hcju/isabelleps huggingface dataset:
true_lit_def: fixes I :: "'b set"   and L :: "'b literal" shows "(I \<TTurnstile>l L) = (if is_pos L then \<lambda>P. P else Not) (atm_of L \<in> I)"
empty: fixes g :: "'c \<Rightarrow> 'a" shows "F.F g {} = \<^bold>1"
resource_of_oracle_state_iso: fixes fg :: "('r \<Rightarrow> 's) \<times> ('s \<Rightarrow> 'r)"   and "oracle" :: "'r \<Rightarrow> 't \<Rightarrow> ('u \<times> 'r) spmf"   and s :: "'s" assumes "state_iso fg" shows "RES (apply_state_iso fg oracle) s = RES oracle (snd fg s)"
learn_lim_sigma: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and i :: "nat" assumes "learn_lim \<psi> U (\<phi> i)" shows "learn_lim \<psi> U (\<sigma> i)"
distinct_set_mset_eq: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "distinct_mset M"   and "distinct_mset N"   and "set_mset M = set_mset N" shows "M = N"
col_trivial_1: fixes A :: "'p"   and B :: "'p" shows "Col A A B"
map_sel_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b le_constraint" shows "lec_poly (map_le_constraint f a) = lec_poly a"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "C.arr f"   and "C.dom f = a" shows "f \<cdot>\<^sub>C a = f"
emptyD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Empty]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Undecided"
starP2_star_of: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'c" shows "(*p2* P) (star_of x) = *p* P x"
toplevel_summands_wf: fixes n :: "nat"   and s :: "'b rexp" shows "local.wf n s = (\<forall>r\<in>toplevel_summands s. local.wf n r)"
foundedEmpty: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool" shows "founded subs P {}"
iwlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "iwlsFSb MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSubstCls MOD \<and> igAbsRen MOD"
assms: shows "\<lbrace>p \<and> b\<rbrace> S \<lbrace>p\<rbrace>\<^sub>u"
trivial_singleton_group: fixes a :: "'b" shows "trivial_group (singleton_group a)"
function_to_finmap_def: fixes J :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'b" shows "function_to_finmap J f f' \<equiv> finite J \<and> (\<forall>i. i \<in> J \<longrightarrow> f' (f i) = i)"
rot90_ii: fixes z :: "complex" shows "rot90 z = z * \<i>"
filter_sup_right_upper_bound: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F" shows "G \<subseteq> filter_sup F G"
a: shows "steps (s, l, r) (A, off) stp = (sa, la, ra)"
rComm1: fixes M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> !P) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
inducts: fixes \<mu> :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and m :: "'b"   and ts :: "'c \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option"   and ts' :: "'c \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option" assumes "multithreaded_base_measure.m\<mu>t \<mu> m ts ts'" obtains t :: "'c" and x :: "'a" and x' :: "'a" and ln' :: "'d \<Rightarrow>f nat" and ln :: "'d \<Rightarrow>f nat" where "finite (dom ts)" and "ts t = \<lfloor>(x, ln)\<rfloor>" and "ts' t = \<lfloor>(x', ln')\<rfloor>" and "\<mu> (x, m) (x', m)" and "\<And>t'. t' \<noteq> t \<Longrightarrow> ts t' = ts' t'"
invfun_def: fixes A :: "'d set"   and B :: "'c set"   and f :: "'d \<Rightarrow> 'c" shows "f\<inverse>\<^bsub>B,A\<^esub> = (\<lambda>y\<in>B. SOME x. x \<in> A \<and> f x = y)"
index_mat_swaprows_mat_3: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "dim_col (swaprows_mat n k l) = n"
or: shows "(\<sigma>, p) \<in> oreachable (opaodv i) (act (recvmsg rreq_rrep_sn)) (other A {i})"
simps_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
case_prod_curry: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). curry f x y) = f"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Trg (t \<^bold>\<star> u) = Trg t"
l1_def: shows "l1 \<equiv> \<lparr>init = l1_init, trans = l1_trans, obs = id\<rparr>"
fmmerge_def: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and xm :: "'e \<rightharpoonup>\<^sub>f 'g"   and ym :: "'e \<rightharpoonup>\<^sub>f 'h" shows "fmmerge f xm ym \<equiv> fmap_of_list (map (\<lambda>k. (k, f (the (fmlookup xm k)) (the (fmlookup ym k)))) (sorted_list_of_fset (fmdom xm |\<inter>| fmdom ym)))"
not_one_le_zero: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
inj_vreal_of_real_impl'': shows "inj vreal_of_real_impl''"
oalist_eq_rawI: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ys :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs"   and "oalist_inv ys"   and "\<And>k. k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys) \<Longrightarrow> lookup_raw xs k = lookup_raw ys k" shows "oalist_eq_raw xs ys"
takeWhile_eq_Nil_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(takeWhile P xs = []) = (xs = [] \<or> \<not> P (hd xs))"
neq_ne: shows "list_neq xs (mean xs) \<noteq> []"
maxchain_def: fixes C :: "'a set" shows "B.maxchain C = (chain_on P C {x \<in> A. \<not> Q x} \<and> (\<nexists>S. chain_on P S {x \<in> A. \<not> Q x} \<and> C \<subset> S))"
octo_of_real_numeral: fixes w :: "num" shows "octo_of_real (numeral w) = numeral w"
module_hom_axioms: shows "module_hom (*a) (*b) (construct B f)"
atm_of_subst_lit: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and L :: "'b literal"   and \<sigma> :: "'c" shows "atm_of (substitution_ops.subst_lit subst_atm L \<sigma>) = subst_atm (atm_of L) \<sigma>"
upper_def: shows "upper \<equiv> map_fun bounds_of_interval id snd"
adds_minus: fixes s :: "'a"   and t :: "'a" assumes "s adds t" shows "t - s + s = t"
inf_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<Sqinter>A \<in> carrier L"
all_partitions_def: fixes A :: "'c set" shows "all_partitions A = {P. P partitions A}"
deleted_not_domA: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" shows "x \<notin> domA (delete x \<Gamma>)"
assms_5: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"
mset_set_set_mset_empty_mempty: fixes D :: "'b multiset" shows "(mset_set (set_mset D) = {#}) = (D = {#})"
eval_wlp_Apply: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> real" shows "wlp (Apply f) P = P \<circ> f"
d_1: shows "D \<in> P_of_state (lnth Sts i)"
Language_init: fixes t :: "'t1" assumes "wf1 t" shows "Language1 (init1 t) = Lang1 t"
assms: shows "(\<Sum>x\<in>S. norm (f x - g x)) < e"
bin_sc_sc_diff: fixes m :: "nat"   and n :: "nat"   and c :: "bool"   and b :: "bool"   and w :: "int" assumes "m \<noteq> n" shows "bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
octo_proj_add: fixes i :: "nat"   and x :: "octo"   and y :: "octo" assumes "i \<le> 7" shows "octo_proj (x + y) i = octo_proj x i + octo_proj y i"
arbitrary_def: fixes \<U> :: "'b set set" shows "arbitrary \<U> \<equiv> True"
rf_encrypt_bad_def: shows "rf_encrypt_bad = (\<lambda>secret (s, bad) plain. spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. prf.random_oracle s r \<bind> (\<lambda>(pad, s'). return_spmf ((r, plain [\<oplus>] pad), s', bad \<or> r = secret))))"
sin_tan_half: fixes x :: "'a" shows "sin ((2::'a) * x) = (2::'a) * tan x / ((1::'a) + (tan x)\<^sup>2)"
fps_compose_add_distrib: fixes a :: "'b fps"   and b :: "'b fps"   and c :: "'b fps" shows "a + b oo c = (a oo c) + (b oo c)"
ind_in_set_bij_betw: fixes B :: "nat set"   and A :: "nat set"   and c :: "nat" assumes "B \<subseteq> A"   and "c \<in> B" shows "bij_betw (ind_in_set A) {i \<in> B. i < c} {i \<in> ind_in_set A ` B. i < ind_in_set A c}"
dynProcModifyReturn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
llist_all2_mono: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and P' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "llist_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> P' x y" shows "llist_all2 P' xs ys"
subgraph_del_vert: fixes u :: "'a" shows "subgraph (del_vert u) G"
n: shows "bij_betw m N (\<Union>l. M (V l))"
S: fixes J :: "'b set"   and S :: "'b \<Rightarrow> 'c set" assumes "J \<in> I"   and "\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)" shows "f -` Pi' J S \<inter> space N \<in> sets N"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>C 'd"   and xa :: "'c \<Rightarrow>\<^sub>C 'd" shows "apply_bcontfun (x + xa) = (\<lambda>xb. apply_bcontfun x xb + apply_bcontfun xa xb)"
Arg2pi_of_real: fixes r :: "real" shows "Arg2pi (complex_of_real r) = (if r < 0 then pi else 0)"
disj_dlHPP_Bisimulation_Up_To_R'_def: fixes d :: "'d"   and PP :: "nat set"   and R' :: "('com list \<times> 'com list) set"   and R :: "('com list \<times> 'com list) set" shows "disj_dlHPP_Bisimulation_Up_To_R' d PP R' R \<equiv> SdlHPPB d PP R' \<and> sym R \<and> trans R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))) \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 \<alpha> p. \<langle>V ! i,m1\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m2\<rangle> \<and> m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1' \<longrightarrow> (\<exists>p' \<alpha>' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow>\<lhd>\<alpha>'\<rhd> \<langle>p',m2'\<rangle> \<and> stepResultsinR p p' (R \<union> R') \<and> (\<alpha>, \<alpha>') \<in> R \<union> R' \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'))"
insert: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
dom_eqI': fixes a :: "'d"   and f :: "'d" assumes "a \<in> D.domains f" shows "a = D.dom f"
simps: fixes G :: "prog"   and I :: "qtname"   and f :: "qtname \<Rightarrow> iface \<Rightarrow> 'b set \<Rightarrow> 'b" shows "iface_rec G I f = (case iface G I of Some i \<Rightarrow> if ws_prog G then f I i ((\<lambda>J. iface_rec G J f) ` set (isuperIfs i)) else undefined)"
qSkel_qSkelInp: fixes delta :: "'h"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "qSkel (qOp delta qinp qbinp) = Branch (qSkelInp qinp) (qSkelBinp qbinp)"
oexp_1_right: fixes \<alpha> :: "V" shows "\<alpha> \<up> 1 = \<alpha>"
max_fun_diff_def: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and g :: "'b \<times> 'c \<Rightarrow> 'd" shows "st.max_fun_diff f g \<equiv> st.maxim {a \<in> Field (s *o t). f a \<noteq> g a}"
B3b: shows "\<^bold>\<forall>x. cod x \<cdot> x \<cong> x"
mr_is_C: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x p = \<lfloor>a\<rfloor>" shows "Cp (list2FWpolicy p) x = Cp a x"
w\<^sub>\<tau>': shows "ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)"
conj_assoc: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> (b::'a::refinement_lattice) \<iinter> (c::'a::refinement_lattice) = a \<iinter> (b \<iinter> c)"
matches_add_match_MatchNot_simp: 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 rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>add_match (MatchNot m) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[], s\<rangle> \<Rightarrow> t"
parent_child_rel_child_in_heap: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "type_wf h"   and "known_ptr parent"   and "(parent, child_ptr) \<in> parent_child_rel h" shows "child_ptr |\<in>| object_ptr_kinds h"
obj_trg: fixes V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and \<mu> :: "'d" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "horizontal_homs.obj V src (trg \<mu>)"
assms_1: shows "rreq_rrep_fresh (rt (\<sigma> sip)) (Rreq hops rreqid dip dsn dsk oip osn sip)"
filter_diff_zmset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b zmultiset"   and N :: "'b zmultiset" shows "filter_zmset P (M - N) = filter_zmset P M - filter_zmset P N"
lemma_real_divide_sqrt_less: fixes u :: "real" assumes "0 < u" shows "u / sqrt 2 < u"
to_fun_monic_monom: fixes b :: "nat \<Rightarrow> int"   and n :: "nat" assumes "b \<in> carrier Zp" shows "monom (UP Zp) \<one> n \<bullet> b = b [^] n"
dom_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "fst ` list.set (AList.update_with_aux v k f ps) = {k} \<union> fst ` list.set ps"
simps_1: fixes \<A> :: "'b \<Rightarrow> bool"   and k :: "'b" shows "lit_semantics \<A> (k\<^sup>+) = \<A> k"
fres_canc2: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" shows "less_eq_prod y (prod.fres (times_prod' y x) x)"
ideal_points_on_circline: fixes H :: "circline" shows "\<forall>z\<in>ideal_points H. z \<in> circline_set H"
possible_empty: fixes x' :: "'b" shows "possible TTree.empty x' = False"
NullLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal NullLiteral : \<tau>" obtains "\<tau> = OclVoid[?]"
alllstsp_alllsts_eq: fixes A :: "'b set" shows "alllstsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> A\<^sup>\<infinity>)"
implements_actJP_jAction: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol"   and t :: "'s Trace"   and n :: "nat"   and a :: "'a" assumes "implements jp"   and "t \<in> jkbpC\<^bsub>n\<^esub>" shows "set (actJP jp t a) = set (jAction MC\<^bsub>n\<^esub> t a)"
multiset_nonemptyE: fixes A :: "'k multiset" assumes "A \<noteq> {#}" obtains x :: "'k" where "x \<in># A"
effect_to_assignments_def: fixes op :: "'c strips_operator" shows "effect_to_assignments op \<equiv> effect__strips op"
slength_0_conv_2: fixes xs :: "[:'a:]" shows "(slength\<cdot>xs = MkI\<cdot>0) = (xs = [::])"
map_option_case: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'c option" shows "map_option f y = (case y of None \<Rightarrow> None | \<lfloor>x\<rfloor> \<Rightarrow> \<lfloor>f x\<rfloor>)"
extremum_unique: fixes a :: "'q multiset" shows "(a \<subseteq># {#}) = (a = {#})"
nat_filtration_borel_measurable_characterization: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "random_variable borel f" shows "(f \<in> borel_measurable (nat_filtration n)) = (f \<circ> pseudo_proj_True n = f)"
Ex_SubstAtomicP_Fls: fixes y :: "tm"   and z :: "tm"   and v :: "tm"   and tm :: "tm"   and y' :: "tm" shows "{y EQ Q_Ex z, SubstAtomicP v tm y y'} \<turnstile> Fls"
scaleC_right: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d"   and r :: "complex"   and b :: "'e" assumes "bounded_sesquilinear prod" shows "prod a (r *\<^sub>C b) = r *\<^sub>C prod a b"
deadlock_free_v2_FD: fixes P :: "'b process" shows "deadlock_free_v2 P = DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
swapEnv_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "goodEnv (rho &[x \<and> y]_xs)"
RSQ_def: shows "RSQ = cbox (case (l, snd p\<^sub>0) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (real_of_int l + \<delta>, real_of_int (snd p\<^sub>0) + \<delta>)"
det_0_negate: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det (- A) = (0::'b)) = (det A = (0::'b))"
gen_lossless_bind_gpvD1: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv" assumes "gen_lossless_gpv co \<I> (gpv \<bind> f)" shows "gen_lossless_gpv co \<I> gpv"
simps_1: fixes i :: "name"   and x :: "tm" shows "subst i x Zero = Zero"
trans_lex_prod: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('c \<times> 'c) set" assumes "trans R1"   and "trans R2" shows "trans (R1 <*lex*> R2)"
cases: fixes a1 :: "com"   and a2 :: "state" assumes "a1 \<down> a2" obtains f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = Do f__" and "a2 = s__" and "f__ s__ \<noteq> {}"   | c\<^sub>1 :: "com" and s\<^sub>0 :: "state" and c\<^sub>2 :: "com" where "a1 = (c\<^sub>1__; c\<^sub>2__)" and "a2 = s\<^sub>0" and "c\<^sub>1__ \<down> s\<^sub>0" and "\<forall>s\<^sub>1. s\<^sub>0 -c\<^sub>1__\<rightarrow> s\<^sub>1 \<longrightarrow> c\<^sub>2__ \<down> s\<^sub>1"   | b :: "state \<Rightarrow> bool" and s :: "state" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = IF ba__ THEN c\<^sub>1'__ ELSE c\<^sub>2'__" and "a2 = sa__" and "ba__ sa__" and "c\<^sub>1'__ \<down> sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c\<^sub>2 :: "com" and c\<^sub>1 :: "com" where "a1 = IF bb__ THEN c\<^sub>1 ELSE c\<^sub>2" and "a2 = sb__" and "\<not> bb__ sb__" and "c\<^sub>2 \<down> sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE bc__ DO ca__" and "a2 = sc__" and "\<not> bc__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE b DO cb__" and "a2 = sd__" and "b sd__" and "cb__ \<down> sd__" and "\<forall>t. sd__ -cb__\<rightarrow> t \<longrightarrow> WHILE b DO cb__ \<down> t"   | c :: "com" and f :: "state \<Rightarrow> state" and s :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = LOCAL f; c; g" and "a2 = s" and "c \<down> f s"
Always_union: fixes F :: "'b program"   and f' :: "'b \<Rightarrow> 'c multiset"   and f :: "'b \<Rightarrow> 'c multiset"   and g' :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "F \<in> Always {s. f' s \<le> f s}"   and "F \<in> Always {s. g' s \<le> g s}" shows "F \<in> Always {s. f' s + g' s \<le> f s + g s}"
permutes_bij_main: fixes A :: "'b set"   and a_to_b :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and b_to_a :: "'c \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> a_to_b a \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> b_to_a b \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> b_to_a (a_to_b a) = a"   and "\<And>b. b \<in> B \<Longrightarrow> a_to_b (b_to_a b) = b" shows "(\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a) ` {p. p permutes B} \<subseteq> {p. p permutes A}"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "less_eq (inf a b) b"
monom_mult_zero_right: fixes c :: "'b"   and t :: "'a" shows "MPoly_Type_Class.punit.monom_mult c t 0 = 0"
NormalizeI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Language.normalize c Q,A"
A: shows "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"
index: shows "fst ` set as \<union> set ui \<subseteq> fst ` set cs"
nonneg: fixes a :: "'c"   and i :: "nat" assumes "a \<in> A" shows "0 \<le> f i a"
C2: fixes x :: "'e"   and y :: "'c" assumes "P' x"   and "(y, x) \<in> S1" shows "P y"
h3: shows "rank (mat_of_cols n B) = card (set B)"
is_le_rel_trans: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_le_rel rel"   and "rel x y"   and "rel y z" shows "rel x z"
mono_D23: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D (S `;` P)) \<subseteq> T (S `;` Q)"
induced_arrow_self: shows "\<chi>a.induced_arrow (l a) (\<chi> a) = l a"
funion_left_commute: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)"
subset_antisym: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A"   and "order.greater_eq A B" shows "A = B"
conf_clinit_shupd: fixes P :: "jvm_prog"   and sh :: "sheap"   and frs :: "frame list"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state" assumes "conf_clinit P sh frs"   and "distinct (C # clinit_classes frs)" shows "conf_clinit P (sh(C \<mapsto> (sfs, i))) frs"
star_inductl_var_eq2: fixes y :: "'a"   and x :: "'a" assumes "y = x ; y" shows "y = x\<^sup>\<star> ; y"
mm_equiv_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" where "strong_low_bisim_mm \<R>" and "(\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R>"
prod_fields_4: shows "prod.num_acc = igba.num_acc"
get_const: fixes m :: "'m" shows "get (\<lambda>_. m) = m"
times_bit0_def: fixes x :: "'a bit0"   and y :: "'a bit0" shows "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"