hcju/isabelleps huggingface dataset:
ivp_solsI: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and s :: "'b"   and t\<^sub>0 :: "real"   and S :: "'b set" assumes " D X = (\<lambda>t. f t (X t)) on U s"   and "X t\<^sub>0 = s"   and "X \<in> U s \<rightarrow> S"   and "t\<^sub>0 \<in> U s" shows "X \<in> Sols f U S t\<^sub>0 s"
pot_comp_cong: fixes u :: "'a"   and v :: "'a"   and to1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and to2 :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "snd (rep_nat_term u) = snd (rep_nat_term v) \<Longrightarrow> to1 u v = to2 u v" shows "pot_comp to1 u v = pot_comp to2 u v"
nprv_allI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and x :: "'var" assumes "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "x \<notin> \<Union> (Fvars ` F)" shows "nprv F (all x \<phi>)"
gap_between_aligned: fixes a :: "'b word"   and b :: "'b word"   and n :: "nat" assumes "a < b"   and "is_aligned a n"   and "is_aligned b n"   and "n < LENGTH('b)" shows "a + (2 ^ n - 1) < b"
assms_3: shows "g \<in> I \<rightarrow> carrier G"
hiding_fronttickFree: fixes s :: "'a event list"   and A :: "'a set" assumes "front_tickFree s" shows "front_tickFree (trace_hide s (ev ` A))"
indspace_findim: shows "\<exists>ms. set ms \<subseteq> indV \<and> R_scalar_mult.RSpan UNIV (\<currency>\<currency>) ms = indV"
less_ge: fixes A :: "'b set set"   and f :: "'b \<Rightarrow> real" assumes "sigma_algebra A"   and "\<forall>a. {w. f w < a} \<in> A" shows "\<forall>a. {w. a \<le> f w} \<in> A"
openin_clauses_3: fixes K :: "'a set set"   and U :: "'a topology" assumes "\<forall>S\<in>K. openin U S" shows "openin U (\<Union> K)"
poly_mult_zero_2: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult p [] = []"
assms_1: fixes xs :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ys :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "set (comb xs ys) = set xs \<union> set ys"
induct: fixes P :: "('d, 'e) xstate \<Rightarrow> bool"   and xstate :: "('d, 'e) xstate" assumes "\<And>x. P (Normal x)"   and "\<And>x. P (Abrupt x)"   and "\<And>x. P (Fault x)"   and "P Stuck" shows "P xstate"
PRECOND_tag_def: fixes P :: "'c" shows "PRECOND_tag P \<equiv> P"
clm_def: fixes ts :: "traffic" shows "clm ts \<equiv> fst (snd (snd (Rep_traffic ts)))"
CompDiag_preserves_Diag_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Diag (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u)"
inf_subs_reduced_red_inf: fixes N :: "'f set" shows "Inf_between UNIV (Red_F N) \<subseteq> Red_Red_I N"
IH2_2: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M1" shows "f \<in> Mapping.keys M2"
simps_3: fixes b :: "'b"   and ts :: "'a FOL_Fitting.term list"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Pred b ts[s/k] = Pred b (ts[s/k])"
assms_3: fixes y :: "'al" assumes "y \<in> A" shows "y \<le>\<^sub>o\<^sub>w x"
eucl_of_list_list_of_eucl: fixes x :: "'c" shows "eucl_of_list (list_of_eucl x) = x"
cexpr_equiv_def: fixes e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_equiv e1 e2 \<equiv> \<forall>\<sigma>. cexpr_sem \<sigma> e1 = cexpr_sem \<sigma> e2"
hom_deg_set_Polys_eq_span: fixes z :: "nat"   and X :: "'b set" shows "hom_deg_set z P[X] = phull (monomial (1::'c) ` deg_sect X z)"
MaxR_opt_code: fixes A :: "'a list" shows "local.MaxR.MaxR_opt (set A) = fold local.MaxR.MaxR_f A None"
wf_SNa_Redu: shows "wf SNa_Redu"
reachable_V: shows "gba.reachable \<subseteq> gba.V"
all_info: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"
flow_by_value: fixes v :: "ennreal" assumes "v < \<alpha>"   and "\<forall>f. \<alpha> = \<top> \<longrightarrow> flow \<Delta> f \<longrightarrow> value_flow \<Delta> f < \<alpha>" obtains f :: "'v \<times> 'v \<Rightarrow> ennreal" where "flow \<Delta> f" and "value_flow \<Delta> f = v"
assms_2: shows "\<not> bdd_below {u n / real n |n. 0 < n}"
of_int_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 1 = one"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "\<eta> (x \<cdot> y) = \<eta> x \<cdot>' \<eta> y"
wfUNIVI: fixes r :: "('b \<times> 'b) set" assumes "\<And>P x. \<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wf r"
Xfix: shows "X \<subseteq> {q \<in> A. f q \<sim> q} \<union> P"
simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list"   and a :: "'b" shows "scan f (x # xs) a = f x a # scan f xs (f x a)"
mult2_s_locally_cancel1: fixes s :: "('b \<times> 'b) set"   and ns :: "('b \<times> 'b) set"   and z :: "'b"   and X :: "'b multiset"   and Y :: "'b multiset" assumes "s O ns \<subseteq> s"   and "ns O s \<subseteq> s"   and "refl ns"   and "trans ns"   and "locally_irrefl s (add_mset z X)"   and "locally_irrefl s (add_mset z Y)"   and "trans s"   and "(add_mset z X, add_mset z Y) \<in> mult2_s ns s" shows "(X, Y) \<in> mult2_s ns s"
swap_disjunct_emb_step_at: fixes p :: "dir list"   and q :: "dir list"   and d2 :: "dir"   and d1 :: "dir"   and t :: "('c, 'd) tm" assumes "length p \<le> length q \<Longrightarrow> take (length p) q \<noteq> p"   and "length q \<le> length p \<Longrightarrow> take (length q) p \<noteq> q" shows "emb_step_at q d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at q d2 t)"
surjective_var: fixes x :: "'a" shows "surjective x = reflexive (x[[x]])"
Assign: fixes a :: "aexp"   and A :: "char list set"   and x :: "char list" assumes "vars a \<subseteq> A" shows "D A (x ::= a) (insert x A)"
weak_lless_imp_not_eq: fixes R :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "x \<sqsubset>\<^bsub>R\<^esub> y" shows "x .\<noteq>\<^bsub>R\<^esub> y"
Int_iff: fixes c :: "'h"   and A :: "'h set"   and B :: "'h set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" assumes "\<exists>(s', T)|\<in>|possible_steps e s r l i. recognises_execution e s' (evaluate_updates T i r) t" shows "recognises_execution e s r ((l, i) # t)"
fresh_bij: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'a" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
reachability_closed_under_execution_step: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "substitution.reachable subst s"   and "substitution.revision_step subst r s s'" shows "substitution.reachable subst s'"
check_no_overlapI: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('a, 'c) flow_entry_match list" assumes "no_overlaps \<gamma> t" shows "check_no_overlap \<gamma> t"
normalize_power: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "normalize (power.power one times a n) = power.power one times (normalize a) n"
simps: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and v :: "'c" shows "Vertex_Graph.terminal_vertex outPorts vertices nodeOf v = (v |\<in>| vertices \<and> Vertex_Graph.terminal_node outPorts (nodeOf v))"
step: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> a.iterates f" shows "f x \<in> a.iterates f"
latin_square_def: fixes s :: "(latin_type \<Rightarrow> nat) set"   and n :: "nat" shows "latin_square s n \<equiv> \<forall>t. bij_betw (without t) s ({0..<n} \<times> {0..<n})"
elem_carrier: fixes A :: "'a set" assumes "A \<in> F" obtains "A \<subseteq> carrier" and "A \<noteq> {}"
union_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a \<in># A + B) = (a \<in># A \<or> a \<in># B)"
homeomorphic_compactness: fixes s :: "'b set"   and t :: "'c set" assumes "s homeomorphic t" shows "compact s = compact t"
cmp_simps'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "S.src (S.cmp\<^sub>U\<^sub>P (\<mu>, \<nu>)) = UP.map\<^sub>0 (src \<nu>)"
functor_ToCT: shows "functor VxVxV.comp (\<cdot>) H.ToCT"
Nodes_levellist_subset_t: fixes t :: "dag"   and levellist :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and i :: "nat" assumes "wf_ll t levellist var"   and "i \<le> length levellist" shows "Nodes i levellist \<subseteq> set_of t"
not_None_eq: fixes x :: "'d option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
lincomb_union: fixes U :: "'c set"   and U' :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "U \<subseteq> carrier V"   and "U' \<subseteq> carrier V"   and "U \<inter> U' = {}"   and "finite U"   and "finite U'"   and "a \<in> U \<union> U' \<rightarrow> carrier K" shows "lincomb a (U \<union> U') = lincomb a U \<oplus>\<^bsub>V\<^esub> lincomb a U'"
Sum1_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> P'"
mbisim_dom_eq: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "dom (thr s1) = dom (thr s2)"
mult_eSuc_right': fixes m :: "eint"   and n :: "eint" shows "m * (n + 1) = m + m * n"
inject_3: fixes x2 :: "ty"   and x1 :: "ty"   and y2 :: "ty"   and y1 :: "ty" shows "(x2 \<otimes> x1 = y2 \<otimes> y1) = (x2 = y2 \<and> x1 = y1)"
token_succeeds_alt_def: fixes x :: "nat" shows "\<UU>.token_succeeds x = (\<forall>\<^sub>\<infinity>n. \<UU>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
inverse_Measurement_System_ext_def: fixes x :: "\<lparr>unit_sys :: 'c, \<dots> :: 'd\<rparr>" shows "inverse x = \<lparr>unit_sys = unit, \<dots> = inverse (mmore x)\<rparr>"
lt_of_comp_def: fixes acomp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" shows "lt_of_comp acomp x y = (case acomp x y of Lt \<Rightarrow> True | _ \<Rightarrow> False)"
pre_iso_test6: shows "\<forall>g\<in>set Hex. pre_iso_test g"
rest_is_non_overlapping: shows "is_non_overlapping (P - {X})"
that: assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "thesis"
perm_empty2: fixes xs :: "'d list" shows "(xs <~~> []) = (xs = [])"
E_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.E p q = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {0} \<and> x mod int q \<in> {(int q - 1) div 2<..int q - 1}}"
eq: fixes x :: "'a" assumes "x \<in> orbit g a" shows "g' (f x) = f (g x)"
remove1_mset_eqE: fixes x1 :: "'a multiset"   and L :: "'a"   and M :: "'a multiset" assumes "remove1_mset L x1 = M" obtains "L \<in># x1" and "x1 = M + {#L#}"   | "L \<notin># x1" and "x1 = M"
le_sqrt_iff: fixes x :: "real"   and y :: "real" assumes "0 \<le> x" shows "(x \<le> sqrt y) = (x\<^sup>2 \<le> y)"
lemLineContainsEndpoint: fixes q :: "'a Point"   and p :: "'a Point" shows "inLine q line joining p to q"
struct_specD1: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h" assumes "punit.struct_spec sel ap ab compl" shows "sel_spec sel"
sigma_sets_single: fixes A :: "'b set" shows "sigma_sets A {A} = {{}, A}"
LIM_not_zero: fixes k :: "'b"   and a :: "'c" assumes "k \<noteq> (0::'b)" shows "\<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> (0::'b)"
eSubstAbs_simp5: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and ys :: "'varSort"   and y :: "'var"   and eA :: "'c withERR" shows "eSubstAbs MOD ys ERR y eA = ERR"
oth_class_taut_5_j: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> \<phi> \<^bold>& \<^bold>\<not>\<psi> \<^bold>\<or> \<^bold>\<not>\<phi> \<^bold>& \<psi> in v]"
le_infE: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a \<sqinter> b" obtains "x \<le> a" and "x \<le> b"
has_integral_spike_eq: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f has_integral y) T = (g has_integral y) T"
sum_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.semiring_0 plus zero times" shows "times r (comm_monoid_add.sum plus zero f A) = comm_monoid_add.sum plus zero (\<lambda>n. times r (f n)) A"
d: fixes y :: "'a" assumes "y \<in> closed_segment a b" shows "f d \<le> f y"
\<alpha>edges_subset: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"   and "wgraph_by_map.invar M_lookup M_invar g" shows "wgraph_by_map.\<alpha>edges_aux M_lookup g \<subseteq> wgraph_by_map.\<alpha>nodes_aux M_lookup g \<times> wgraph_by_map.\<alpha>nodes_aux M_lookup g"
to_nat_inj: fixes a :: "'a"   and b :: "'a" assumes "to_nat a = to_nat b" shows "a = b"
assms_3: fixes a :: "'a" assumes "a \<in> A" shows "\<exists>b\<in>B. a \<le> b"
closed_UNIV: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open UNIV"
indicator_times_eq_if_2: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and S :: "'d set" shows "f x * indicator S x = (if x \<in> S then f x else (0::'c))"
not_not: fixes P :: "bool" shows "(\<not> \<not> P) = P"
prob_space_PiM: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)" shows "prob_space (Pi\<^sub>M I M)"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "ide (e \<cdot> m)" shows "ide (L e \<cdot> L m)"
eval_liftFm_helper: fixes init :: "real list"   and i :: "nat"   and I :: "real list"   and amount :: "nat"   and F :: "atom fm"   and xs :: "real list" assumes "length init = i"   and "length I = amount" shows "eval F (init @ xs) = eval (liftFm i amount F) (init @ I @ xs)"
Skip: 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 Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> Q Skip Q,A"
simps: fixes G :: "'b graph"   and nP :: "'b \<Rightarrow> nat" shows "SINVAR_BLPbasic.sinvar G nP = (\<forall>(e1, e2)\<in>edges G. nP e1 \<le> nP e2)"
remove: fixes x :: "'b"   and A :: "'b multiset" assumes "x \<in># A" shows "\<Sum>\<^sub># A = x + \<Sum>\<^sub># (remove1_mset x A)"
Acts_extend: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h" shows "Acts (extend h F) = extend_act h ` Acts F"
sel_4: fixes x1 :: "symbol \<times> symbol list"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat" shows "item_end (Item x1 x2 x3 x4) = x4"
df: fixes x :: "complex"   and n :: "nat" assumes "x \<in> S" shows "(f n has_field_derivative f' n x) (at x within S)"
f': fixes x :: "'a" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
match_result_sound: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "case match_result env s v0 pes err_v of Rval (e, env') \<Rightarrow> \<forall>bv. evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e bv \<longrightarrow> evaluate_match ck env s v0 pes err_v bv | Rerr err \<Rightarrow> evaluate_match ck env s v0 pes err_v (s, Rerr err)"
reflexive_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "reflexive x"   and "reflexive y" shows "reflexive (x \<sqinter> y)"
heap_hupd_other: fixes aa :: "loc"   and a :: "loc"   and obj :: "obj"   and s :: "state" assumes "aa \<noteq> a" shows "heap (hupd(aa\<mapsto>obj) s) a = heap s a"
Conjunctive_def: shows "Apply.Conjunctive = {x. \<forall>X. x (Inf X) = Inf (x ` X)}"
w0_V: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "vmc_path_no_deadend.w0 P \<in> V\<^bsub>G\<^esub>"
mono_body_fixp: fixes F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)" shows "lfp.fixp_fun F = F (lfp.fixp_fun F)"
reflexive_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "reflexive R \<equiv> \<forall>x. R x x"
set_Cs: shows "order.greater_eq (f \<turnstile> folding_g.\<C>) (set (C0 # Cs))"
set_alls: fixes B :: "nat"   and a :: "nat list" shows "set (alls B a) = {(x, s). length x = length a \<and> (\<forall>i<length a. x ! i \<le> B) \<and> s = a \<bullet> x}"
c: shows "c = Fraction_Field.Fract c0 (1::'a)"
fls_subdegree_fls_to_fps: fixes f :: "'a fps" shows "fls_subdegree (fps_to_fls f) = int (subdegree f)"
strengthen_step: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat"   and e :: "('e \<Rightarrow> 'f) \<Rightarrow> bool"   and prb' :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb"   and "strengthen_extends prb rv e prb'" shows "RedBlack prb'"
plus_right_mono: fixes z :: "'c"   and y :: "'c"   and x :: "'c" assumes "z \<le> y" shows "x + z \<le> x + y"
singleton_subspace_set_in_sets: fixes A :: "('d \<Rightarrow>\<^sub>F 'e) set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and I :: "'d set set" assumes "A \<in> sets (Pi\<^sub>F {J} M)"   and "finite J"   and "J \<in> I" shows "A \<in> sets (Pi\<^sub>F I M)"
dyn_equiv_stat: shows "dynamically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> empty_ord.Red_F_\<G> = statically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> empty_ord.Red_F_\<G>"
lexordp_eq_simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "ord.lexordp_eq less (x \<Colon> xs) [] = False"
rfd_rfb_adj: fixes f :: "('d \<times> 'c) set" shows "fd\<^sub>\<R> f \<stileturn> bb\<^sub>\<R> f"
that: fixes F :: "'a formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "F, \<Gamma>' \<Rightarrow> \<Delta>'"   and "atoms F \<subseteq> atoms_mset (\<Gamma> + \<Delta>)"   and "atoms F \<subseteq> atoms_mset (\<Gamma>' + \<Delta>')" shows "thesis"
sub_arg: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "sub s t" shows "sub s (App t u)"
hash_rt_F\<^sub>m_alt_def: fixes h :: "'a \<Rightarrow> 'c"   and rhm :: "'b \<Rightarrow> 'd" shows "hash_rt_F\<^sub>m h rhm = hash_blindable (map_prod h (map rhm))"
Un_insert_left: fixes a :: "'c"   and B :: "'c set"   and C :: "'c set" shows "{a} \<union> B \<union> C = {a} \<union> (B \<union> C)"
ordered_term_axioms: shows "ordered_term to_pair_unit fst (\<preceq>) (\<prec>) (\<preceq>) (\<prec>)"
set_of_pred_def: shows "set_of_pred = Collect \<circ> pred.eval"
ordLess_iff: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "(r <o r') = (Well_order r \<and> Well_order r' \<and> (\<nexists>f'. embed r' r f'))"
DERIV_neg_dec_right: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "(f has_real_derivative l) (at x)"   and "l < 0" shows "\<exists>d>0. \<forall>h>0. h < d \<longrightarrow> f (x + h) < f x"
simps_3: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and xs :: "'d list" shows "successive P (x # y # xs) = (\<not> P y \<and> successive P xs \<or> \<not> P x \<and> successive P (y # xs))"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "map_option f (pure x2) = pure (f x2)"
simps: fixes A :: "'a mat"   and c :: "'a vec" shows "mat_append_row A c = mat_of_rows (dim_col A) (rows A @ [c])"
Ord_add_mult_iff: fixes \<beta> :: "V"   and \<gamma> :: "V"   and \<beta>' :: "V"   and \<alpha> :: "V"   and \<alpha>' :: "V" assumes "\<beta> \<in> elts \<gamma>"   and "\<beta>' \<in> elts \<gamma>"   and "Ord \<alpha>"   and "Ord \<alpha>'"   and "Ord \<gamma>" shows "(\<gamma> * \<alpha> + \<beta> \<in> elts (\<gamma> * \<alpha>' + \<beta>')) = (\<alpha> \<in> elts \<alpha>' \<or> \<alpha> = \<alpha>' \<and> \<beta> \<in> elts \<beta>')"
assms_4: fixes x :: "'a" assumes "x \<in> s" shows "g' x = g x"
same_prefix_nil: fixes xs :: "'c list"   and ys :: "'c list" shows "prefix (xs @ ys) xs = (ys = [])"
sign_r_pos_sgnx_iff: fixes p :: "real poly"   and a :: "real" shows "sign_r_pos p a = (0 < sgnx (poly p) (at_right a))"