x2)))"
abs_fresh_5: fixes b :: "W.tvar"   and a :: "W.var"   and x :: "'a" shows "b \<sharp> [a].x = b \<sharp> x"
RT: assumes "a" shows "hn_refine (\<Gamma>1 * hn_val bool_rel a a') b' \<Gamma>2b R b"
span_cols_upt_k_Gram_Schmidt_column_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and j :: "nat" assumes "k < ncols A"   and "j < ncols A" shows "span (cols_upt_k A k) = span (cols_upt_k (Gram_Schmidt_column_k A j) k)"
supset_ld_is_ld: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and A :: "'c set"   and B :: "'c set" assumes "Module.module R M"   and "module.lin_dep R M A"   and "A \<subseteq> B" shows "module.lin_dep R M B"
to_bl_nth: fixes n :: "nat"   and w :: "'a word" assumes "n < size w" shows "to_bl w ! n = w !! (size w - Suc n)"
in_Sup_multisetD: fixes x :: "'k"   and A :: "'k multiset set" assumes "x \<in># Sup A" shows "\<exists>X\<in>A. x \<in># X"
that: fixes x :: "'a"   and y :: "'b" assumes "m = NondetT x"   and "m' = NondetT y"   and "M x y" shows "thesis"
inf_of_singletonI: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_partial_order L"   and "x \<in> carrier L" shows "is_glb L x {x}"
assms_1: shows "p \<TTurnstile> netglobal P"
vp_e: shows "v' \<in> E e \<rho>'"
emeasure_space_le_1: shows "emeasure M (space M) \<le> 1"
below_in_P_simplex_like: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ComplexLikePoset less_eq less P"   and "x \<in> P" shows "ordering.simplex_like less_eq less {y \<in> P. less_eq y x}"
assms_1: shows "\<forall>x\<in>M. \<forall>y\<in>M. f x = f y"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>\<mu> f : G a \<rightarrow>\<^sub>D G' b\<guillemotright>"
sub: shows "set (bts_dfs alist ts) \<subseteq> set (elements xs)"
subst_lift: fixes t :: "dB"   and k :: "nat"   and s :: "dB" shows "lift t k[s/k] = t"
step: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> segment f a b"   and "f x \<noteq> b" shows "f x \<in> segment f a b"
simps_2: fixes i :: "nat"   and l :: "nat" shows "a i (Suc l) = (1::'a) / d l \<cdot>\<^sub>v (d (Suc l) \<cdot>\<^sub>v a i l - \<mu>' i l \<cdot>\<^sub>v gso' l)"
makes_compatible_invariant: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and k :: "nat"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "(cms\<^sub>1, mem\<^sub>1) \<leadsto>\<^bsub>k\<^esub> (cms\<^sub>1', mem\<^sub>1')" obtains cms\<^sub>2'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" and mem\<^sub>2'' :: "'Var \<Rightarrow> 'Val" and mems' :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" where "map snd cms\<^sub>1' = map snd cms\<^sub>2'' \<and> (cms\<^sub>2, mem\<^sub>2) \<leadsto>\<^bsub>k\<^esub> (cms\<^sub>2'', mem\<^sub>2'') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2'', mem\<^sub>2'') mems'"
simps_4: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "D (IF b THEN c1 ELSE c2) = {(s, t). if bval b s then (s, t) \<in> D c1 else (s, t) \<in> D c2}"
minus_flow_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>" shows "flow_attainability.minus_flow \<Delta> f g = (\<lambda>(x, y). if edge \<Delta> x y then f (x, y) - g (x, y) else if edge \<Delta> y x then g (y, x) - f (y, x) else 0)"
continuous_on_max: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. max (f x) (g x))"
simps: fixes a1 :: "'d"   and a2 :: "'d \<Rightarrow> real"   and a3 :: "intv" shows "intv_elem a1 a2 a3 = ((\<exists>u x d. a1 = x \<and> a2 = u \<and> a3 = Const d \<and> u x = real d) \<or> (\<exists>d u x. a1 = x \<and> a2 = u \<and> a3 = Intv d \<and> real d < u x \<and> u x < real (d + 1)) \<or> (\<exists>c u x. a1 = x \<and> a2 = u \<and> a3 = Greater c \<and> real c < u x))"
not_mod_2_eq_0_eq_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(modulo a (numeral.numeral one plus (num.Bit0 num.One)) \<noteq> zero) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
dom_def: fixes m :: "'c \<Rightarrow> 'd option" shows "dom m = {a. m a \<noteq> \<bottom>}"
2: shows "x\<^sup>T ; r \<le> x\<^sup>\<star> ; r"
to_fract_eq_0_iff: fixes x :: "'b" shows "(to_fract x = 0) = (x = (0::'b))"
select_convs_8: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getWatch1 \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getWatch1"
power_int_def: fixes x :: "'b"   and n :: "int" shows "x powi n = (if 0 \<le> n then x ^ nat n else inverse x ^ nat (- n))"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d llist" shows "lmap g (lmap f v) = lmap (g \<circ> f) v"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" shows "list_member (x # xs) y = (equal x y \<or> list_member xs y)"
fdia_star_unfold_var: fixes y :: "'a"   and x :: "'a" shows "d y +  |x\<rangle>  |x\<^sup>\<star>\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
r_psi_def: shows "r_psi \<equiv> let f = Cn 3 r_less [recf.Id 3 2, Cn 3 r_length [Cn 3 r_prefixes [recf.Id 3 0, recf.Id 3 1]]] in Cn 2 r_nth [Cn 2 r_prefixes [Mn 2 f, recf.Id 2 0], recf.Id 2 1]"
is_inf_related: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y" shows "is_inf x y x"
map_pair_subset: fixes f :: "'f \<times> 'h \<Rightarrow> 'f \<times> 'g"   and xs :: "('f \<times> 'h) list" shows "set (map_pair f xs) \<subseteq> f ` set xs"
weakCongruence_def: fixes P :: "ccs"   and Q :: "ccs" shows "P \<cong> Q \<equiv> P \<leadsto><weakBisimulation> Q \<and> Q \<leadsto><weakBisimulation> P"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and llist :: "'d llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (\<forall>x21 x22. llist = LCons x21 x22 \<longrightarrow> P (f2 x21 x22)))"
integer_test_bit_def: fixes x :: "integer"   and n :: "integer" shows "integer_test_bit x n = (if n < 0 then undefined x n else bit x (nat_of_integer n))"
polyadd_eq_const_degreen: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and c :: "'b"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "p +\<^sub>p q = (c)\<^sub>p" shows "degreen p m = degreen q m"
dvd_refl: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a a"
j: shows "\<forall>j\<in>set (j # xs). j < min CARD('nr) CARD('nc)"
view_partitioned_def: shows "view_partitioned \<equiv> \<forall>execs ms mt n u. equivalent_states ms mt u \<longrightarrow> B2 (run n ms execs) (run n mt (purge execs u)) (\<lambda>rs rt. vpeq u rs rt \<and> current rs = current rt)"
Abs_multiset_inject: fixes x :: "'k \<Rightarrow> nat"   and y :: "'k \<Rightarrow> nat" assumes "x \<in> multiset"   and "y \<in> multiset" shows "(Abs_multiset x = Abs_multiset y) = (x = y)"
g_step_def: fixes c_ls :: "nat"   and key :: "nat" shows "g_step c_ls key = (let n = c_fst key; x = c_snd key; n1 = c_fst n mod 7 in if n1 = 0 then c_cons (c_pair key 0) c_ls else if n1 = 1 then c_cons (c_pair key (Suc x)) c_ls else if n1 = 2 then c_cons (c_pair key (c_fst x)) c_ls else if n1 = 3 then c_cons (c_pair key (c_snd x)) c_ls else if n1 = 4 then g_comp c_ls key else if n1 = 5 then g_pair c_ls key else if n1 = 6 then g_rec c_ls key else c_ls)"
collapse_1: fixes "term" :: "('c, 'd) Term.term" assumes "is_Var term" shows "Var (the_Var term) = term"
\<delta>: fixes p :: "fm" assumes "iszlfm p" shows "d_\<delta> p (\<delta> p) \<and> 0 < \<delta> p"
inj_on_finite: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
normal_hd_coeffs1: fixes \<phi> :: "atom fm" assumes "qfree \<phi>"   and "Z.normal \<phi>" shows "Z.normal (hd_coeffs1 \<phi>)"
onlD: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and P :: "'i \<times> 'h \<Rightarrow> bool"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "onl \<Gamma> P (\<xi>, p)" shows "\<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
statesNames_finite: fixes s :: "(integer \<times> edge list) iarray" shows "finite (statesNames s)"
proj2_Col_coeff: fixes a :: "proj2"   and r :: "proj2"   and t :: "proj2" assumes "proj2_Col a r t"   and "a \<noteq> r"   and "t \<noteq> a" shows "t = proj2_abs (proj2_Col_coeff a r t *\<^sub>R proj2_rep a + proj2_rep r)"
sp_append_one: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es :: "'d edge list"   and e :: "'d edge"   and v2 :: "'d" shows "Graph.subpath g v1 (es @ [e]) v2 = (Graph.subpath g v1 es (src e) \<and> e \<in> edges g \<and> tgt e = v2)"
induced_arrow_cones_map_1: fixes a' :: "'c" assumes "C.ide a'" shows "(\<lambda>f. D.cones_map f \<chi>) \<in> C.hom a' a \<rightarrow> D.cones a'"
Bc_large: shows "\<forall>\<^sub>F n in sequentially. B * int (nth_prime n) = c n * a n - c (n + 1) \<and> \<bar>real_of_int (c (n + 1))\<bar> < real_of_int (a n) / 2"
assms: shows "cmod (z - z0) \<noteq> \<bar>r\<bar>"
dist_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and d :: "nat"   and v' :: "nat" shows "Graph.dist c v d v' \<equiv> \<exists>p. Graph.isPath c v p v' \<and> length p = d"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>e n. \<lbrakk>n \<noteq> 0 \<Longrightarrow> P 0 e; n \<noteq> 0 \<Longrightarrow> P (e + 1) (n div base)\<rbrakk> \<Longrightarrow> P e n" shows "P a0 a1"
asAbs_equal_imp_alphaAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and qB :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs qA"   and "asAbs qA = asAbs qB" shows "qA $= qB"
wp: shows "\<guillemotleft> P \<guillemotright> \<tturnstile> t \<guillemotleft> Q \<guillemotright>"
sterms_no_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "p1 \<oplus> p2 \<notin> sterms \<Gamma> p"
suffix_take: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "ys = take (length ys - length xs) ys @ xs"
carrier_possible_subset: fixes t :: "'b ttree"   and A :: "'b set"   and x :: "'b" assumes "carrier t \<subseteq> A"   and "possible t x" shows "x \<in> A"
algebraicI: fixes p :: "'b poly"   and x :: "'b" assumes "\<And>i. coeff p i \<in> \<int>"   and "p \<noteq> 0"   and "poly p x = (0::'b)" shows "algebraic x"
1: shows "\<exists>!G. \<langle>X, G\<rangle> \<^bold>\<in> hlam A B C F"
avar_def2: fixes G :: "prog"   and i' :: "val"   and a' :: "val"   and s :: "state" shows "avar G i' a' s = ((the (snd (snd (the_Arr (globs (snd s) (Inl (the_Addr a'))))) (Inr (the_Intg i'))), \<lambda>v (x, s'). (raise_if (\<not> G,s'\<turnstile>v fits fst (the_Arr (globs (snd s) (Inl (the_Addr a'))))) ArrStore x, upd_gobj (Inl (the_Addr a')) (Inr (the_Intg i')) v s')), abupd (raise_if (\<not> the_Intg i' in_bounds fst (snd (the_Arr (globs (snd s) (Inl (the_Addr a')))))) IndOutBound \<circ> np a') s)"
map_diagonalize: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "E.map (\<F>C.D f) = E.map f"
0: shows "finite {m. f (lookup (mapping_of p) m) \<noteq> (0::'a)}"
drlex_pp_def: shows "drlex_pp \<equiv> map_fun pp.mapping_of (map_fun pp.mapping_of id) drlex_pm"
wf_subtype_env: fixes \<Gamma> :: "binding list"   and P :: "POPLmark.type"   and Q :: "POPLmark.type" assumes "\<Gamma> \<turnstile> P <: Q" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
final1_simulation: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "case s1 of (x1, m) \<Rightarrow> final1 x1" shows "\<exists>s2'. r2.silent_moves t s2 s2' \<and> t \<turnstile> s1 \<approx> s2' \<and> (case s2' of (x2, m) \<Rightarrow> final2 x2)"
square_free_smult: fixes c :: "'a"   and f :: "'a poly" assumes "c \<noteq> (0::'a)"   and "square_free f" shows "square_free (smult c f)"
R_R\<^sub>N: fixes A :: "'a set"   and u :: "nat"   and l :: "nat" assumes "finite A"   and "p \<in> {0..1}" shows "R A u l = R\<^sub>N (card A) u l"
prv_allE: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "prv (all x \<phi>)" shows "prv (subst \<phi> t x)"
mod_exE: fixes P :: "'b \<Rightarrow> assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> (\<exists>\<^sub>Ax. P x)" obtains x :: "'b" where "h \<Turnstile> P x"
free_name_def: fixes t :: "'a" shows "free_name t = (case unfree t of Some name \<Rightarrow> name)"
prems_3: shows "\<I> \<turnstile>res res \<surd>"
cos_is_zero: shows "\<exists>!x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.gt_same ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s) \<and> head t = head s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym) (args t) (args s)))"
prem: fixes x :: "'b"   and y :: "'c" shows "g (h (x, y)) = x"