hcju/isabelleps huggingface dataset:
expand_rec_rule: fixes \<Phi> :: "'b node \<times> 'b node set \<Rightarrow> bool"   and x :: "'b node \<times> 'b node set"   and M :: "'b node \<times> 'b node set \<Rightarrow> (nat \<times> 'b node set) nres" assumes "\<Phi> x"   and "\<And>f x. \<lbrakk>\<And>x. f x \<le> expand x; \<And>x. \<Phi> x \<Longrightarrow> f x \<le> M x; \<Phi> x\<rbrakk> \<Longrightarrow> (case x of (n, ns) \<Rightarrow> if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else f (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, {n} \<union> ns) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in if \<exists>q. \<phi> = prop\<^sub>r(q) \<or> \<phi> = nprop\<^sub>r(q) then if not\<^sub>r \<phi> \<in> old n then RETURN (name n, ns) else f (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = true\<^sub>r then f (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = false\<^sub>r then RETURN (name n, ns) else if \<exists>\<nu> \<mu>. \<phi> = \<nu> and\<^sub>r \<mu> \<or> \<phi> = X\<^sub>r \<nu> then f (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) else f (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). f (n\<lparr>name := nm, new := new2 \<phi> \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds)))) \<le> M x" shows "expand x \<le> M x"
simps_30: fixes vd :: "'b"   and v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b" shows "(SKIP {vd} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
fold_combine_plus_def: shows "PMF_Impl.fold_combine_plus = comm_monoid_set.F (Mapping.combine (+)) Mapping.empty"
trivial_group_alt: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (\<exists>a. carrier G \<subseteq> {a})"
sup_subE: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "\<exists>x. x \<in> c"   and "Subspace.subspace F E"   and "vectorspace E" shows "Subspace.subspace H E"
disjnt_self_iff_empty: fixes S :: "'b set" shows "disjnt S S = (S = {})"
degree_bound: fixes ws :: "'a poly list"   and vs :: "'a poly list" assumes "mset ws \<subseteq># mset vs"   and "length ws + length ws \<le> length vs" shows "degree (prod_list ws) \<le> degree_bound vs"
c_unfinished: shows "\<forall>i<Suc k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"
vs2: shows "vs2 = getFrN (map snd txs) (map fst txs) [\<phi>2] (length txs)"
converse_join: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(sup r s)\<inverse>\<inverse> = sup r\<inverse>\<inverse> s\<inverse>\<inverse>"
all_processes_snapshotted_in_final_state: fixes t :: "('a, 'b, 'c) event list"   and p :: "'a" assumes "trace init t final" shows "ps final p \<noteq> None"
FDOM_eff_subset_FDOM_valid_states: fixes p :: "('d, 'e) fmap"   and e :: "('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and s :: "('d, 'g) fmap" assumes "(p, e) \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' e \<subseteq> fmdom' s"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   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 get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"   and "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs"   and "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs remove_shadow_root remove_shadow_root_locs" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr"
vv: shows "v \<in> lang (seq [Star (Times (Atom y) (Atom x))])"
symrun_interp_decreases_setinc: fixes \<Gamma> :: "'b constr list"   and \<Gamma>' :: "'b constr list" assumes "set \<Gamma> \<subseteq> set \<Gamma>'" shows "\<lbrakk>\<lbrakk> \<Gamma>' \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<subseteq> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
Abs_preserves_good: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "goodAbs (Abs xs x X)"
assms_2: fixes x\<^sub>j :: "nat"   and l\<^sub>i :: "'a" assumes "min_rvar_incdec dir s x\<^sub>i = Inr x\<^sub>j"   and "l\<^sub>i = the (LB dir s x\<^sub>i)"   and "check' dir x\<^sub>i s = pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s" shows "P (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)"
tendsto_add_ereal_general2: fixes x :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal"   and y :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
intersect_num_in_set: fixes b1 :: "'b set"   and b2 :: "'b set" assumes "b1 \<in># \<B>'"   and "b2 \<in># remove1_mset b1 \<B>'" shows "b1 |\<inter>| b2 \<in> target.intersection_numbers"
concat_append: fixes xs :: "'b list list"   and ys :: "'b list list" shows "concat (xs \<cdot> ys) = concat xs \<cdot> concat ys"
absolutely_integrable_mult_Fejer_kernel_reflected3: fixes n :: "nat"   and c :: "real" shows "(\<lambda>x. Fejer_kernel n x * c) absolutely_integrable_on {- pi..pi}"
lemma2_6_1_multiset: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset" shows "r \<down>m (M + N) = r \<down>m M \<union> r \<down>m N"
add_mset_diff_bothsides: fixes a :: "'a"   and M :: "'a multiset"   and A :: "'a multiset" shows "add_mset a M - add_mset a A = M - A"
dropWhile_eq_Nil_conv: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(dropWhile P xs = []) = (\<forall>x\<in>set xs. P x)"
61313: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - (- - y \<sqinter> x)"
n_def: shows "n \<equiv> LEAST k. k \<noteq> 0 \<and> x \<in> elts (y * ord_of_nat k)"
simps_5: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> and\<^sub>n \<psi>) = combine mk_and (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
assms: fixes s :: "'a"   and t :: "'b" shows "A (s, t)"
conc_def: fixes p :: "'a \<times> 'a list \<times> 'a"   and q :: "'a \<times> 'a list \<times> 'a" shows "conc p q = (fst p, fst (snd p) @ fst q # fst (snd q), snd (snd q))"
simps_28: fixes x51 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x52 :: "exp"   and x53 :: "exp"   and x31 :: "nat"   and x32 :: "exp" shows "EPrim x51 x52 x53 \<noteq> ELam x31 x32"
relAPP_def: fixes f :: "('h \<times> 'i) set \<Rightarrow> 'g"   and x :: "('h \<times> 'i) set" shows "\<langle>x\<rangle>f \<equiv> f x"
assms_6: shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"
dominate_trans: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat"   and n3 :: "nat" assumes "cfg_doms G"   and "cfg_doms.dominate G n1 n2"   and "cfg_doms.dominate G n2 n3" shows "cfg_doms.dominate G n1 n3"
h9b: shows "0 < n \<longrightarrow> MPoly_Type.degree (sum f {0..n}) x = n"
self_evaluation_map_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c" shows "self_evaluation_map V H \<a> \<i> src trg \<equiv> bicategory V H \<a> \<i> src trg"
S_def: shows "S \<equiv> {z. Re z = x \<and> Im z \<in> {y1..y2}}"
Rel_Rel_match: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "Transfer.Rel R x y" shows "Rel_match R x y"
filternew_flows_state_alt2: fixes \<T> :: "'a stateful_policy" shows "filternew_flows_state \<T> = {e \<in> flows_state \<T>. e \<notin> backflows (flows_fix \<T>)}"
c: shows "max (conv_abscissa f) (conv_abscissa g) < ereal c"
elem_le_zlcms: fixes "is" :: "int list"   and i :: "int" assumes "\<forall>i\<in>set is. i \<noteq> 0"   and "i \<in> set is" shows "i \<le> zlcms is"
a_repr_independence: fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H +> x"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "H +> x = H +> y"
diff_numeral_special_12: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus one) (uminus one) = zero"
jumpF_polyL_0_2: fixes q :: "real poly"   and a :: "real" shows "jumpF_polyL q 0 a = 0"
u_3: fixes y :: "'a" assumes "y \<in> cball x eu" shows "s (flow0 y (u y)) = 0"
lemma6: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and \<pi>\<^sub>A :: "Syntax.term list"   and \<pi>' :: "Syntax.term list"   and eV' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>\<pi>\<^sub>A, eV\<ggreater> V\<rightarrow> \<lless>\<pi>', eV'\<ggreater>" obtains e' :: "Syntax.term" and eP' :: "Syntax.term" and \<pi> :: "Syntax.term list" where "\<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>" and "\<forall>\<pi>\<^sub>P. \<lless>\<pi>\<^sub>P, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>P @ \<pi>, eP'\<ggreater>" and "{$$} \<turnstile> e', eP', eV' : \<tau> \<and> \<pi>\<^sub>A = \<pi> @ \<pi>' \<or> (\<exists>s s'. closed s \<and> closed s' \<and> \<pi> = [s] \<and> \<pi>\<^sub>A = [s'] @ \<pi>' \<and> s \<noteq> s' \<and> hash s = hash s')"
cSup_upper2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set"   and y :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "less_eq y x"   and "preorder.bdd_above less_eq X" shows "less_eq y (Sup X)"
pr_coincidence: shows "\<iota> = \<r>[\<one>]"
subst_upd_pr_var: fixes s :: "'b \<Rightarrow> 'c"   and x :: "'d \<Longrightarrow> 'c"   and v :: "('d, 'b) uexpr" shows "s(&x \<mapsto>\<^sub>s v) = s(x \<mapsto>\<^sub>s v)"
matrix_add_ldistrib: fixes A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'd) vec"   and C :: "(('a, 'b) vec, 'd) vec" shows "A ** (B + C) = A ** B + A ** C"
integrable_real_mult_indicator: fixes A :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. f x * indicat_real A x)"
same_one_same_zero_1: fixes R :: "'a ring" assumes "R \<in> C" shows "\<one>\<^bsub>union_ring C\<^esub> = \<one>\<^bsub>R\<^esub>"
eq_entails: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "P = Q" shows "P \<tturnstile> Q"
assms: fixes i :: "'c"   and j :: "'a" assumes "i \<in> Basis"   and "j \<in> Basis" shows "continuous F (\<lambda>x. blinfun_apply (f x) j \<bullet> i)"
synthetic_div_pCons: fixes a :: "'a"   and p :: "'a poly"   and c :: "'a" shows "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
is_symbol_distinct: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.is_symbol \<NN> \<TT> x" shows "CFG.is_terminal \<TT> x \<noteq> CFG.is_nonterminal \<NN> x"
eucl_of_list_inner_eq: fixes xs :: "real list"   and ys :: "real list" assumes "length xs = DIM('c)"   and "length ys = DIM('c)" shows "eucl_of_list xs \<bullet> eucl_of_list ys = inner_lv_rel xs ys"
x_in_\<pi>: shows "x \<in> \<pi> k {} I"
ih: fixes r :: "'a rexp" shows "Derivs s (lang r) = \<Union> (lang ` pderivs s r)"
preprocess'_Tableau_Poly_Mapping_Some': fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat"   and p :: "linear_poly"   and v :: "nat" assumes "Poly_Mapping (preprocess' cs start) p = Some v" shows "\<exists>h. poly h = p \<and> \<not> is_monom (poly h) \<and> qdelta_constraint_to_atom h v \<in> snd ` set (Atoms (preprocess' cs start))"
prems_3: shows "(x, y) \<in> set (pair # pairs)"
ideal_apow_vp: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "I = vp K v\<^bsup> Vr K v n_val K v (Ig K v I)\<^esup>"
bernoulli_in_Rats: fixes n :: "nat" shows "bernoulli n \<in> \<rat>"
rotate_4: shows "round_robin_reschedule t0 (t' # queue) n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"
expectation_of_bool_is_pmf: fixes M :: "bool pmf" shows "measure_pmf.expectation M of_bool = pmf M True"
simps_2: fixes f :: "'b \<rightarrow> 'b \<rightarrow> 'b"   and x :: "'b"   and xs :: "['b]" shows "foldl1\<cdot>f\<cdot>(x : xs) = foldl\<cdot>f\<cdot>x\<cdot>xs"
IC_bd: fixes s :: "real"   and t :: "real"   and p :: "nat"   and q :: "nat"   and i :: "nat"   and j :: "nat" assumes "s \<le> t"   and "correct p t"   and "correct q t" shows "\<bar>IC p i t - IC q j t\<bar> \<le> \<bar>IC p i s - IC q j s\<bar> + 2 * \<rho> * (t - s)"
isometry_on_compose: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "isometry_on X f"   and "isometry_on (f ` X) g" shows "isometry_on X (\<lambda>x. g (f x))"
prod_list_het0: fixes x :: "real list" assumes "x \<noteq> [] \<and> het x = 0" shows "prod_list x = mean x ^ length x"
out_out_two_sides: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and Y :: "'a"   and I :: "'a"   and U :: "'a"   and V :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.TS Bet A B X Y"   and "Tarski_neutral_dimensionless.Col Bet I A B"   and "Tarski_neutral_dimensionless.Col Bet I X Y"   and "Tarski_neutral_dimensionless.Out Bet I X U"   and "Tarski_neutral_dimensionless.Out Bet I Y V" shows "Tarski_neutral_dimensionless.TS Bet A B U V"
in_face_cycle_setD: fixes y :: "'b"   and x :: "'b" assumes "y \<in> GM.face_cycle_set x"   and "x \<in> arcs G" shows "y \<in> arcs G"
NegNeg_I: fixes H :: "fm set"   and A :: "fm" assumes "H \<turnstile> A" shows "H \<turnstile> Neg (Neg A)"
small_vreal_ls: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> real_of_vreal a < real_of_vreal b}"
progress_time_conv: fixes \<pi> :: "'b prefix"   and \<pi>' :: "'b prefix" assumes "pts \<pi> = pts \<pi>'" shows "progress \<pi> = progress \<pi>'"
o_def: fixes f :: "'j \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "f \<circ> g = (\<lambda>x. f (g x))"
type_eq_Ity: fixes v :: "val" shows "(type v = Ity) = (\<exists>i. v = Iv i)"
rv_less_iff: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "measure_space M" shows "(f \<in> rv M) = (\<forall>a. {w. f w < a} \<in> measurable_sets M)"
I_or: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "'a list" shows "DLO.I (or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = DLO.I (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
rat_abs_code: fixes p :: "rat" shows "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
west_def: shows "west = proj2_abs (vector [- 1, 0, 1])"
fusion_spec_local: fixes a :: "'d"   and init :: "'d set"   and p :: "'d \<times> 'f \<Rightarrow> bool"   and r :: "'d \<times> 'f \<Rightarrow> 'e \<Rightarrow> bool"   and p' :: "'f \<Rightarrow> bool"   and r' :: "'f \<Rightarrow> 'e \<Rightarrow> bool" assumes "a \<in> init" shows "([: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {. p .} \<circ> [: r :]) \<parallel> ({. p' .} \<circ> [: r' :]) = [: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {.u, x. p (u, x) \<and> p' x.} \<circ> [: (u, x) \<leadsto> y . r (u, x) y \<and> r' x y :]"
rExt: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', R, S) \<in> X" shows "(\<Psi>' \<otimes> \<Psi>'', R, S) \<in> X \<or> \<Psi>' \<otimes> \<Psi>'' \<rhd> R \<sim> S"
saturatep_saturate_eq: fixes \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset set" shows "inference_system.saturatep \<Gamma> (\<lambda>x. x \<in> CC) = (\<lambda>x. x \<in> inference_system.saturate \<Gamma> CC)"
YFtorObj: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "YFtor C @@ X = Hom\<^bsub>C\<^esub>[\<midarrow>,X]"
gL_rel_mult: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "D x y" shows "partial_semigroup.gL_rel times D y (times x y)"
False: shows "\<not> (uint a1 = 8 \<or> uint a1 = 10)"
left_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "Groups.group f z inverse" shows "f (inverse a) a = z"
cmod_eq_Re: fixes z :: "complex" assumes "is_real z" shows "cmod z = \<bar>Re z\<bar>"
vars_code: fixes lp :: "linear_poly" shows "vars lp = fset (fmdom (linear_poly_map lp))"
assms_2: shows "m1 \<in> carrier (\<G> \<eta>)"
subgroup_incl: fixes I :: "('c \<Rightarrow> 'a) set"   and J :: "('c \<Rightarrow> 'a) set" assumes "subgroup I (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "subgroup J (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "I \<subseteq> J" shows "subgroup I (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<lparr>carrier := J\<rparr>)"
card_of_empty2: fixes A :: "'a set" assumes "|A| =o |{}|" shows "A = {}"
grey_protects_whiteE_2: fixes g :: "'e"   and y :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and w :: "'e" assumes "(g grey_protects_white y) s"   and "(y points_to w) s"   and "white w s" shows "(g grey_protects_white w) s"
proj_ik\<^sub>s\<^sub>t_is_proj_rcv_set: fixes n :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "ik\<^sub>s\<^sub>t (proj_unl n A) = {t. (ln n, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set A \<or> (\<star>, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set A}"
bernoulli_denom_correct: fixes m :: "nat" obtains a :: "int" where "coprime a (int (bernoulli_denom m))" and "bernoulli m = real_of_int a / real (bernoulli_denom m)"
extensional_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "\<gamma> K \<phi> = \<gamma> K \<psi>"
igSwapInpIPresIGWlsInpSTR_def: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSwapInpIPresIGWlsInpSTR MOD \<equiv> \<forall>zs z1 z2 delta inp. igWlsInp MOD delta (igSwapInp MOD zs z1 z2 inp) = igWlsInp MOD delta inp"
simps_13: fixes f1 :: "'a"   and f2 :: "num \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "num \<Rightarrow> 'a \<Rightarrow> 'a"   and x2 :: "num" shows "rec_num f1 f2 f3 (num.Bit0 x2) = f2 x2 (rec_num f1 f2 f3 x2)"
IH_2: fixes x21 :: "bexp"   and x22 :: "int"   and x :: "nat"   and pc' :: "nat" assumes "pc < array_length ins"   and "array_get ins pc = TestI x21 x22"   and "\<not> bval x21 s"   and "x = nat (int (pc + 1) + x22)"   and "pc < x"   and "x < pc_bound ins"   and "pc' \<in> set (exec' ins s x)" shows "pc' < pc_bound ins"
is_methodD: fixes membr :: "qtname \<times> memberdecl" assumes "is_method membr" shows "\<exists>declC m. membr = (declC, mdecl m)"
fmrestrict_set_twice: fixes S :: "'d set"   and T :: "'d set"   and m :: "('d, 'e) fmap" shows "fmrestrict_set S (fmrestrict_set T m) = fmrestrict_set (S \<inter> T) m"
r_result_saturating: fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat"   and d :: "nat" assumes "eval r_result [t, i, x] \<down>= Suc v" shows "eval r_result [t + d, i, x] \<down>= Suc v"
continuous_finite_range_constant_eq: fixes S :: "'c set" shows "connected S = (\<forall>f. continuous_on S f \<and> finite (f ` S) \<longrightarrow> f constant_on S)"
injections_equiv: fixes Y :: "'c set"   and X :: "'d list" assumes "finite Y"   and "distinct X" shows "set (injections_alg X Y) = injections (set X) Y"
assms_4: shows "unique_idx (gs @ ab gs [] bs data) data"
UU: shows "topspace X \<subseteq> \<Union> \<U>"
S: shows "prefix S (transaction_receive T @ transaction_selects T @ transaction_checks T)"
NF_E: fixes a :: "'d"   and A :: "('d \<times> 'd) set"   and b :: "'d" assumes "a \<in> NF A" obtains "(a, b) \<notin> A"
semicomplete_iff_conditionally_complete_extremed: fixes A :: "'a set" assumes "A \<noteq> {}" shows "semicomplete A (\<sqsubseteq>) = (conditionally_complete A (\<sqsubseteq>) \<and> extremed A (\<sqsubseteq>))"
sum_strict_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "finite A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less (f x) (g x)" shows "less (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero g A)"
disc_5: fixes x2 :: "'f" shows "\<not> is_Yield (Skip x2)"
distrib_right: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semiring_ow U plus times"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "times (plus a b) c = plus (times a c) (times b c)"