hcju/isabelleps huggingface dataset:
rmge2_init: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "rmge2 (f \<triangleright> n) = (if \<forall>i<Suc n. the (f i) < 2 then Some 0 else Some (the (f (GREATEST i. i < Suc n \<and> 2 \<le> the (f i)))))"
jipsen2l: fixes y :: "'a"   and x :: "'a" shows "(y \<leftarrow> (x \<rightarrow> y)) \<rightarrow> y = x \<rightarrow> y"
normalize_ports_generic: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and \<gamma> :: "('b common_primitive \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and normalize_pos :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr list"   and a :: "action"   and p :: "'c"   and rewrite_neg :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<And>m. \<lbrakk>normalized_nnf_match m; \<not> has_disc_negated disc False m\<rbrakk> \<Longrightarrow> match_list \<gamma> (normalize_pos m) a p = matches \<gamma> m a p"   and "\<And>m. normalized_nnf_match m \<Longrightarrow> matches \<gamma> (rewrite_neg m) a p = matches \<gamma> m a p"   and "\<And>m. normalized_nnf_match m \<Longrightarrow> \<not> has_disc_negated disc False (rewrite_neg m)" shows "match_list \<gamma> (normalize_ports_generic normalize_pos rewrite_neg m) a p = matches \<gamma> m a p"
mset_le_not_refl: fixes M :: "'b multiset" shows "\<not> M < M"
B_eval1: fixes g :: "dB" shows "\<B> \<degree> g \<rightarrow>\<^sub>\<beta> Abs (Abs (lift (lift g 0) 0 \<degree> (Var 1 \<degree> Var 0)))"
Not2IE: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> G" shows "\<^bold>\<not> (\<^bold>\<not> F) \<triangleright> \<Gamma> \<turnstile> G"
fv_Lam: fixes x :: "Vars.var"   and e :: "exp" shows "fv (Lam [x]. e) = fv e - {x}"
defined_not_I: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x" shows "\<tau> \<Turnstile> \<delta> not x"
bound_child_parent_not_eq_last_diff_paths: fixes s :: "nat set"   and k :: "nat" assumes "s \<noteq> {}"   and "\<forall>x. x \<in> s \<longrightarrow> x < k" shows "Sup s < k"
h_def: shows "h = (\<lambda>(x, y). if reachable x \<and> reachable y then g (x, y) else 0)"
tan_less_zero: fixes x :: "real" assumes "- pi / 2 < x"   and "x < 0" shows "tan x < 0"
simps_2: fixes n :: "nat"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and m :: "'c nres" shows "bounded_WHILE (Suc n) cond step m = bounded_WHILE n cond step m \<bind> (\<lambda>x. if cond x then step x else RETURN x)"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) replete_setcat.comp F'G'.\<phi>C F'G'.\<phi>D F' F'G'.F.G F'G'.F.\<phi> F'G'.F.\<psi> (meta_adjunction.\<eta> (\<cdot>\<^sub>C) F' F'G'.F.\<phi>) (meta_adjunction.\<epsilon> (\<cdot>\<^sub>B) F' F'G'.F.G F'G'.F.\<psi>) (transformation_by_components.map F'G'.DopxC.comp replete_setcat.comp F'G'.Hom_FopxC.map (meta_adjunction.\<Phi>o (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) F' F'G'.F.G F'G'.F.\<phi>)) (inverse_transformation.map F'G'.DopxC.comp replete_setcat.comp (F'G'.HomD.map \<circ> product_functor.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>B) (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>C) F'G'.Dop.map F'G'.F.G) (transformation_by_components.map F'G'.DopxC.comp replete_setcat.comp F'G'.Hom_FopxC.map (meta_adjunction.\<Phi>o (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) F' F'G'.F.G F'G'.F.\<phi>)))"
iT_Mult_inext_nth: fixes I :: "nat set"   and k :: "nat"   and n :: "nat" assumes "I \<noteq> {}" shows "(I \<otimes> k) \<rightarrow> n = (I \<rightarrow> n) * k"
inverse_monom_def: fixes monom :: "real \<times> real list" shows "inverse_monom monom = (case monom of (c, es) \<Rightarrow> (inverse c, map uminus es))"
mgu_same_empty: fixes t :: "('b, 'a) Term.term" shows "mgu t t = Some Var"
at_ide_is_diagram: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'d \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "diagram_in_functor_category A B J D"   and "partial_magma.ide A a" shows "diagram J B (\<lambda>j. currying.uncurry J A B D (j, a))"
Sym_L: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (t EQ u) H \<turnstile> A" shows "insert (u EQ t) H \<turnstile> A"
lconf_upd: fixes G :: "prog"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and vn :: "'a" assumes "G,s\<turnstile>l[\<Colon>\<preceq>]L"   and "G,s\<turnstile>v\<Colon>\<preceq>T"   and "L vn = Some T" shows "G,s\<turnstile>l(vn \<mapsto> v)[\<Colon>\<preceq>]L"
smult_1: fixes d :: "'a" shows "Polynomial.smult d 1 = [:d:]"
simps_21: fixes v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "bexp"   and ve :: "'b acom"   and vf :: "'b acom"   and vg :: "'b" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> IF vd THEN ve ELSE vf {vg}) = False"
assms_2: shows "a \<in> F \<^bsub>\<o>\<^esub> A"
mult_unit_dvd_iff: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "is_unit b" shows "(a * b dvd c) = (a dvd c)"
invalid_assn_cong: fixes x :: "'a"   and x' :: "'a"   and y :: "'b"   and y' :: "'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R' :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "x \<equiv> x'"   and "y \<equiv> y'"   and "R x' y' \<equiv> R' x' y'" shows "invalid_assn R x y = invalid_assn R' x' y'"
mem_interior_closure_convex_shrink: fixes S :: "'c set"   and c :: "'c"   and x :: "'c"   and e :: "real" assumes "convex S"   and "c \<in> interior S"   and "x \<in> closure S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> interior S"
THROW_def: fixes e :: "'c" shows "THROW e \<equiv> RETURN (Inl e)"
rel_spmfE: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf" assumes "rel_spmf P p q" obtains pq :: "('c \<times> 'd) spmf" where "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y" and "p = map_spmf fst pq" and "q = map_spmf snd pq"
mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "inf a b \<le> inf c d"
ground_aux_simps_3: fixes t :: "tm"   and u :: "tm"   and S :: "atom set" shows "ground_aux (Eats t u) S = (ground_aux t S \<and> ground_aux u S)"
subsetCI: fixes A :: "'d set"   and B :: "'d set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<notin> B\<rbrakk> \<Longrightarrow> False" shows "A \<subseteq> B"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Grave of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f2"
funpow_0: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "(f ^^ 0) x = x"
to_cnf2_eq_Cons: fixes x :: "ordinal"   and a :: "ordinal"   and b :: "nat"   and list :: "(ordinal \<times> nat) list" assumes "to_cnf2 x = (a, b) # list" shows "a = oLog \<omega> x"
append_eq_Cons_conv: fixes ys :: "'a list"   and zs :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
alphaAbs_qAbs_iff_ex_qFresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
prems_2: shows "(l # ls) ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>, \<R>)"
that_2: assumes "x = y"   and "r < s" shows "thesis"
alllsts_UNIV: fixes s :: "'b llist" shows "s \<in> UNIV\<^sup>\<infinity>"
rrsmultD2: fixes x :: "('f, 'g) aezfun"   and r :: "('f, 'g) aezfun"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "x \<notin> FH" shows "(r \<currency> f) x = (0::'v)"
catalan_aux_simps_2: fixes k :: "nat"   and n :: "nat"   and acc :: "nat" assumes "k < n" shows "catalan_aux n k acc = catalan_aux n (Suc k) (acc * (2 * (2 * k + 1)) div (k + 2))"
rep_list_in_ideal_sig_inv_set: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "r \<in> sig_inv_set' j" shows "rep_list r \<in> ideal (set (take j fs))"
phl_while: fixes p :: "'a"   and s :: "'a"   and w :: "'a"   and x :: "'a" assumes "p \<cdot> s \<le> s \<cdot> p \<cdot> s"   and "p \<cdot> w \<le> w \<cdot> p \<cdot> w"   and "\<lbrace>p \<cdot> s\<rbrace>x\<lbrace>p\<rbrace>" shows "\<lbrace>p\<rbrace>(s \<cdot> x)\<^sup>\<dagger> \<cdot> w\<lbrace>p \<cdot> w\<rbrace>"
is_cparser_exactly: fixes xs :: "char list" assumes "1 \<le> length xs" shows "is_cparser (exactly xs)"
finite_imageI': fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "finite {(x, y). P x y}" shows "finite {f x y |x y. P x y}"
C: shows "\<nexists>c A'. c \<in> carrier_vec n \<and> A' \<subseteq> A \<and> Suc (card A') = dim_span (insert b A) \<and> (\<forall>a\<in>A'. c \<bullet> a = (0::'a)) \<and> lin_indpt A' \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)"
einterpret_fold_const_fa: fixes xs :: "'a list"   and fa :: "'a \<Rightarrow> floatarith"   and vs :: "real list" assumes "length xs = DIM('b)" shows "einterpret (map (\<lambda>i. fold_const_fa (fa i)) xs) vs = einterpret (map fa xs) vs"
omega_in_\<alpha>: shows "\<omega> \<in>\<^sub>\<circ> \<alpha>"
space_subprob_algebra: fixes A :: "'b measure" shows "space (subprob_algebra A) = {M. subprob_space M \<and> sets M = sets A}"
one: shows "Missing_List.min_list (x # xs) = min x (Missing_List.min_list xs)"
Inf_dual: shows "\<partial> \<circ> Inf = Sup \<circ> (`) \<partial>"
intro: fixes bind :: "'c \<Rightarrow> (unit \<Rightarrow> 'b) \<Rightarrow> 'b"   and return :: "unit \<Rightarrow> 'c"   and ASSERT :: "bool \<Rightarrow> 'c"   and ASSUME :: "bool \<Rightarrow> 'c" assumes "\<And>f. bind bot f = bot"   and "\<And>f. bind top f = top"   and "\<And>u f. bind (return u) f = f u"   and "ASSERT \<equiv> iASSERT return"   and "ASSUME \<equiv> iASSUME return" shows "generic_Assert bind return ASSERT ASSUME"
inv_diff: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "inv t1"   and "inv t2" shows "inv (Set2_Join.diff join t1 t2)"
invol_dual_var: fixes x :: "'a" shows "\<partial> (\<partial> x) = x"
summable_powser_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat"   and z :: "'b" shows "summable (\<lambda>n. f (n + m) * z ^ n) = summable (\<lambda>n. f n * z ^ n)"
star: fixes x :: "'a" assumes "x \<in> S" shows "open_segment a x \<subseteq> rel_interior S"
simps_4: fixes x :: "quat" shows "Im3 (- x) = - Im3 x"
Collect_case_prod_Sigma: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
path_component_eq_eq: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "(path_component S x = path_component S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y)"
action_needed_vars_def: fixes a :: "('g, 'h) fmap \<times> 'i"   and s :: "('g, 'h) fmap" shows "action_needed_vars a s \<equiv> {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
concurrent_history_steps_simulates_store_buffer_steps: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'f"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "'f"   and ts\<^sub>h :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S>\<^sub>h :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts', m', \<S>')"   and "ts \<sim>\<^sub>h ts\<^sub>h " shows "\<exists>ts\<^sub>h' \<S>\<^sub>h'. (ts\<^sub>h, m, \<S>\<^sub>h) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>h', m', \<S>\<^sub>h') \<and> ts' \<sim>\<^sub>h ts\<^sub>h' "
neq_le_trans: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
rep_eq: fixes x :: "'c" shows "measure_pmf (return_pmf x) = return (count_space UNIV) x"
assms_2: shows "p \<le> \<^bold>|w\<^bold>|"
inject_2: fixes x21 :: "'b"   and x22 :: "real"   and x23 :: "'b kdt"   and x24 :: "'b kdt"   and y21 :: "'b"   and y22 :: "real"   and y23 :: "'b kdt"   and y24 :: "'b kdt" shows "(Node x21 x22 x23 x24 = Node y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
cblinfun_image_bot: fixes U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "U *\<^sub>S \<bottom> = \<bottom>"
alec: fixes m :: "nat"   and n :: "nat" shows "a (real m / 2 ^ n) \<le> c (real m / 2 ^ n)"
dot_square_norm: fixes x :: "'b" shows "x \<bullet> x = (norm x)\<^sup>2"
simps: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "witness Y r rs = Conv ((if length rs = 0 then id_matrix else if length rs = 1 then all1_matrix else copy_first_matrix) Y r) (witness' r rs)"
one: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
lookup_zero: fixes k :: "'d" shows "poly_mapping.lookup 0 k = (0::'c)"
ALPHA_def: shows "ALPHA = (\<lambda>n. real_of_int q * (\<Prod>j = 1..n. real_of_int (a j)) * (\<Sum>j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1))))"
boundedI: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" assumes "b \<subseteq># a"   and "c \<subseteq># a" shows "b \<union># c \<subseteq># a"
input2state_append: fixes i :: "value list"   and a :: "value" shows "input2state (i @ [a]) = (input2state i)(length i $:= Some a)"
prob_space_imp_subprob_space: fixes M :: "'c measure" assumes "prob_space M" shows "subprob_space M"
minus_mod_eq_div_mult: fixes a :: "'a"   and b :: "'a" shows "a - a mod b = a div b * b"
null_option_def: shows "null \<equiv> \<lfloor>UML_Types.bot_class.bot\<rfloor>"
simps_5: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
nth_via_drop: fixes n :: "nat"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "drop n xs = y # ys" shows "xs ! n = y"
inv_eq_1_iff: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "Group.group G"   and "x \<in> carrier G" shows "(inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>) = (x = \<one>\<^bsub>G\<^esub>)"
po_eq_conv: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.po below" shows "(x = y) = (below x y \<and> below y x)"
simps_3: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))"
left: fixes t1 :: "('c, 'd) tm"   and t2 :: "('c, 'd) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t1"
filterlim_at_bot_dense: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x < Z)"
A1: shows "\<forall>x1 x2. \<exists>y1 y2. P x1 x2 y1 y2"
to_vec_eq_iff: fixes M :: "('c, 'd) sq_matrix"   and N :: "('c, 'd) sq_matrix" shows "(to_vec M = to_vec N) = (M = N)"
elims: fixes x :: "'b Literal"   and y :: "'b" assumes "atom x = y" obtains A :: "'b" where "x = Pos Aa__" and "y = Aa__"   | A :: "'b" where "x = Neg A" and "y = A"
integral_component_ubound: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and k :: "'b"   and B :: "real" assumes "f integrable_on cbox a b"   and "\<forall>x\<in>cbox a b. f x \<bullet> k \<le> B"   and "k \<in> Basis" shows "integral (cbox a b) f \<bullet> k \<le> B * content (cbox a b)"
S2_def: shows "S2 = {y \<in> supp f. - y + x \<in> supp g}"
map_spmf_bind_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'j spmf"   and g :: "'j \<Rightarrow> 'i spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
distinct_sum_list_conv_Sum: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'a list" assumes "class.comm_monoid_add plus zero"   and "distinct xs" shows "monoid_add.sum_list plus zero xs = comm_monoid_add.sum plus zero (\<lambda>x. x) (set xs)"
Liminf_entails_Bot: shows "Liminf_llist Ns \<Turnstile> {B}"
strand_fv_bvars_disjoint_unfold: fixes S :: "('b, 'a) strand_step list"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and Y :: "'a list"   and G :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S"   and "\<forall>Y\<langle>\<or>\<noteq>: G\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "set Y \<inter> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X) = {}"
simps_1: fixes cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and m :: "'c nres" shows "bounded_WHILE 0 cond step m = m"
fid: fixes x :: "'a" assumes "x \<in> sphere a r"   and "x \<in> T" shows "f x = x"
lookup_default_map_values: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and k :: "'c"   and d :: "'d"   and m :: "('c, 'd) mapping" shows "Mapping.lookup_default (f k d) (Mapping.map_values f m) k = f k (Mapping.lookup_default d m k)"
left_in_interval: fixes cl :: "'b potype"   and a :: "'b"   and b :: "'b" assumes "PO cl"   and "a \<in> pset cl"   and "b \<in> pset cl"   and "interval (order cl) a b \<noteq> {}" shows "a \<in> interval (order cl) a b"
execn_to_execn_mark_guards_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and f :: "'f"   and x :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t = Fault f" shows "\<exists>f'. \<Gamma>\<turnstile> \<langle>mark_guards x c,s\<rangle> =n\<Rightarrow> Fault f'"
WT_callee: fixes x :: "'a"   and s :: "'d" assumes "x \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g callee s x \<surd>"
consec_def: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<equiv> i \<noteq> bot \<and> j \<noteq> bot \<and> maximum i + 1 = minimum j"
A_1: shows "u \<in> [up M]\<^bsub>v,n\<^esub>"
class_methodsI: fixes dcl :: "char list"   and cl :: "cl"   and fds :: "fd list"   and meth_defs :: "meth_def list" shows "class_methods (cld_def dcl cl fds meth_defs) meth_defs"
meets_wd: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and p :: "'b"   and q :: "'b" assumes "class.interval meets \<I>"   and "meets p q" shows "\<I> p \<and> \<I> q"
eval_fds_zeta: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds fds_zeta s = zeta s"
cset_nondetM_id: shows "cset_nondetM return_id bind_id merge_id merge_id"
P12: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<le> b l\<rightarrow> c) = (b \<le> a r\<rightarrow> c)"
fps_nonzero_subdegree_nonzeroI: fixes f :: "'a fps" assumes "0 < subdegree f" shows "f \<noteq> 0"
ICons_1: fixes c :: "nat"   and ys :: "'a list"   and a :: "nat" assumes "take c xs = take c ys"   and "a \<le> c" shows "take a xs = take a ys"
span_list_def: fixes vs :: "rat vec list" shows "gs.span_list vs \<equiv> {uu_. \<exists>c. uu_ = gs.lincomb_list c vs}"
mult_left_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "times one x = x"
onode_deliverTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and s :: "'e"   and R :: "nat set"   and ii :: "nat"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and i' :: "nat"   and s' :: "'e"   and R' :: "nat set"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, NodeS i s R), ii:deliver(d), \<sigma>', NodeS i' s' R') \<in> onode_sos S" obtains "ii = i" and "((\<sigma>, s), deliver d, \<sigma>', s') \<in> S" and "\<forall>j. j \<noteq> i \<longrightarrow> \<sigma>' j = \<sigma> j" and "i' = i" and "R' = R"
sup_INF: 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 bot :: "'a"   and top :: "'a"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and B :: "'b set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "sup a (Inf (f ` B)) = Inf ((\<lambda>b. sup a (f b)) ` B)"
inj_onI: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
zero_closed: shows "0\<^sub>M \<in> U\<^sub>M"
prems_2: shows "suffix n w \<Turnstile>\<^sub>n af \<phi> (w [Suc i \<rightarrow> n])[X]\<^sub>\<nu>"
assms_1: shows "\<forall>\<^sub>F y in F. \<forall>x\<in>X. f y x = g y x"
assms: shows "card ((argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) = 1"
cut_lessE: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>< t" obtains "x \<in> I" and "x < t"
swapStarFresh: fixes x :: "name"   and p :: "(name \<times> name) list"   and T :: "'d" assumes "x \<sharp> p" shows "x \<sharp> p \<bullet> T = x \<sharp> T"
MtransC_MtransT: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and cf'' :: "'b" assumes "PL.MtransC tval aval cf cf'"   and "PL.MtransT tval aval cf' cf''" shows "PL.MtransT tval aval cf cf''"
group_ow_axioms_def: fixes U :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" shows "group_ow_axioms U f z inverse \<equiv> (z \<in> U \<and> (\<forall>a. a \<in> U \<longrightarrow> inverse a \<in> U)) \<and> (\<forall>a. a \<in> U \<longrightarrow> f z a = a) \<and> (\<forall>a. a \<in> U \<longrightarrow> f (inverse a) a = z)"
ex_next_free_stutter_free_canonical: fixes \<phi> :: "'b pltl"   and A :: "('b \<Rightarrow> bool) set" assumes "atoms_pltl \<phi> \<subseteq> A"   and "finite A" shows "\<exists>\<psi>. next_free \<psi> \<and> atoms_pltl \<psi> \<subseteq> A \<and> (\<forall>\<sigma>. stutter_free \<sigma> \<and> canonical_sequence A \<sigma> \<longrightarrow> \<sigma> \<Turnstile>\<^sub>p \<psi> = \<sigma> \<Turnstile>\<^sub>p \<phi>)"
well_dec_Ma: shows "local.well_base (image_mset (local.decode 0) (hmsetmset Ma))"
usedBy_uop: fixes x :: "'e \<Longrightarrow> 'f"   and e :: "('g, 'f) uexpr"   and f :: "'g \<Rightarrow> 'h" assumes "x \<natural> e" shows "x \<natural> uop f e"