hcju/isabelleps huggingface dataset:
mat_leqb_eqc_satisfies2: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and x :: "nat \<Rightarrow> rat"   and i :: "nat" assumes "x \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"   and "dim_vec b \<le> i"   and "i < dim_vec (b @\<^sub>v c)"   and "dim_row A = dim_vec b"   and "dim_vec c \<le> dim_col A" shows "matrix_to_lpolies (two_block_non_interfering A A\<^sup>T) ! i \<lbrace> x \<rbrace> = (b @\<^sub>v c) $ i"
\<sigma>_def: shows "\<sigma> = auto_run \<xi> qs"
Lim_at_imp_Lim_at_within: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x :: "'b"   and S :: "'b set" assumes "f \<midarrow>x\<rightarrow> l" shows "(f \<longlongrightarrow> l) (at x within S)"
simps_8: fixes p :: "fm"   and q :: "fm" shows "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
a: shows "paid_A ! n = filter (\<lambda>x. Suc x < length init) (paid_A' ! n)"
subset_closed_def: fixes C :: "'b set set" shows "subset_closed (C::'b set set) \<equiv> \<forall>S::'b set\<in>C. \<forall>s\<subseteq>S. s \<in> C"
i_Exec_Stream_Init_nth: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c n = f_Exec_Comp trans_fun (input \<Down> n) c"
map_iteratei_I: fixes invar :: "'c \<Rightarrow> bool"   and iti :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator (iti m) (\<alpha> m)" shows "map_iteratei \<alpha> invar iti"
d: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list"   and k :: "nat"   and la :: "complex" assumes "spectral_radius_1_jnf_max A n m lam n_as"   and "(k, la) \<in> set n_as"   and "cmod la = 1" shows "la ^ spectral_radius_1_jnf_max.d n_as la = 1 \<and> spectral_radius_1_jnf_max.d n_as la \<noteq> 0"
ceiling_zero: shows "\<lceil>0::'b\<rceil> = 0"
Gauss_Jordan_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "Gauss_Jordan A = Gauss_Jordan_upt_k A (ncols A - 1)"
prefixclosed_def: fixes tr :: "'c list set" shows "prefixclosed tr \<equiv> \<forall>l1\<in>tr. \<forall>l2. l2 \<preceq> l1 \<longrightarrow> l2 \<in> tr"
finite_measure_axioms: fixes s :: "'a" shows "finite_measure (DTMC.T s)"
n_kat_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<cdot> t z = (0::'a)) = (!x \<cdot> y \<le> y \<cdot> !z)"
poly_of_vars_simps_1: fixes \<phi> :: "char list \<Rightarrow> nat"   and x :: "char list"   and xs :: "char list multiset" assumes "poly_embed \<phi>" shows "poly_embed.poly_of_vars \<phi> (add_mset x xs) = Var (\<phi> x) * poly_embed.poly_of_vars \<phi> xs"
piecewise_continuous_on_combine: fixes a :: "'a"   and b :: "'a"   and J :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "'a" assumes "piecewise_continuous_on a b J f"   and "piecewise_continuous_on b c J f" shows "piecewise_continuous_on a c J f"
elims: fixes x :: "('dyn, 'ubx1, 'ubx2) unboxed"   and y :: "('dyn, 'c, 'd) unboxed" assumes "box_operand x = y" obtains d :: "'dyn" where "xa__ = OpDyn d" and "y = OpDyn d"   | x :: "'ubx1" where "xa__ = OpUbx1 xb__" and "y = OpDyn (box_ubx1 xb__)"   | x :: "'ubx2" where "xa__ = OpUbx2 x" and "y = OpDyn (box_ubx2 x)"
dg_prod_Cod_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "\<D>\<^sub>\<circ> (dg_prod I \<AA>\<lparr>Cod\<rparr>) = dg_prod I \<AA>\<lparr>Arr\<rparr>"
simps_1: fixes bs :: "'b list"   and c :: "int \<times> int" shows "\<lparr>C c\<rparr>\<^sub>p\<^bsup>bs\<^esup> = INum c"
init_fin_bisim_iff: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and s1 :: "status"   and x1 :: "'a"   and m1 :: "'d"   and s2 :: "status"   and x2 :: "'b"   and m2 :: "'e" shows "FWbisimulation_base.init_fin_bisim final1 final2 bisim t ((s1, x1), m1) ((s2, x2), m2) = (s1 = s2 \<and> bisim t (x1, m1) (x2, m2) \<and> (s2 = Finished \<longrightarrow> final1 x1 \<and> final2 x2))"
list_encode_eq: fixes x :: "nat list"   and y :: "nat list" shows "(list_encode x = list_encode y) = (x = y)"
valid_sops_expr_eval_expr_in_range: fixes t :: "nat"   and e :: "expr" assumes "valid_sops_expr t e" shows "\<forall>t'\<in>fst (eval_expr t e). t' < t + used_tmps e"
fmfilter_alt_defs_5: fixes B :: "'d fset" shows "fmrestrict_fset B = fmfilter (\<lambda>a. a |\<in>| B)"
assms: shows "(s, a, s') \<in> automaton.trans (pnet np n)"
Quotient3_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a" shows "Quotient3 R Abs Rep = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)))"
monotone_case_prod_applyI: fixes orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" assumes "monotone orda ordb (\<lambda>(a, b). f a b y)" shows "monotone orda ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y)"
path_edge: fixes a :: "'edge" assumes "valid_edge a" shows "sourcenode a -[a]\<rightarrow>* targetnode a"
simps_4: fixes b1 :: "bool"   and v :: "'b"   and va :: "value list" shows "GExp_Lexorder.less_gexp_aux (Bc b1) (In v va) = True"
fds_nth_scale_power: fixes c :: "nat"   and f :: "'b fds"   and n :: "nat" assumes "0 < c" shows "fds_nth (fds_scale c f) (n ^ c) = fds_nth f n"
IH: fixes out :: "'c"   and c :: "('a, 'c, 'e) rpv"   and input :: "'e"   and gpv' :: "('b, 'd, 'f) gpv" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "rel_gpv'' A C R (c input) gpv'" shows "finite_gpv \<I>' gpv'"
subset_cls_val_process': fixes A :: "'b set"   and p :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and n :: "nat"   and w :: "'c" assumes "finite A"   and "support_set p \<subseteq> A" shows "cls_val_process Mkt p (Suc n) w = (\<Sum>x\<in>A. prices Mkt x (Suc n) w * p x (Suc n) w)"
mod_dist_5: fixes P :: "assn"   and Q :: "assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P \<or>\<^sub>A Q) = (h \<Turnstile> P \<or> h \<Turnstile> Q)"
FAss\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "val list"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and h\<^sub>2' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "fs' = fs((F, D) \<mapsto> v)"   and "h\<^sub>2' = h\<^sub>2(a \<mapsto> (C, fs'))" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h\<^sub>2', l\<^sub>2)\<rangle>"
simps_2: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface" shows "iiface_constrain ipassmt (Match (IIface ifce)) = ipassmt_iface_constrain_srcip_mexpr ipassmt ifce"
jordan_nf_matrix_poly_bound: fixes A :: "'a mat"   and n :: "nat"   and n_as :: "(nat \<times> 'a) list"   and N :: "nat" assumes "A \<in> carrier_mat n n"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; 0 < n\<rbrakk> \<Longrightarrow> norm a \<le> 1"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; norm a = 1\<rbrakk> \<Longrightarrow> n \<le> N"   and "jordan_nf A n_as" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (N - 1))"
is_EQ_def: shows "is_EQ \<equiv> Ordering_case\<cdot>FF\<cdot>TT\<cdot>FF"
supp_distinguishing_weak_formula: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P \<equiv>\<cdot> Q" shows "supp (distinguishing_weak_formula P Q) \<subseteq> supp P"
prod_ord_pairD: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "prod_ord_pair x P xs ys"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "P k (lookup_pair x xs k) (lookup_pair x ys k)"
typed_secure: fixes m :: "Mode \<Rightarrow> Eg1.var set"   and c :: "(Eg1.var, aexp, bexp) Stmt"   and \<Gamma>' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>' :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set" assumes "has_type (\<Gamma>_of_mds m) (\<S>_of_mds m) {} c \<Gamma>' \<S>' P'"   and "mds_yields_stable_types m" shows "com_sifum_secure (c, m)"
dist_triangle: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "dist x z \<le> dist x y + dist y z"
map_disc_iff: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c option" shows "(map_option f a = None) = (a = None)"
\<sigma>_eq_0_output_correct: fixes \<alpha>0 :: "nat"   and r :: "nat"   and u0 :: "nat"   and v0 :: "nat" shows "((\<^bold>g [^] \<alpha>0) [^] r) [^] u0 \<otimes> (\<^bold>g [^] \<alpha>0) [^] v0 \<otimes> x0 \<otimes> inv (((\<^bold>g [^] r) [^] u0 \<otimes> \<^bold>g [^] v0) [^] \<alpha>0) = x0"
simps_2: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'b" shows "parTs \<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> = parTs"
exhaust: fixes y :: "bdt" obtains "y = Zero"   | "y = One"   | x31 :: "bdt" and x32 :: "nat" and x33 :: "bdt" where "y = Bdt_Node x31 x32 x33"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "real \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "'a"   and ereal :: "ereal" shows "P (case ereal of ereal x \<Rightarrow> f1 x | PInfty \<Rightarrow> f2 | MInfty \<Rightarrow> f3) = ((\<forall>x1. ereal = ereal x1 \<longrightarrow> P (f1 x1)) \<and> (ereal = PInfty \<longrightarrow> P f2) \<and> (ereal = MInfty \<longrightarrow> P f3))"
proc_ord2a: fixes P :: "'c process"   and Q :: "'c process"   and s :: "'c event list"   and X :: "'c event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F P) = ((s, X) \<in> F Q)"
"case": fixes f :: "cl \<Rightarrow> char list \<Rightarrow> vd list \<Rightarrow> 'a"   and x1 :: "cl"   and x2 :: "char list"   and x3 :: "vd list" shows "(case meth_sig_def x1 x2 x3 of meth_sig_def x xa xb \<Rightarrow> f x xa xb) = f x1 x2 x3"
matroid_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "matroid \<E> (indep_in \<E>)"
mat_zero_def: shows "\<delta> \<equiv> \<lambda>j i. 0::'c"
simps_1: fixes uu :: "'c multiset" shows "repeat_mset 0 uu = {#}"
Program_def: shows "Program = {(init, acts, allowed). Id \<in> acts \<and> Id \<in> allowed}"
simps: fixes P1 :: "'c \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> bool"   and a :: "'c + 'd" shows "pred_sum P1 P2 a = ((\<exists>aa. a = Inl aa \<and> P1 aa) \<or> (\<exists>b. a = Inr b \<and> P2 b))"
permutation_mat_left: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n nc"   and "p permutes {..<n}" shows "permutation_mat n p * A = Matrix.mat n nc (\<lambda>(i, j). A $$ (inv p i, j))"
induct: fixes P :: "ctx \<Rightarrow> bool"   and ctx :: "ctx" assumes "P \<box>"   and "\<And>ctx lam. P ctx \<Longrightarrow> P (CAppL ctx lam)"   and "\<And>lam ctx. P ctx \<Longrightarrow> P (CAppR lam ctx)"   and "\<And>name ctx. P ctx \<Longrightarrow> P (CLam [name].ctx)" shows "P ctx"
add_mult_less: fixes \<gamma> :: "V"   and \<alpha> :: "V"   and \<nu> :: "V"   and \<beta> :: "V" assumes "\<gamma> \<in> elts \<alpha>"   and "\<nu> \<in> elts \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" shows "\<alpha> * \<nu> + \<gamma> \<in> elts (\<alpha> * \<beta>)"
assms: fixes l :: "'a"   and x :: "'b"   and y :: "'b" assumes "(l, x, y) \<in> edges X"   and "x \<in> vertices X"   and "y \<in> vertices X" shows "(l, f x, f y) \<in> Y"
eq_onp_to_eq: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "eq_onp P x y" shows "x = y"
In_def: fixes x2 :: "'d"   and x1 :: "'d" shows "x2\<lparr>x1\<rparr> \<equiv> Abs_action (In_Rep x2 x1)"
hb_consistent_append_D1: fixes xs :: "'a list"   and ys :: "'a list" assumes "hb_consistent (xs @ ys)" shows "hb_consistent xs"
WbisT_matchC_MC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>wT d" shows "matchC_MC WbisT c d"
single_valued_confluent: fixes r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "single_valued r"   and "(x, y) \<in> r\<^sup>*"   and "(x, z) \<in> r\<^sup>*" shows "(y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
TS_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.TS Bet A B P Q \<equiv> \<not> Tarski_neutral_dimensionless.Col Bet P A B \<and> \<not> Tarski_neutral_dimensionless.Col Bet Q A B \<and> (\<exists>T. Tarski_neutral_dimensionless.Col Bet T A B \<and> Bet P T Q)"
apply_vareq: fixes x :: "'c"   and t :: "real" shows "blinfun_apply (vareq x t) = blinfun_apply (ode_d1 (flow0 x t))"
eq_onp_eqD: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "eq_onp P = Q" shows "P x = Q x x"
ptrm_1_Fvars: fixes \<sigma> :: "'fmla" assumes "\<sigma> \<in> ptrm (Suc 0)" shows "Fvars \<sigma> = {out, inp}"
HVZK_def: shows "local.\<Sigma>_protocols_base.HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>R. local.\<Sigma>_protocols_base.R h w e = local.\<Sigma>_protocols_base.S h e) \<and> (\<forall>h\<in>valid_pub. \<forall>(a, z)\<in>set_spmf (S2 h e). check h a e z)"
tensor_char: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "f \<otimes>\<^sub>S g = \<F>C.mkarr (\<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>\<F>C.rep g\<^bold>\<rfloor>)"
mono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
l_cancel_one': fixes x :: "'a vec"   and a :: "'a vec" assumes "x \<in> carrier_vec n"   and "a \<in> carrier_vec n" shows "(x = x + a) = (a = 0\<^sub>v n)"
simps_46: fixes x2 :: "int" shows "vars (Const x2) = {}"
interleave_cases: fixes a3 :: "'b list"   and a1 :: "'b list"   and a2 :: "'b list" assumes "a3 \<in> a1 \<otimes> a2" obtains "a1 = []" and "a2 = []" and "a3 = []"   | xs :: "'b list" and ys :: "'b list" and zs :: "'b list" and x :: "'b" where "a1 = x__ # xsa__" and "a2 = ys__" and "a3 = x__ # zs__" and "zs__ \<in> xsa__ \<otimes> ys__"   | xs :: "'b list" and ys :: "'b list" and zs :: "'b list" and x :: "'b" where "a1 = xs" and "a2 = x # ys" and "a3 = x # zs" and "zs \<in> xs \<otimes> ys"
subst_fmla_id: fixes A :: "fmla"   and a :: "name" shows "A(a::=Var a) = A"
remove_shadow_root_get_child_nodes_different_pointers: fixes ptr :: "(_) object_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr"   and "w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr" shows "r h h'"
7: shows "set (project visible s\<^sub>2) = set (project visible s\<^sub>3)"
VertauschungS: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<EE> :: "'b formula"   and \<DD> :: "'b formula"   and \<Lambda> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<Theta> @ \<EE> # \<DD> # \<Lambda>" shows "\<Gamma> \<Rightarrow> \<Theta> @ \<DD> # \<EE> # \<Lambda>"
multiset_eq_iff: fixes M :: "'k multiset"   and N :: "'k multiset" shows "(M = N) = (\<forall>a. count M a = count N a)"
b: shows "nil \<sqinter> (c::'a::refinement_lattice) ; c\<^sup>\<omega> \<sqinter> c ; c\<^sup>\<omega> ; c\<^sup>\<omega> = c\<^sup>\<omega> \<sqinter> c ; c\<^sup>\<omega> ; c\<^sup>\<omega>"
prefixToLevelElementsElementLevel: fixes e :: "'b"   and level :: "nat"   and M :: "('b \<times> bool) list" assumes "e \<in> set (elements (prefixToLevel level M))" shows "elementLevel e M \<le> level"
iptables_bigstepD_9: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Call chain)]"   and "s = Undecided"   and "matches \<gamma> m p"   and "\<Gamma> chain = Some rs" obtains "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m' :: "'c match_expr" where "rs = rs\<^sub>1 @ Rule m' Return # rs\<^sub>2" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> Undecided" and "t = Undecided"
mem_correct_tracing: shows "mem_correct (state_mem_defs.lookup_trace (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (Mapping.lookup m k)))) (state_mem_defs.update_trace (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (Mapping.update k v m)))) ((\<lambda>_. True) \<circ> snd)"
hyp2_rep_def: fixes p :: "hyp2" shows "hyp2_rep p \<equiv> cart2_pt (Rep_hyp2 p)"
tObsC_def: shows "tObsC \<equiv> tMap (envObsC \<circ> es)"
path_independent_on_symmetricI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "\<And>B C. \<lbrakk>B \<subseteq> A; C \<subseteq> A\<rbrakk> \<Longrightarrow> f (B \<union> C) = f (f B \<union> f C)" shows "path_independent_on A f"
pCons_eq_0_iff: fixes a :: "'a"   and p :: "'a Polynomial.poly" shows "(pCons a p = 0) = (a = (0::'a) \<and> p = 0)"
exH_eq_H: shows "exH_n = H_k (n - 1)"
simps: fixes a1 :: "real \<Rightarrow> real"   and a2 :: "real msstream" shows "local.asymp_powser a1 a2 = (\<exists>f cs f' c. a1 = f' \<and> a2 = MSSCons c cs \<and> local.asymp_powser f cs \<and> f' \<in> O[F](\<lambda>_. 1) \<and> (\<forall>\<^sub>F x in F. f' x = c + f x * h x))"
Rep_OrdSum_eq: fixes x :: "('b, 'c) OrdSum"   and y :: "('b, 'c) OrdSum" assumes "Rep_OrdSum x = Rep_OrdSum y" shows "x = y"
rderivs_replicate_mod: fixes i :: "nat"   and as :: "'a"   and r :: "'b rexp"   and m :: "nat" assumes "\<guillemotleft>rderivs (replicate i as) r\<guillemotright> = \<guillemotleft>r\<guillemotright>"   and "0 < i" shows "\<guillemotleft>rderivs (replicate m as) r\<guillemotright> = \<guillemotleft>rderivs (replicate (m mod i) as) r\<guillemotright>"
nt_mem: fixes a :: "'a"   and n :: "int" assumes "sfg A a" shows "n\<triangleright>a\<^bsub>A\<^esub> \<in> carrier A"
cases: fixes x :: "'a list" obtains xa :: "'a" and uu_ :: "'a list" where "x = xa # uu_"   | "x = []"
inner_verts_Cons: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u (e # es) v" shows "inner_verts (e # es) = (if es \<noteq> [] then snd e # inner_verts es else [])"
leg1_is_map: shows "is_left_adjoint s\<^sub>1"
rtrancl_Int1: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set" assumes "(a, b) \<in> (A \<inter> B)\<^sup>*" shows "(a, b) \<in> A\<^sup>*"
coeffs_nth: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0"   and "n \<le> degree p" shows "coeffs p ! n = coeff p n"
m1step: shows "(\<exists>p \<alpha>. \<langle>x:=\<^bsub>\<iota>\<^esub> e,m1\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m2\<rangle>) \<and> (\<forall>m''. (\<exists>p \<alpha>. \<langle>x:=\<^bsub>\<iota>\<^esub> e,m1\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m''\<rangle>) \<longrightarrow> m'' = m2)"
intros: fixes p :: "'a \<times> 'b" shows "fstsp p (fst p)"
rw_add_alt: shows "rw_add = (\<lambda>p q. \<not> rw_add_strict q p)"
pos_zcount_in_zmset: fixes M :: "'a zmultiset"   and x :: "'a" assumes "0 < zcount M x" shows "x \<in>#\<^sub>z M"
assms: shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T \<Leftarrow> U"
mult_same_comm: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> Units G" shows "a [^] m \<otimes> inv (a [^] n) = inv (a [^] n) \<otimes> a [^] m"
Diff: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a - b \<in> sets M"
simps_2: fixes c :: "'c"   and uu :: "'d" shows "collect_clks (LT c uu) = {c}"
approx_8: shows "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"
refl: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "\<langle>fst (fst lc), snd (fst lc), snd lc\<rangle> \<in> loc_reach lc"
rm1_size: shows "int (size (remove1_mset b1 \<B>)) = \<v> - 1"
j_m: fixes jj :: "nat" assumes "jj \<le> i + 1" shows "int (N ^ jj) \<le> int (N ^ m)"
hastyE_1: fixes c :: "Const"   and t :: "Ty" assumes "v_const c hasty t" obtains "c isof t"
contlubI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))" shows "contlub f"
in_X: fixes y :: "'v" assumes "y \<notin> X" shows "h_diff i (y, x) = 0"
absorb1: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "max a b = a"
simps: fixes k :: "nat" shows "\<I> k = natUnion (\<J> k)"
simps_5: fixes m :: "nat"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "closed m (Or p q) = (closed m p \<and> closed m q)"
\<F>_subformulas\<^sub>\<mu>: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F> \<phi> w \<subseteq> subformulas\<^sub>\<mu> \<phi>"
SUP_nat_binary: fixes A :: "'c"   and B :: "'c" shows "sup A (SUP x\<in>Collect ((<) 0). B) = sup A B"
Inv_Inv: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Inv C (monoidal_language.Inv C t) = t"
inject_1: fixes x1 :: "Keys"   and y1 :: "Keys" shows "(kKS x1 = kKS y1) = (x1 = y1)"
assms_1: shows "desargues_config A B C A' B' C' M N P R"
Let_supp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "supp (Terms.Let \<Gamma> e) = supp e \<union> supp \<Gamma> - atom ` domA \<Gamma>"