hcju/isabelleps huggingface dataset:
Cons: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es :: "char list exp list"   and es' :: "char list exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>Val v # es',s\<^sub>2\<rangle>"
residue_times_zero_r: fixes k :: "nat"   and y :: "int" shows "0 \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y = 0"
prems_3: shows "(p', ins, outs, c) \<in> set procs"
convex_closure_inter_two: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T"   and "rel_interior S \<inter> rel_interior T \<noteq> {}" shows "closure (S \<inter> T) = closure S \<inter> closure T"
select_convs_1: fixes carrier :: "'d set"   and eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
not_none_is_obs: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "IFC_def.att prob (\<pi> i) \<noteq> None" shows "IFC_def.is_kth_obs prob \<pi> (card (IFC_def.obs_ids prob \<pi> \<inter> {..<i})) i"
fair_clss_seq_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and Ns :: "'b literal multiset set llist" assumes "effective_redundancy_criterion \<Gamma> Rf Ri" shows "effective_redundancy_criterion.fair_clss_seq \<Gamma> Rf Ri Ns = (let N' = Liminf_llist Ns - Rf (Liminf_llist Ns) in concls_of (inference_system.inferences_from \<Gamma> N' - Ri N') \<subseteq> Sup_llist Ns \<union> Rf (Sup_llist Ns))"
select_convs_2: fixes src :: "'d"   and tgt :: "'d"   and more :: "'e" shows "tgt \<lparr>src = src, tgt = tgt, \<dots> = more\<rparr> = tgt"
\<iota>_eq_trm: shows "\<iota> = \<t>[\<one> \<otimes> \<one>]"
addpre_empty: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "addpre r {} = r"
Ord_linear2: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" obtains "k \<^bold>\<in> l"   | "l \<le> k"
scale_left_imp_eq: fixes a :: "'b"   and x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" assumes "a \<noteq> (0::'b)"   and "a *s x = a *s y" shows "x = y"
card_of_ordLeq_infinite: fixes A :: "'d set"   and B :: "'e set" assumes "ordLeq3 (card_of A) (card_of B)"   and "infinite A" shows "infinite B"
elims: fixes x :: "('a, 'b) fmap"   and xa :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and y :: "('a, 'b) fmap" assumes "exec_plan x xa = y" obtains s :: "('a, 'b) fmap" where "x = s__" and "xa = []" and "y = s__"   | s :: "('a, 'b) fmap" and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" where "x = s" and "xa = a # as" and "y = exec_plan (state_succ s a) as"
i: shows "\<forall>x\<in>P. set_entails_clause (cl_ecl ` S) (cl_ecl x)"
multpw_splitL: fixes X :: "'a multiset"   and Y1 :: "'a multiset"   and Y2 :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(X, Y1 + Y2) \<in> multpw ns" obtains X1 :: "'a multiset" and X2 :: "'a multiset" where "X = X1 + X2" and "(X1, Y1) \<in> multpw ns" and "(X2, Y2) \<in> multpw ns"
True_start_steps_star: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and rr :: "bool list"   and w :: "'b list" shows "((True # start A, rr) \<in> NAe.steps (RegExp2NAe.star A) w) = (\<exists>us v. w = concat us @ v \<and> (\<forall>u\<in>set us. NAe.accepts A u) \<and> (\<exists>r. (start A, r) \<in> NAe.steps A v \<and> rr = True # r))"
A_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and QA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and QB :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Simulation_Invariants A B sim PA QA PB QB"   and "PA a"   and "A a b" shows "QA b"
simps_1: fixes uu :: "'b"   and B :: "'b"   and uv :: "'c" shows "Maps.Cod (Maps.MkArr uu B uv) = B"
update_by_fun_gr_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "update_by_fun_gr_pair_ko_ntm ko \<equiv> comparator.update_by_fun_gr_pair (nat_term_compare_inv ko)"
f_last_message_hold_length: fixes xs :: "'b message_af list"   and k :: "nat" shows "length (xs \<longmapsto> k) = length xs"
fps_compose_nth: fixes a :: "'a fps"   and b :: "'a fps"   and n :: "nat" shows "(a oo b) $ n = (\<Sum>i = 0..n. a $ i * b ^ i $ n)"
Pd_Xd: fixes x :: "'a"   and y :: "'a"   and d :: "'b" assumes "(x, y) \<in> Pd'_tax d" shows "Xd x = d \<and> Xd y = d"
wfc_traces: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and u :: "'b"   and xs :: "'a list"   and ys :: "'a list" assumes "weakly_future_consistent P I D R"   and "u \<in> range D \<inter> (- I) `` range D"   and "(xs, ys) \<in> R u" shows "(xs \<in> traces P) = (ys \<in> traces P)"
rep_eq: shows "count {#} = (\<lambda>a. 0)"
vwb_lens_def: fixes x :: "'c \<Longrightarrow> 'd" shows "vwb_lens x \<equiv> wb_lens x \<and> mwb_lens x"
ide_in_iso_class: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.ide f" shows "f \<in> Maps.iso_class f"
invalid_assn_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt B x y" shows "hn_invalid A x y \<Longrightarrow>\<^sub>t hn_invalid B x y"
set_return_pmf: fixes x :: "'b" shows "set_pmf (return_pmf x) = {x}"
Max_by_dest: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "A \<noteq> {}" shows "Max_by f A \<in> A \<and> f (Max_by f A) = Max (f ` A)"
combine_infmatrix_assoc: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "f (0::'b) (0::'b) = (0::'b)"   and "associative f" shows "associative (combine_infmatrix f)"
left_commute: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" shows "b \<inter># (a \<inter># c) = a \<inter># (b \<inter># c)"
inorder_n1: fixes t :: "'a bro" shows "inorder (n1 t) = inorder t"
that: shows "A \<turnstile>\<^sub>R res1 \<sim> res2"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (F.map\<^sub>0 a) = F.map\<^sub>0 a"
evaln_FinA: fixes G :: "prog"   and s :: "st"   and a :: "abopt"   and c :: "stmt"   and n :: "nat"   and v :: "vals"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>In1r (FinA a c)\<succ>\<midarrow>n\<rightarrow> (v, s') = False"
int: shows "integrable M (\<lambda>x. D x * log b (D x))"
zcf_to_fun: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "zcf p = to_fun p \<zero>"
permute_flip_cancel2: fixes a :: "'f"   and b :: "'f"   and x :: "'e" shows "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
Liminf_bot: fixes f :: "'d \<Rightarrow> 'b" shows "Liminf bot f = top"
steps: shows "\<Gamma>\<turnstile> ([c], [], s) \<rightarrow>\<^sup>* ([], [], t)"
assms_2: shows "constructive_security_obsf real middle sim_outer \<I>_real \<I>_middle \<I>_common \<A> adv2"
inv_char: fixes f :: "'a" assumes "hom'.iso f" shows "hom'.inv f = local.inv f"
a1: shows "(s, {tick} \<union> ev ` events_of P) \<in> F P"
in_set_independent_pdevs_nonzero: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "x \<in> set (independent_pdevs xs)" shows "x \<noteq> 0"
l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs \<equiv> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs"
induct: fixes P :: "'a list list \<Rightarrow> 'a list list \<Rightarrow> bool"   and a0 :: "'a list list"   and a1 :: "'a list list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
function_ring_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme" shows "function_ring S R = \<lparr>carrier = S \<rightarrow>\<^sub>E carrier R, mult = function_mult S R, one = function_one S R, zero = function_zero S R, add = function_add S R, smult = function_scalar_mult S R\<rparr>"
unwinding_theorem_BSIA: fixes \<rho> :: "'e V_rec \<Rightarrow> 'e set"   and ur :: "('s \<times> 's) set" assumes "lrbe \<rho> ur"   and "osc ur" shows "BSIA \<rho> \<V> Tr\<^bsub>(induceES SES)\<^esub>"
finsert_mono: fixes C :: "'b fset"   and D :: "'b fset"   and a :: "'b" assumes "C |\<subseteq>| D" shows "finsert a C |\<subseteq>| finsert a D"
image_chamber_system: shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> \<C>"
bottom_below_in_relative: fixes P :: "'a set"   and y :: "'a"   and x :: "'a" assumes "has_bottom (P .\<^bold>\<le> y)"   and "x \<in> P"   and "y \<succeq> x" shows "bottom (P .\<^bold>\<le> x) = bottom (P .\<^bold>\<le> y)"
Healthy_if: fixes P :: "'d upred"   and H :: "'d upred \<Rightarrow> 'd upred" assumes "P is H" shows "H P = P"
insertI_2_3: shows "\<forall>t\<in>{s<..<c}. \<not> P t"
\<delta>\<^sub>1_def_2: shows "p\<^sub>1 = find_closest_bf p (p\<^sub>2 # ps)"
unfoldr_append_generator_Inl: fixes g1 :: "('a, 'b) generator"   and g2 :: "('a, 'c) generator"   and s1 :: "'b"   and s2 :: "'c" shows "list.unfoldr (append_generator g1 g2) (Inl (s1, s2)) = list.unfoldr g1 s1 @ list.unfoldr g2 s2"
srelated_find_match: fixes cs :: "(Term_Class.term \<times> pterm) list"   and t :: "pterm"   and penv :: "(name, pterm) fmap"   and pat :: "Term_Class.term"   and rhs :: "pterm"   and senv :: "(name, sterm) fmap" assumes "find_match cs t = Some (penv, pat, rhs)"   and "srelated'.P_env penv senv" shows "find_match (map (map_prod id pterm_to_sterm) cs) (pterm_to_sterm t) = Some (senv, pat, pterm_to_sterm rhs)"
sep_implD: fixes x :: "'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(x \<longrightarrow>* y) h" shows "\<forall>h'. h ## h' \<and> x h' \<longrightarrow> y (h + h')"
ide_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "ide (src \<mu>)"
sel_1: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c" assumes "ParityGame G" shows "lhd (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0) = v0"
word_less_eq_iff_unsigned: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'b word"   and b :: "'b word" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(a \<le> b) = less_eq (semiring_1.unsigned one plus zero a) (semiring_1.unsigned one plus zero b)"
const_p_fac: fixes m :: "nat"   and p :: "nat"   and a :: "nat" assumes "0 < m"   and "prime p" shows "multiplicity p (p ^ a * m choose p ^ a) = multiplicity p m"
diff_diff_add: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a b) c = minus a (plus b c)"
r_child_def: fixes l :: "int"   and p :: "int" shows "r_child l p = 2 * p - l + 2"
nterm_to_term_wellformed: fixes \<Gamma> :: "name list"   and t :: "nterm" shows "wellformed' (length \<Gamma>) (nterm_to_term \<Gamma> t)"
assms_5: shows "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)"
isomorphic_to_zero_left: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModule G smult V"   and "V = 0"   and "\<exists>T. FGModuleIso G smult V smult' T W" shows "W = 0"
Suc_2: shows "\<forall>j<Suc m. \<exists>no. \<forall>n\<ge>no. P n j"
sheaf_of_rings_axioms: shows "sheaf_of_rings S is_open imS.im_sheaf imS.im_sheaf_morphisms b imS.add_im_sheaf imS.mult_im_sheaf imS.zero_im_sheaf imS.one_im_sheaf"
nprod_mem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and l :: "nat" assumes "\<forall>i\<le>n. f i \<in> carrier R"   and "l \<le> n" shows "e\<Pi>\<^bsub>R,l\<^esub> f \<in> carrier R"
inter_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(a \<in># A \<inter># B) = (a \<in># A \<and> a \<in># B)"
image_from_trivial_group: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "trivial_group G" shows "h ` carrier G = {\<one>\<^bsub>H\<^esub>}"
r_cancel_one': fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (K [X])"   and "a \<in> carrier (K [X])" shows "(x = a \<oplus>\<^bsub>K [X]\<^esub> x) = (a = \<zero>\<^bsub>K [X]\<^esub>)"
degree_gen_shleg_poly: fixes k :: "nat" shows "degree (local.gen_shleg_poly k) = 2 * n - k"
cons: fixes c :: "'f"   and a :: "'g"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set"   and w :: "'g list"   and c'' :: "'f" assumes "(c, a, c') \<in> t"   and "(c', w, c'') \<in> trcl t" shows "(c, a # w, c'') \<in> trcl t"
chainTransitionAppend_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'"   and "x \<sharp> P" shows "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"
set_minus_filter_out: fixes xs :: "'a list"   and y :: "'a" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
scf_lowering_axioms: shows "scf_lowering agents alts scf {A1, A2, A3} {a, b, c}"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_uminus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vreal_uminus" shows "b = c"
ennreal_sum_less_top: fixes I :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I < top) = (\<forall>i\<in>I. f i < top)"
nths_drop: fixes n :: "nat"   and xs :: "'c list"   and I :: "nat set" shows "nths (drop n xs) I = nths xs ((+) n ` I)"
comp_fun_idem_insert_mapping: shows "comp_fun_idem insert_mapping"
not_le: fixes x :: "'b"   and y :: "'b" shows "(\<not> x \<le> y) = (y < x)"
ffun_member_minus: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" shows "((x, y) \<in>\<^sub>f f - g) = ((x, y) \<in>\<^sub>f f \<and> \<not> (x, y) \<in>\<^sub>f g)"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "ide (V a)"
subd_diff: fixes p :: "nat"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" shows "subd p (c1 - c2) = subd p c1 - subd p c2"
intros_1: fixes Env :: "env"   and B :: "lname set" shows "Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
wtSB_tpOf: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and x :: "Preliminaries.var" assumes "wtSB \<pi>" shows "tpOf (\<pi> x) = tpOfV x"
trivial_limit_at_right_real: fixes x :: "'a" shows "at_right x \<noteq> bot"
eval: shows "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"
consistent_signs_at_roots_def: fixes p :: "real poly"   and qs :: "real poly list" shows "consistent_signs_at_roots p qs = sgn_vec qs ` roots p"
dg_Par_components_1: fixes \<alpha> :: "V" shows "dg_Par \<alpha>\<lparr>Obj\<rparr> = Vset \<alpha>"
is_maprop: fixes x :: "'a" shows "is_map x = (x ; - 1' = - x)"
proj3_def: shows "\<pi>\<^sub>3 \<equiv> \<lambda>(uu_, dsk, uu_, uu_, uu_). dsk"
GSMP_disjoint_term: fixes l :: "'a"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and l' :: "'a"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec" shows "t \<notin> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<or> t \<notin> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) \<or> t \<in> Sec \<or> {} \<turnstile>\<^sub>c t"
natural_isomorphism_axioms: shows "natural_isomorphism \<F>C.CC.comp (\<cdot>\<^sub>S) D.FoT\<^sub>C.map D.FoT\<^sub>C.map D.FoT\<^sub>C.map"
executableI_step: fixes p :: "'state"   and a :: "'action" assumes "p \<in> nodes"   and "en a p" shows "a \<in> executable"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "f (b1 + b2) - g (b1 + b2) = f b1 - g b1 + (f b2 - g b2)"
assms_2: shows "(b, c) \<in> A\<^sup>\<down>"
bi_uniqueI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique R"   and "right_unique R" shows "bi_unique R"
except_plus: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "except (p + q) S = except p S + except q S"
cond_idem: fixes b :: "'c upred"   and P :: "('b, 'c) uexpr" shows "P \<triangleleft> b \<triangleright> P = P"
IH: shows "(Abs ^^ n) (var_dist (rev [0..<n]) (liftn n t 0)) \<leftrightarrow> t"
k: shows "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
cases: fixes f :: "nat \<Rightarrow> bool" assumes "wf_set_bits_int f" obtains n :: "nat" where "\<forall>n'\<ge>n__. \<not> f n'"   | n :: "nat" where "\<forall>n'\<ge>n. f n'"
assms_2: shows "\<forall>p. Ana (pair p) = ([], [])"
rel_refl_strong: fixes x :: "'a list"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set x \<Longrightarrow> Ra z z" shows "list_all2 Ra x x"
multiplicative_function_mult: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (\<lambda>n. f n * g n)"
restrict_def: fixes A :: "nat set"   and v :: "'b option list" shows "restrict A v = map (\<lambda>i. if i \<in> A then v ! i else None) [0..<length v]"
simps_2: fixes v :: "Vars.var"   and e :: "exp"   and as :: "assn" shows "asToHeap (ACons v e as) = (v, e) # asToHeap as"
subseq_sum_path_weights_le: fixes xs :: "('a \<times> 'lbl \<times> 'b) list"   and ys :: "('c \<times> 'lbl \<times> 'd) list" assumes "map (\<lambda>(s, l, t). l) xs \<preceq> map (\<lambda>(s, l, t). l) ys" shows "sum_weights (map (\<lambda>(s, l, t). l) xs) \<le> sum_weights (map (\<lambda>(s, l, t). l) ys)"
Equality_I: fixes H :: "fm set"   and y :: "tm"   and x :: "tm" assumes "H \<turnstile> y SUBS x"   and "H \<turnstile> x SUBS y" shows "H \<turnstile> x EQ y"
Suffixes_star: fixes A :: "'a list set" assumes "A \<noteq> {}" shows "Suffixes (star A) = Suffixes A @@ star A"
nat_power_diff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.nat_power_field one times plus zero nat_power divide inverse minus uminus"   and "0 < n" shows "nat_power n (minus a b) = divide (nat_power n a) (nat_power n b)"
simple: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "multiplicity bl = 1"
C: shows "\<forall>X\<in>A. X = r X \<union> s X \<union> t X \<and> set xs \<subseteq> range p \<union> range q \<and> r X \<subseteq> range p \<and> s X \<subseteq> range q \<and> t X \<subseteq> - range p \<and> t X \<subseteq> - range q \<and> (map (inv p) (filter (\<lambda>x. x \<in> range p) xs), inv p ` r X) \<in> failures P \<and> (map (inv q) (filter (\<lambda>x. x \<in> range q) xs), inv q ` s X) \<in> failures Q"
rawpsubstT_suc: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> trm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubstT (suc r) txs = suc (rawpsubstT r txs)"
bin_rest_BIT: fixes x :: "int"   and b :: "bool" shows "x BIT b div 2 = x"