hcju/isabelleps huggingface dataset:
conj_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.conj \<gamma> \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<and> \<gamma>' t n"
orbit_eq_singleton_iff: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "(orbit f x = {x}) = (f x = x)"
convergent_one_step_axioms_def: fixes t0 :: "real"   and t :: "nat \<Rightarrow> real" shows "convergent_one_step_axioms t0 t \<equiv> t0 = t 0"
matchs_alt_def: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and ps :: "Term_Class.term list"   and vs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "length ps = length vs" shows "term.matchs unapp unconst ps vs = map_option (foldl (++\<^sub>f) fmempty) (those (Term_Utils.map2 (term.match unapp unconst) ps vs))"
forget_2: fixes c :: "coname"   and M :: "trm"   and x :: "name"   and P :: "trm" assumes "c \<sharp> M" shows "M{c:=x.P} = M"
nonneg_dist_dens_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and dst :: "pdf_dist"   and e2 :: "cexpr"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : dist_param_type dst"   and "\<Gamma> \<turnstile>\<^sub>c e2 : dist_result_type dst"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V" shows "nonneg_cexpr V \<Gamma> (dist_dens_cexpr dst e1 e2)"
length_tl: fixes xs :: "'a list" shows "length (tl xs) = length xs - 1"
strategy_attracts_via_trivial: fixes v0 :: "'a"   and W :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "v0 \<in> W" shows "strategy_attracts_via p \<sigma> v0 A W"
64281: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) = - (y \<squnion> - - ((top \<squnion> y) \<sqinter> - x))"
slice_intra_edges_no_nodes_in_slice: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and as :: "'edge list" assumes "slice_edges S cs as = []"   and "\<forall>a\<in>set as. intra_kind (kind a)"   and "\<forall>c\<in>set cs. sourcenode c \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "\<forall>nx\<in>set (sourcenodes as). nx \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
leftderivations_def: shows "leftderivations = leftderivations1\<^sup>*"
swap_subst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y" shows "X #[Y / y]_ys #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
P1: shows "gcd (gcd a b) c dvd gcd a (gcd b c)"
BC_def: shows "BC = [\<one>, \<zero>, \<one>, \<one>, \<one>, \<one>, \<zero>, \<zero>]"
execute_parallel_plan_precondition_cons_1: fixes s :: "'a \<Rightarrow> bool option"   and a :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s (a # ops)"   and "are_all_operator_effects_consistent (a # ops)"   and "are_all_operators_non_interfering (a # ops)" shows "are_all_operators_applicable (s ++ map_of (effect_to_assignments a)) ops"
f: shows "f \<in> {x \<in> A. f x \<noteq> \<one>} \<rightarrow> carrier G"
R1a1_defs_1: shows "R1a1 \<equiv> {(s, t). s = med1a1 t}"
rel_nondetT'_phantom: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_nondetT' A = rel_nondetT' \<top>"
hyp2: shows "(0::'a) \<le> \<Sum>\<^sub># (image_mset f A)"
simps: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree" shows "mkt_bal_l n l r = (if ht l = ht r + 2 then case l of MKT ln ll lr x \<Rightarrow> if ht ll < ht lr then case lr of MKT lrn lrl lrr x \<Rightarrow> mkt lrn (mkt ln ll lrl) (mkt n lrr r) else mkt ln ll (mkt n lr r) else mkt n l r)"
in_set_with_P: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "P x" shows "x \<in> {y. P y}"
am2: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y = ad y \<cdot> ad x"
upper_split_interval2: fixes X :: "'b interval"   and m :: "'b" shows "upper (snd (split_interval X m)) = max (upper X) m"
bind_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
wp_under_wlp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "nearly_healthy (wlp body)"   and "wp_under_wlp body" shows "wp_under_wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
smap_ctr: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b stream"   and x :: "'a"   and s' :: "'a stream" shows "(smap f s = x ## s') = (f (shd s) = x \<and> smap f (stl s) = s')"
degree_aform_err_ln_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "ln_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
j: shows "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
th: shows "\<forall>(\<phi>, n, s)\<in>set (rsplit0 a). numbound0 s \<and> isrlfm \<phi>"
simps_1: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c"   and y21a :: "'b"   and y22a :: "'b"   and y23a :: "'c" shows "(MkArr x21 x22 x23 = MkArr y21a y22a y23a) = (x21 = y21a \<and> x22 = y22a \<and> x23 = y23a)"
mult_neg_numeral_simps_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n) = uminus (numeral.numeral one plus (m * n))"
dedekind_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<sqinter> z = (x \<sqinter> z * y\<^sup>T) * (y \<sqinter> x\<^sup>T * z) \<sqinter> z"
chainE: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "class.po below"   and "po.chain below Y" shows "below (Y i) (Y (Suc i))"
mk_of_class_def: fixes ty :: "typ"   and c :: "String.literal" shows "mk_of_class ty c = Ct (const_of_class c) (itselfT ty \<rightarrow> propT) $ mk_type ty"
Xhyp: shows "X \<in> Sols f U S t\<^sub>0 s"
intros_23: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2 # cs, css, Normal s) \<rightarrow> ([c\<^sub>1], (cs, c\<^sub>2 # cs) # css, Normal s)"
2: shows "LBINT x=ereal a..ereal b. g' x *\<^sub>R f (g x) = interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f"
push_adm_edges_2: fixes e :: "nat \<times> nat" assumes "push_precond f' l e" shows "adm_edges (push_effect f' e) l \<subseteq> adm_edges f' l"
rel_transfer: fixes Sa :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sc :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((Sa ===> Sc ===> (=)) ===> llist_all2 Sa ===> llist_all2 Sc ===> (=)) llist_all2 llist_all2"
multiplicity_times_same: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p"   and "p \<noteq> (0::'a)" shows "multiplicity p (p * x) = Suc (multiplicity p x)"
measurable_subprob_algebra_generated: fixes N :: "'c measure"   and \<Omega> :: "'c set"   and G :: "'c set set"   and M :: "'d measure"   and K :: "'d \<Rightarrow> 'c measure" assumes "sets N = sigma_sets \<Omega> G"   and "Int_stable G"   and "G \<subseteq> Pow \<Omega>"   and "\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)"   and "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"   and "\<And>A. A \<in> G \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"   and "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M" shows "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
span_is_subset2: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set" assumes "module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.module.span R M S \<subseteq> carrier M"
SeqFormP_Mem: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm" assumes "atom s \<sharp> (k, x, y)"   and "atom k \<sharp> (x, y)" shows "{TermP x, TermP y} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) (Q_Mem x y)))"
FF_dim_1: shows "dim_row (mat_of_rows n fs) = length fs"
extr_mono_bad: fixes bad :: "agent set"   and bad' :: "agent set"   and IK :: "msg set"   and H :: "chan set" assumes "bad \<subseteq> bad'" shows "extr bad IK H \<subseteq> extr bad' IK H"
multiplicity_le_imp_dvd: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "\<And>p. prime p \<Longrightarrow> multiplicity p x \<le> multiplicity p y" shows "x dvd y"
mset_subset_eq_count: fixes A :: "'k multiset"   and B :: "'k multiset"   and a :: "'k" assumes "A \<subseteq># B" shows "count A a \<le> count B a"
R1_imp_P1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>" shows "f \<in> \<P>"
OPT2_split11: fixes x :: "'a"   and y :: "'a"   and R :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "x \<noteq> y"   and "R \<in> {[x, y], [y, x]}"   and "set xs \<subseteq> {x, y}"   and "set ys \<subseteq> {x, y}" shows "OPT2 (xs @ [x, x] @ ys) R = OPT2 (xs @ [x, x]) R @ OPT2 ys [x, y]"
perm_assoc_switch: fixes as :: "'a list"   and bs :: "'a list"   and cs :: "'a list" assumes "as [\<sim>] bs"   and "bs <~~> cs" shows "\<exists>bs'. as <~~> bs' \<and> bs' [\<sim>] cs"
naturality: fixes f :: "'d" assumes "D.arr f" shows "D.\<rho>' (D.cod f) \<cdot>\<^sub>D D.map f = (f \<otimes>\<^sub>D \<I>\<^sub>D) \<cdot>\<^sub>D D.\<rho>' (D.dom f)"
rdvd_abs1: fixes d :: "int"   and t :: "real" shows "(\<bar>real_of_int d\<bar> rdvd t) = (real_of_int d rdvd t)"
unrest_semir_dash: fixes x :: "'d \<Longrightarrow> 'f"   and Q :: "('e, 'f) urel"   and P :: "('g, 'e) urel" assumes "$x\<acute> \<sharp> Q" shows "$x\<acute> \<sharp> P ;; Q"
induct: fixes P :: "('b match_expr \<Rightarrow> 'b match_expr option) \<Rightarrow> 'b rule list \<Rightarrow> bool"   and a0 :: "'b match_expr \<Rightarrow> 'b match_expr option"   and a1 :: "'b rule list" assumes "\<And>uu_. P uu_ []"   and "\<And>f m a rs. \<lbrakk>f m = None \<Longrightarrow> P f rs; \<And>x2. f m = Some x2 \<Longrightarrow> P f rs\<rbrakk> \<Longrightarrow> P f (Rule m a # rs)" shows "P a0 a1"
residual_network_def: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "residual_network f = \<lparr>edge = \<lambda>x y. edge \<Delta> x y \<or> edge \<Delta> y x \<and> y \<noteq> source \<Delta>, capacity = \<lambda>(x, y). if edge \<Delta> x y then capacity \<Delta> (x, y) - f (x, y) else if y = source \<Delta> then 0 else f (y, x), source = source \<Delta>, sink = sink \<Delta>, \<dots> = network.more \<Delta>\<rparr>"
NPreflow_axioms: shows "NPreflow c s t pp_init_f"
inject_2: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and y21 :: "POPLmarkRecord.type"   and y22 :: "trm" shows "((\<lambda>:x21. x22) = (\<lambda>:y21. y22)) = (x21 = y21 \<and> x22 = y22)"
distinct_7: fixes x51 :: "expr"   and x52 :: "cmd" shows "Skip \<noteq> while (x51) x52"
LPO_S_subst: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'b) Term.term" assumes "(s, t) \<in> LPO_S" shows "(s \<cdot> \<sigma>, t \<cdot> \<sigma>) \<in> LPO_S"
ltl_models_equiv_prop_entailment: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> = {\<psi>. w \<Turnstile>\<^sub>n \<psi>} \<Turnstile>\<^sub>P \<phi>"
seqE: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" obtains "A.arr f" and "A.arr g" and "A.dom g = A.cod f"
rec_1: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" shows "rec_nat f1 f2 0 = f1"
problem_plan_bound_charles_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'c) fmap) set" shows "problem_plan_bound_charles PROB \<equiv> Sup (MPLS_charles PROB)"
foldr_collect_interrupt_subset: fixes Ts :: "'b set"   and Ts' :: "'b set"   and ias :: "'b interrupt_action list" assumes "Ts \<subseteq> Ts'" shows "foldr collect_interrupt ias Ts \<subseteq> foldr collect_interrupt ias Ts'"
mor_Op: fixes C :: "('e, 'd, 'f) Category_scheme" shows "mor\<^bsub>Op C\<^esub> = mor\<^bsub>C\<^esub>"
equiv_def: fixes x :: "nat \<Rightarrow> nat"   and y :: "nat \<Rightarrow> nat" shows "preorder.equiv x y = (x \<lesssim> y \<and> y \<lesssim> x)"
min_mon0_closed: shows "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
digitToList_empty: fixes d :: "('c, 'd) Digit" shows "digitToList d \<noteq> []"
Dp_Mp_eq: fixes f :: "int poly" shows "f = Mp f + smult m (Dp f)"
subtopology_subtopology: fixes X :: "'d topology"   and S :: "'d set"   and T :: "'d set" shows "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
wf_prog_lift: fixes A :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and B :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "wf_prog A P"   and "\<And>wf_md C M b Ts Ca T m bd. \<lbrakk>wf_prog wf_md P; P \<turnstile> Ca sees M, b :  Ts\<rightarrow>T = m in Ca; set Ts \<subseteq> types P; bd = (M, b, Ts, T, m); A P Ca bd\<rbrakk> \<Longrightarrow> B P Ca bd" shows "wf_prog B P"
1: fixes u :: "('a, 'b) pp"   and i :: "nat" shows "fst (rep_nat_term (u, i)) = rep_nat_pp u"
Ri_def: fixes N :: "'a literal multiset set" shows "sr.Ri N = {\<gamma> \<in> gr.ord_\<Gamma>. sr.redundant_infer N \<gamma>}"
a_adopt_node_locs_def: shows "Shadow_DOM.adopt_node_locs = adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
boundOutputEqFresh: fixes x :: "name"   and B :: "('d, 'e, 'f) boundOutput"   and y :: "name"   and C :: "('d, 'e, 'f) boundOutput" assumes "\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>y\<rparr>C"   and "x \<sharp> B" shows "y \<sharp> C"
path_summary_conv_loc_imps_fw: fixes s :: "'sum"   and loc1 :: "'loc"   and loc2 :: "'loc"   and c :: "('loc, 't) configuration" assumes "s \<in>\<^sub>A path_summary loc1 loc2" shows "\<exists>M xs. loc_imps_fw c loc1 loc2 M xs \<and> sum_weights (map (\<lambda>(s, l, t). l) xs) = s"
tso_lock_invL_def: fixes m :: "'h" shows "mut_m.tso_lock_invL m = (\<lambda>s. (atS (mutator m) mut_m.tso_lock_locs s \<longrightarrow> sys_mem_lock s\<down> = Some (mutator m)) \<and> (atS (mutator m) (- mut_m.tso_lock_locs) s \<longrightarrow> sys_mem_lock s\<down> \<noteq> Some (mutator m)))"
greatest_ge_nonzero_row: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" assumes "\<not> is_zero_row_upt_k i k A" shows "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"
einterval_same: fixes a :: "ereal" shows "einterval a a = {}"
ACIcl_map_respects: fixes r :: "'d rexp"   and s :: "'d rexp"   and f :: "'d \<Rightarrow> 'e" assumes "r ~~ s" shows "map_rexp f r ~~ map_rexp f s"
tms_def: shows "tms \<equiv> image_mset to_mini_formula"
frechet_continuous: fixes I :: "('sf, 'sc, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm" assumes "is_interp I"   and "dfree \<theta>" shows "continuous_on UNIV (blin_frechet (good_interp I) (simple_term \<theta>))"
cod_eqI': fixes a :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "a \<in> Maps.codomains f" shows "a = Maps.cod f"
prems_3: fixes xs :: "'a list" assumes "ls @ xs = ls'"   and "xs \<noteq> []" shows "thesis"
EndPhase1or2_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "EndPhase1or2 s s' p = (EndPhase1 s s' p \<or> EndPhase2 s s' p)"
lin_dep_def: fixes S :: "'a vec set" shows "NC.lin_dep S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> True \<and> NC.lincomb a A = 0\<^sub>v nc \<and> v \<in> A \<and> a v \<noteq> (0::'a))"
ag_inv_zero: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "-\<^sub>a\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>A\<^esub>"
A: shows "is_tangle_diagram xs \<and> is_tangle_diagram ys \<longrightarrow> is_tangle_diagram (xs \<otimes> ys)"
ZFProdFnInRangeExplode: fixes A :: "ZF set"   and B :: "ZF set"   and f :: "ZF \<times> ZF \<Rightarrow> ZF" assumes "A \<in> range explode"   and "B \<in> range explode" shows "f ` (A \<times> B) \<in> range explode"
mndet_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "mndet A P \<equiv> if A = {} then STOP else Abs_process (\<Union>x\<in>A. F (x \<rightarrow> P x), \<Union>x\<in>A. D (x \<rightarrow> P x))"
mat_ext_vars1: fixes A :: "'b mat" shows "mat_extension dims vars1 A = tensor_P A (1\<^sub>m K)"
uminus_literal_def: fixes L :: "'a literal" shows "- L = (if is_pos L then Neg else Pos) (atm_of L)"
steps_non_empty: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "\<not> Graph_Defs.steps E []"
walk_dropWhile: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> bool" assumes "walk xs" shows "walk (dropWhile f xs)"
assms: shows "\<bar>Inf (ereal ` A)\<bar> \<noteq> \<infinity>"
less_lengthI: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set"   and p :: "nat" assumes "xs \<in> list n A"   and "p < n" shows "p < length xs"
sup_state_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list"   and c :: "ty list \<times> ty err list" assumes "P \<turnstile> a \<le>\<^sub>i b"   and "P \<turnstile> b \<le>\<^sub>i c" shows "P \<turnstile> a \<le>\<^sub>i c"
help2: fixes p :: "nat"   and INV :: "'a \<Rightarrow> enat"   and t :: "'a"   and s :: "'a"   and n :: "nat" assumes "enat p + INV t + 1 \<le> INV s"   and "INV s = enat n" shows "INV t < INV s"
in_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and h :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> O[F](h)) = (g \<in> O[F](h))"
le_less: fixes y :: "('c, 'd) Term.term"   and x :: "('c, 'd) Term.term" shows "y \<unrhd> x = (y \<rhd> x \<or> x = y)"
expr'_rel_eq: fixes e\<^sub>1 :: "'b expr'"   and e\<^sub>2 :: "'b expr'" shows "expr'_rel (=) e\<^sub>1 e\<^sub>2 = (e\<^sub>1 = e\<^sub>2)"
rel_setI: fixes A :: "'c set"   and B :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
pmaxsimpchain_def: fixes xs :: "'a set list" shows "pmaxsimpchain xs \<equiv> Ball (set xs) maxsimp \<and> padjacentchain xs"
maybe_and_idempotent: fixes a :: "trilean" shows "a \<and>? a = a"
gt_antisym: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   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 t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "epo.gt ground_heads_var gt_sym extf t s" shows "\<not> epo.gt ground_heads_var gt_sym extf s t"
lambda_zero: shows "(\<lambda>h. 0::'d) = (*) (0::'d)"
cx_def: fixes x :: "'a"   and y :: "'a" shows "cx x y = - y + x * ((1::'a) - x\<^sup>2 - y\<^sup>2)"
rel_at_def: fixes X :: "('a \<times> 'a) set" shows "rel_at X = Id \<inter> - X"
node_props_eq_node_props_formaldef: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d"   and receiver_violation :: "bool" assumes "SecurityInvariant sinvar default_node_properties receiver_violation" shows "SecurityInvariant.node_props_formaldef default_node_properties = SecurityInvariant.node_props default_node_properties"
walk_first_edge': fixes v :: "'c"   and xs :: "'c list" assumes "T.walk (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow>\<^bsub>T\<^esub> hd xs"
ivl_integral_def: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" shows "ivl_integral a b f = integral {a..b} f - integral {b..a} f"
sum: fixes g :: "'d \<Rightarrow> 'b"   and S :: "'d set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
pair_measure_count_space: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)"
pos_poly_def: fixes p :: "'a poly" shows "pos_poly p = ((0::'a) < lead_coeff p)"
const_valid: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "const X" shows "const (\<upsilon> X)"
"3_1": fixes s :: "'a match_expr" assumes "\<forall>m\<in>set (v2 # va). \<forall>a\<in>set (normalize_match m). normalized_n_primitive (disc, sel) f a"   and "s \<in> set (normalize_match (andfold_MatchExp (v2 # va)))" shows "normalized_n_primitive (disc, sel) f s"
Pj_mem: fixes N :: "'a set"   and x :: "'a" assumes "G \<triangleright> N"   and "x \<in> carrier G" shows "Pj G N x = N \<bullet> x"
le: shows "r' - l' \<le> (ir - il) / 2 ^ i"
bisim_final: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i" assumes "bisimulation_final trsys1 trsys2 bisim tlsim final1 final2"   and "bisim s1 s2" shows "final1 s1 = final2 s2"
infdist_Un_min: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"
ag_inv_inv: fixes x :: "'a" assumes "x \<in> carrier A" shows "-\<^sub>a (-\<^sub>a x) = x"
action_proj_idempot'': fixes P :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" assumes "prob_dom P \<subseteq> vs" shows "prob_proj P vs = P"
SCAs0'_\<eta>_fo_SMCAs: shows "map S CAs0' \<cdot>cl \<eta> = map (S_M S M) CAs"
crel_vs_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'c"   and s :: "(heap, 'd) state" shows "state_dp_consistency.crel_vs R v s \<equiv> \<forall>M. state_dp_consistency.cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> state_dp_consistency.cmem M' \<and> P M')"
cvector_fraction_eq_iff: fixes u :: "complex"   and v :: "complex"   and a :: "'a"   and x :: "'a" shows "((u / v) *\<^sub>C a = x) = (if v = 0 then x = (0::'a) else u *\<^sub>C a = v *\<^sub>C x)"
assign_conforms_def: fixes s :: "st"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and T :: "ty"   and E :: "prog \<times> lenv" shows "s\<le>|f\<preceq>T\<Colon>\<preceq>E = ((\<forall>s' w. Norm s'\<Colon>\<preceq>E \<longrightarrow> fst E,s'\<turnstile>w\<Colon>\<preceq>T \<longrightarrow> s\<le>|s' \<longrightarrow> assign f w (Norm s')\<Colon>\<preceq>E) \<and> (\<forall>s' w. error_free s' \<longrightarrow> error_free (assign f w s')))"
simps: fixes v :: "name"   and y :: "tm"   and u :: "tm"   and w :: "tm"   and au :: "name" assumes "atom v \<sharp> (y, u, w, au)"   and "atom au \<sharp> (y, u, w)" shows "MakeFormP y u w = y EQ Q_Disj u w OR y EQ Q_Neg u OR SyntaxN.Ex v (SyntaxN.Ex au (AbstFormP (Var v) Zero u (Var au) AND y EQ Q_Ex (Var au)))"
acyclic_5a_5e: fixes x :: "'a" shows "acyclic_5a x = acyclic_5e x"