hcju/isabelleps huggingface dataset:
assms_2: shows "\<forall>\<sigma>. cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> = None \<or> f (snd (the (M \<sigma>))) < f \<sigma>"
assms: shows "\<exists>\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
degree_Poly: fixes xs :: "'b list" shows "degree (Poly xs) \<le> length xs"
i_prems: shows "set (prems_of \<iota>) \<subseteq> fst ` NL"
pvalid_with_def: fixes p :: "(symbol \<times> character list) list"   and x :: "item"   and u :: "nat"   and \<gamma> :: "symbol list" shows "pvalid_with p x u \<gamma> = (wellformed_tokens p \<and> wellformed_item x \<and> u \<le> length p \<and> charslength p = item_end x \<and> charslength (take u p) = item_origin x \<and> is_derivation (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> derives (item_\<alpha> x) (terminals (drop u p)))"
list_ex_length: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "list_ex P xs = (\<exists>n<length xs. P (xs ! n))"
lexord_lex: fixes x :: "'a list"   and y :: "'a list"   and r :: "('a \<times> 'a) set" shows "((x, y) \<in> lex r) = ((x, y) \<in> lexord r \<and> length x = length y)"
assms_5: shows "rem_comps_spec (gs @ ab gs [] bs data) (rc, data)"
select_convs_4: fixes Label :: "String.literal"   and Arity :: "nat"   and Guards :: "vname gexp list"   and Outputs :: "vname aexp list"   and Updates :: "(nat \<times> vname aexp) list"   and more :: "'a" shows "Outputs \<lparr>Label = Label, Arity = Arity, Guards = Guards, Outputs = Outputs, Updates = Updates, \<dots> = more\<rparr> = Outputs"
eq_onp_mono_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(eq_onp P \<le> eq_onp Q) = (P \<le> Q)"
Necessary_expl: shows "\<lfloor>\<lambda>w. \<forall>x. Necessary x w = (\<^bold>\<box>(Abstract x \<^bold>\<or> Concrete x)) w\<rfloor>"
StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then if x \<tau> = null \<or> y \<tau> = null then \<lfloor>\<lfloor>x \<tau> = null \<and> y \<tau> = null\<rfloor>\<rfloor> else \<lfloor>\<lfloor>oid_of (x \<tau>) = oid_of (y \<tau>)\<rfloor>\<rfloor> else invalid \<tau>"
freshEnv_psubst_ident: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "good X"   and "goodEnv rho"   and "\<And>zs z. freshEnv zs z rho \<or> fresh zs z X" shows "X #[rho] = X"
inv_into_into: fixes x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
to_nat_plus_one_less_card: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "\<forall>a. plus a one \<noteq> zero \<longrightarrow> mod_type.to_nat Rep a + 1 < CARD('a)"
update_convs_1: fixes Pol' :: "('c \<times> 'c) set \<Rightarrow> ('c \<times> 'c) set"   and Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "Pol_update Pol' \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = \<lparr>Pol = Pol' Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr>"
left_neutral: fixes a :: "'c" shows "bot \<squnion> a = a"
cnf2bhv_bhv2cnf: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 \<le> n" shows "dynamic_component.cnf2bhv tCMP active c t (dynamic_component.bhv2cnf tCMP active c t n) = n"
image_mset_mset_set: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "f `# mset_set A = mset_set (f ` A)"
intros_21: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and op0 :: "lop"   and e2 :: "exp0" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_log op0 v1 e2 = None" shows "evaluate ck env s1 (Log op0 e1 e2) (s2, Rerr (Rabort Rtype_error))"
before_in_swap2: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat"   and x :: "'c"   and y :: "'c" assumes "dist_perm xs ys"   and "Suc n < length xs"   and "x \<noteq> y" shows "x < y in swap n xs = (\<not> x < y in xs \<and> y = xs ! n \<and> x = xs ! Suc n \<or> x < y in xs \<and> \<not> (y = xs ! Suc n \<and> x = xs ! n))"
prj_simps_15: shows "C.dom gfg.Prj\<^sub>0\<^sub>0 = Chn (g \<star> f \<star> g)"
carrier_product_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" shows "carrier (product_group I G) = (\<Pi>\<^sub>E i\<in>I. carrier (G i))"
sigma_algebra_disjoint_iff: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M))"
simps_3: fixes l1 :: "'b tree"   and a1 :: "'b"   and r1 :: "'b tree"   and l2 :: "'b tree"   and a2 :: "'b"   and r2 :: "'b tree" shows "T_merge \<langle>l1, a1, r1\<rangle> \<langle>l2, a2, r2\<rangle> = (if a1 \<le> a2 then T_merge \<langle>l2, a2, r2\<rangle> r1 else T_merge \<langle>l1, a1, r1\<rangle> r2) + 1"
exhaust: fixes y :: "color" obtains "y = R"   | "y = B"
has_contour_integral_diff: fixes f1 :: "complex \<Rightarrow> complex"   and i1 :: "complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex"   and i2 :: "complex" assumes "(f1 has_contour_integral i1) g"   and "(f2 has_contour_integral i2) g" shows "((\<lambda>x. f1 x - f2 x) has_contour_integral i1 - i2) g"
keys_of_alist: fixes xs :: "('c \<times> 'd) list" shows "Mapping.keys (Mapping.of_alist xs) = set (map fst xs)"
set_map: fixes f :: "'e \<Rightarrow> 'c"   and v :: "'d \<Rightarrow> 'e" shows "range (f \<circ> v) = f ` range v"
Image_subset_eq: fixes r :: "('b \<times> 'a) set"   and A :: "'b set"   and B :: "'a set" shows "(r `` A \<subseteq> B) = (A \<subseteq> - r\<inverse> `` (- B))"
LIMSEQ_def: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
gcd_mult_right_right_cancel: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "coprime a c" shows "gcd a (b * c) = gcd a b"
mcont_sup2: fixes y :: "'a" shows "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. x \<squnion> y)"
inverse_Quantity_ext_def: fixes x :: "('c, 'd, 'e) Quantity_scheme" shows "inverse x = \<lparr>mag = inverse (mag x), dim = inverse (Quantity.dim x), \<dots> = inverse (Quantity.more x)\<rparr>"
preimage_includes_then_almost_invariant: fixes A :: "'a set" assumes "A \<in> sets M"   and "A \<subseteq> T --` A" shows "A \<Delta> T --` A \<in> null_sets M"
hermitean_eigenval_real: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex" assumes "hermitean H"   and "eigenval k H" shows "is_real k"
neqE: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
idom_tOp_nonzeros: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c" assumes "Idomain R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>"   and "y \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<noteq> \<zero>\<^bsub>R\<^esub>"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Bossy.Cd d X" shows "x \<in> X \<inter> Field (BPd d)"
E_sup_iterate: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "f \<in> borel_measurable St" shows "E_sup s f = (\<Squnion>D\<in>K s. \<integral>\<^sup>+ t. E_sup t (\<lambda>\<omega>. f (t ## \<omega>)) \<partial>measure_pmf D)"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c stream"   and b :: "'d stream" shows "stream_all2 R a b = (R (shd a) (shd b) \<and> stream_all2 R (stl a) (stl b))"
remdups_adj_Nil_iff: fixes xs :: "'g list" shows "(remdups_adj xs = []) = (xs = [])"
rett_instr_privilege: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state"   and s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list" assumes "snd (rett_instr i s) = False \<and> s' = snd (fst (rett_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
assms: shows "- \<infinity> < subcocycle_avg_ereal u"
null_sets_iff_AE: fixes A :: "'a set"   and M :: "'a measure" shows "(A \<in> null_sets (completion M)) = ((AE x in completion M. x \<notin> A) \<and> A \<subseteq> space (completion M))"
elims: fixes x :: "('c, 'd) list_op"   and y :: "'c set" assumes "list_op_deps x = y" obtains ref :: "'c" and uu_ :: "'d" where "x = Insert (Some ref__) uu_" and "y = {ref__}"   | uv_ :: "'d" where "x = Insert None uv_" and "y = {}"   | ref :: "'c" where "x = Delete ref" and "y = {ref}"
cfun_eqvtI: fixes p :: "perm"   and f :: "'c \<rightarrow> 'd"   and f' :: "'c \<rightarrow> 'd" assumes "\<And>x. p \<bullet> f\<cdot>x = f'\<cdot>(p \<bullet> x)" shows "p \<bullet> f = f'"
that: fixes x :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "thesis"
convergent_add_const_right_iff: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" shows "convergent (\<lambda>n. f n + c) = convergent f"
e_sub: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v' :: "val" assumes "v \<in> E e \<rho>"   and "v' \<sqsubseteq> v" shows "v' \<in> E e \<rho>"
equivalence_in_bicategory_axioms: shows "internal_equivalence f g' \<phi> \<psi>'"
Tarjan_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "Tarjan G \<equiv> param_DFS G"
fst_scaleR: fixes r :: "real"   and A :: "'b \<times> 'c" shows "fst (r *\<^sub>R A) = r *\<^sub>R fst A"
f_accessible_refl_inter_simp: fixes Q :: "'a set"   and r :: "('a, 'b) ta_rule set" shows "Q \<inter> f_accessible r Q = Q"
generate_salt_def: fixes DB_zero :: "bit list" shows "generate_salt DB_zero = show_rightmost_bits DB_zero sLen"
row_def: fixes A :: "'a mat"   and i :: "nat" shows "Matrix.row A i = Matrix.vec (dim_col A) (\<lambda>j. A $$ (i, j))"
well_founded_Restrp: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" shows "well_founded A (r \<restriction> B) = well_founded (A \<inter> B) r"
strictly_preserves_unity: shows "F \<F>C.unity = \<I>\<^sub>D"
assms: shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
zero_not_one: shows "\<zero>\<^bsub>K [X]\<^esub> \<noteq> \<one>\<^bsub>K [X]\<^esub>"
sorted_inorder_list_induct_subtree: fixes ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list" assumes "sorted_less (inorder_list (ls @ (sub, sep) # rs))" shows "sorted_less (inorder sub)"
round_up_uminus_eq: fixes p :: "int"   and x :: "real" shows "round_up p (- x) = - round_down p x"
free_vars_cexpr_comp': fixes e :: "cexpr"   and e' :: "cexpr" shows "free_vars (e \<circ>\<^sub>c e') \<subseteq> free_vars e \<union> free_vars e'"
I: fixes A :: "fm"   and i :: "name"   and x :: "tm" shows "A(i::=x) IMP SyntaxN.Ex i A \<in> special_axioms"
add_pow_ldistr_int: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "[k] \<cdot> a \<otimes> b = [k] \<cdot> (a \<otimes> b)"
pref_lq: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v"   and "v \<le>p w" shows "u\<inverse>\<^sup>>v \<le>p u\<inverse>\<^sup>>w"
prv_psubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "prv \<phi>" shows "prv (psubst \<phi> txs)"
hp: shows "\<forall>m\<in>monomials p. lookup m var \<le> MPoly_Type.degree p var"
i_Exec_Stream_nth_calc_Previous: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c n = trans_fun (input n) ((i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n)"
sums_If_finite_set: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'c" assumes "finite A" shows "(\<lambda>r. if r \<in> A then f r else (0::'c)) sums sum f A"
plus_Mp_1: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (poly_mod.Mp m f + g) = poly_mod.Mp m (f + g)"
ccSUP_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
bounded: fixes V :: "'c set"   and f :: "'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real" assumes "Function_Norm.continuous V f norm" shows "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * norm x"
gen_ge_dim: fixes A :: "'a vec set" assumes "finite A"   and "A \<subseteq> carrier_vec n"   and "local.span A = carrier_vec n" shows "local.dim \<le> card A"
assms: shows "Lfilled n lholed (es @ es') LI"
s_lm: fixes e :: "nat"   and x :: "nat"   and y :: "nat"   and z :: "nat" shows "nat_to_pr (s_ce e x) (c_pair y z) = nat_to_pr e (c_pair (c_pair x y) z)"
poly_lift_hom_degree_bound: fixes S :: "('e, 'f) ring_scheme"   and h :: "'a \<Rightarrow> 'e"   and f :: "nat \<Rightarrow> 'a" assumes "cring S"   and "h \<in> ring_hom R S"   and "f \<in> carrier (UP R)" shows "deg S (poly_lift_hom R S h f) \<le> deg R f"
that: fixes x' :: "'u" assumes "min s P = Some x'"   and "x' \<in> \<alpha> s"   and "P x'"   and "\<forall>x\<in>\<alpha> s. P x \<longrightarrow> x' \<le> x" shows "thesis"
sorted_wrt_take: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (take n xs)"
assms_2: shows "p = 0 \<or> q = 0"
dec_hd: fixes u :: "'b list"   and G :: "'b list set" assumes "u \<noteq> \<epsilon>"   and "u \<in> \<langle>G\<rangle>" shows "hd (Dec G u) \<in> G"
i_2: shows "normal_exponent x = Float.exponent f - int i"
eq_elimL: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "eq \<phi> \<psi> \<pi>l"   and "\<phi> \<pi>l" obtains "\<psi> \<pi>l"
left_options_neg: fixes u :: "game" shows "left_options (neg_game u) = zimage neg_game (right_options u)"
set_choose_sing: fixes x :: "'a" shows "set_choose {x} = x"
assms_1: shows "Controller_L y (fin_inf_append [Zero] l) l z"
P3_4: shows "fst a \<subseteq>\<^sub>f s2"
t_dec: shows "trace init (take i t) (S t i) \<and> S t i \<turnstile> t ! i \<mapsto> S t (Suc i)"
invar_C_ss_V: fixes u :: "nat" assumes "u \<in> C" shows "u \<in> V"
pmf_Pi: fixes A :: "'a set"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "pmf (Pi_pmf A dflt p) f = (if \<forall>x. x \<notin> A \<longrightarrow> f x = dflt then \<Prod>x\<in>A. pmf (p x) (f x) else 0)"
vars_replace_coeff: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b mpoly" assumes "f (0::'b) = (0::'a)" shows "vars (replace_coeff f p) \<subseteq> vars p"
type_wf_def: shows "ObjectClass.type_wf \<equiv> l_type_wf_def\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t.a_type_wf"
ANDConcEnd: fixes xs :: "('b set set, 'c) Combinators list"   and xa :: "('b set set, 'c) Combinators" assumes "allNetsDistinct (xs @ [xa])"   and "singleCombinators xs" shows "allNetsDistinct xs"
rotate_lr: fixes k :: "nat"   and xs :: "'b list" shows "rotater k (rotate k xs) = xs"
degree_of_int_poly: fixes p :: "int poly" shows "degree (of_int_poly p) = degree p"
path_Append: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n'' :: "'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n as n''"   and "CFGExit_wf.path n'' as' n'" shows "CFGExit_wf.path n (as @ as') n'"
cases: fixes x :: "'b match_expr" obtains "x = MatchAny"   | m :: "'b match_expr" where "x = MatchNot m"   | m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = MatchAnd m1 m2"   | uu_ :: "'b" where "x = Match uu_"
antisym: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "x \<subseteq># y"   and "y \<subseteq># x" shows "x = y"
list_succ_funpow_conv: fixes xs :: "'b list"   and x :: "'b"   and n :: "nat" assumes "distinct xs"   and "x \<in> set xs" shows "(list_succ xs ^^ n) x = xs ! ((index xs x + n) mod length xs)"
finite_lookup: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "finite {k. lookup f k \<noteq> (0::'b)}"
segment_inc_if: fixes b :: "'a"   and a :: "'a" assumes "b \<in> carrier D"   and "a \<in> segment D b" shows "a \<prec> b"
face_cycle_eq: fixes b :: "'b"   and a :: "'b" assumes "b \<in> H.face_cycle_set a" shows "H.face_cycle_set b = H.face_cycle_set a"
iFROM_set_def: shows "iFROM_set \<equiv> {[n\<dots>] |n. True}"
inferences_from_def: fixes CC :: "'a literal multiset set" shows "inferences_from CC = {\<gamma> \<in> \<Gamma>. infer_from CC \<gamma>}"
weakBisimulationp_weakBisimulation_eq: shows "weakBisimulationp = weakBisimJudge"
delta_pf_updated: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "cls_val_process Mkt (delta_pf N der matur) (Suc n) w = geom_proc (Suc n) w * delta_price N der matur n w"
on_curve_infinity: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "ell_field R" shows "ell_field.on_curve R a b Infinity"
H_1: shows "F \<subseteq> cball (0::'a) R"
simps: fixes f :: "nat \<Rightarrow> nat"   and V :: "nat set"   and E :: "nat set set" shows "map_ugraph f (V, E) = (f ` V, (`) f ` E)"
cm_alpha1ialpha4mi: shows "m O (e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>) \<subseteq> m \<union> ov \<union> s \<union> d \<union> b \<union> f\<inverse> \<union> f \<union> e"
map_add_map_of_foldr: fixes m :: "'a \<Rightarrow> 'b option"   and ps :: "('a \<times> 'b) list" shows "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
disc_eq_case_2: fixes step :: "('e, 'f) step" shows "is_Skip step = (case step of Skip uu_ \<Rightarrow> True | _ \<Rightarrow> False)"
\<tau>diverge_simulation2: fixes s2 :: "'s2"   and s1 :: "'s1" assumes "s2 -\<tau>2\<rightarrow> \<infinity>"   and "s1 \<approx> s2" shows "s1 -\<tau>1\<rightarrow> \<infinity>"
sub_put_unis: fixes k :: "nat"   and p :: "('a, 'b) form"   and c :: "'a"   and i :: "nat" shows "put_unis k p[App c []/i] = put_unis k (p[App c []/i + k])"
m_upd_def: shows "m_upd = Mapping.filter (\<lambda>_. ts_tp_lt' ts (tp - len)) m"
Complex_scale4: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "c \<noteq> 0" shows "Complex (a / c) (b / c) = Complex a b / cor c"
\<C>_vars_correct: fixes x :: "'Var" shows "\<C>_vars x = vars_of_type (dma_type x)"
1: shows "((\<lambda>x. \<MM> (real (nat \<lfloor>x\<rfloor>)) - ln (real (nat \<lfloor>x\<rfloor>))) \<longlongrightarrow> c) at_top"
root_lb: fixes p :: "real poly" assumes "p \<noteq> 0" obtains lb :: "real" where "\<forall>x. poly p x = 0 \<longrightarrow> lb < x" and "\<forall>x\<le>lb. sgn (poly p x) = sgn_neg_inf p"
AA22: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> 'd" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> N \<and> \<langle>A\<rangle>_v \<longrightarrow> \<circle>Q"   and "|~ P \<and> Unchanged w \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>N \<and> \<box>(\<box>P \<longrightarrow> \<diamond>\<langle>A\<rangle>_v) \<longrightarrow> P \<leadsto> Q"
max_stepsize_mono: fixes j :: "nat"   and n :: "nat" assumes "j \<le> n" shows "max_stepsize j \<le> max_stepsize n"
irc_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set"   and a :: "'d" assumes "irc_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "a \<notin> f (insert a B)" shows "f (insert a B) = f B"
power_commuting_commutes: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x * y = y * x" shows "x ^ n * y = y * x ^ n"
norm_signed00: fixes xs :: "bit list" shows "norm_signed (\<zero> # \<zero> # xs) = norm_signed (\<zero> # xs)"
\<alpha>ah_cons_cases: fixes x :: "'b"   and e :: "'b set \<times> 'b set"   and w' :: "('b set \<times> 'b set) list"   and m :: "'b" assumes "x \<in> \<alpha>ah (e # w') m" obtains "m \<in> fst e" and "x \<in> fst e \<union> snd e \<union> mon_pl w'"   | "m \<notin> fst e" and "x \<in> \<alpha>ah w' m"
that: fixes ns :: "'node list" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "thesis"
assms_4: shows "(\<Sigma>, \<Sigma>') \<in> br_step \<delta>"
Sup_empty: shows "\<Squnion> {} = []\<^sub>\<circ>"
P_t_eq_P_OB: fixes k :: "'key"   and obs :: "'observation list" assumes "k \<in> keys"   and "K k \<noteq> 0"   and "obs \<in> OB ` msgs" shows "prob ((\<lambda>x. ((t \<circ> OB) x, fst x)) -` {(t obs, k)} \<inter> msgs) = real (card (t -` {t obs} \<inter> OB ` msgs)) * prob ((\<lambda>x. (OB x, fst x)) -` {(obs, k)} \<inter> msgs)"
toplevel_summands_in_lang: fixes w :: "'a list"   and n :: "nat"   and s :: "'b rexp" shows "(w \<in> lang n s) = (\<exists>r\<in>toplevel_summands s. w \<in> lang n r)"
list_rel_simp_2: fixes l :: "'c list"   and R :: "('c \<times> 'd) set" shows "((l, []) \<in> \<langle>R\<rangle>list_rel) = (l = [])"
Eps_cong: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Eps P = Eps Q"
rep_eq: fixes x :: "'a sq_mtx"   and xa :: "'a sq_mtx" shows "to_vec (x - xa) = to_vec x - to_vec xa"
case_4: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a" shows "(case Stuck of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f4"
map\<^sub>0_simp: fixes a :: "'b" assumes "B.obj a" shows "GoH.map\<^sub>0 a = \<tau>.G.map\<^sub>0 (H.map\<^sub>0 a)"
path_is_empty_impl_def: fixes s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "path_is_empty_impl s \<equiv> GS.S s = []"
integral_ivl_bound: fixes t0 :: "real"   and h :: "real"   and f :: "real \<Rightarrow> 'b"   and l :: "'b"   and u :: "'b" assumes "\<And>x h'. \<lbrakk>h' \<in> {t0..h}; x \<in> {t0..h}\<rbrakk> \<Longrightarrow> (h' - t0) *\<^sub>R f x \<in> {l..u}"   and "t0 \<le> h"   and "f integrable_on {t0..h}" shows "integral {t0..h} f \<in> {l..u}"