hcju/isabelleps huggingface dataset:
has_fps_expansion_exp: fixes c :: "'a" shows "(\<lambda>x. exp (c * x)) has_fps_expansion fps_exp c"
CFG_return_edge_fun: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list"   and vmap :: "'var \<Rightarrow> 'val option"   and vmap' :: "'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "f' vmap vmap' = vmap'(ParamDefs (targetnode a) [:=] map vmap outs)"
a3: shows "\<forall>nP e1 e2. \<not> P (nP e1) (nP e2) \<longrightarrow> \<not> P \<bottom> (nP e2)"
not_in_hyp: shows "\<forall>n<dim_vec C. C $ n \<noteq> 0 \<longrightarrow> B ! n \<in> A"
assms: shows "Proj (- C) *\<^sub>S A \<le> B"
c2: shows "homotopic_with_canon (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "\<And>factor_sq factor_pr limit n i. \<lbrakk>\<And>x xa. \<lbrakk>i \<le> limit \<and> 2 \<le> i; i dvd n; x = n div i; i dvd x; xa = x div i\<rbrakk> \<Longrightarrow> P (factor_sq * i) factor_pr (nat (root_nat_floor 3 xa)) xa i; \<And>x. \<lbrakk>i \<le> limit \<and> 2 \<le> i; i dvd n; x = n div i; \<not> i dvd x; sqrt_nat x = []\<rbrakk> \<Longrightarrow> P factor_sq (factor_pr * i) (nat (root_nat_floor 3 x)) x (Suc i); \<lbrakk>i \<le> limit \<and> 2 \<le> i; \<not> i dvd n\<rbrakk> \<Longrightarrow> P factor_sq factor_pr limit n (Suc i)\<rbrakk> \<Longrightarrow> P factor_sq factor_pr limit n i" shows "P a0 a1 a2 a3 a4"
space_bind_empty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure" assumes "space M = {}" shows "space (M \<bind> f) = {}"
seq_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b local.term set"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.comp C g f) = (free_monoidal_category.comp C g f \<noteq> partial_magma.null (free_monoidal_category.comp C))"
declassified_secrets_are_homogeneous: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "s \<in> declassified\<^sub>l\<^sub>s\<^sub>t \<A> \<I>" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t s \<A> Sec"
distinct_1: shows "AsmNoReadOrWrite \<noteq> AsmNoWrite"
\<pi>_def_on_edges_to_S: fixes v :: "'v"   and u :: "'v" assumes "v \<in> S (A Q \<pi>)"   and "u \<noteq> r"   and "(u, v) \<in> edges rg" shows "\<pi> u \<noteq> None"
eval_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.eval (R\<lparr>carrier := K\<rparr>) = local.eval"
Under_decr: fixes A :: "'d set"   and B :: "'d set"   and r :: "('d \<times> 'd) set" assumes "A \<subseteq> B" shows "Under r B \<subseteq> Under r A"
leD: fixes y :: "'a"   and x :: "'a" assumes "le y x" shows "\<not> lt x y"
lub_const: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b" assumes "class.po below" shows "po.lub below (range (\<lambda>i. c)) = c"
in_ivp_sols_subset: fixes t\<^sub>0 :: "real"   and U :: "'b \<Rightarrow> real set"   and s :: "'b"   and T :: "'b \<Rightarrow> real set"   and X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "t\<^sub>0 \<in> U s"   and "U s \<subseteq> T s"   and "X \<in> Sols f T S t\<^sub>0 s" shows "X \<in> Sols f U S t\<^sub>0 s"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = tensor_from_lookup [n] (\<lambda>x. if x = [i] then 1::'a else (0::'a))"
Lam_def: fixes i :: "nat" shows "Lam i = snd (n_as ! fst (j_to_jb_index n_as i))"
assms_2: shows "pair_complete A (\<sqsubseteq>)"
uminus_power_if: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) n = (if even n then power.power one times a n else uminus (power.power one times a n))"
infinity_ilessE: fixes m :: "nat"   and R :: "bool" assumes "\<infinity> < enat m" shows "R"
polyn_degree_unique: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d"   and "polyn_expr R X (fst c) c = polyn_expr R X (fst d) d" shows "c_max S c = c_max S d"
list_ex_simps_2: fixes P :: "'a \<Rightarrow> bool" shows "list_ex P [] = False"
pfixLe_trans: fixes x :: "nat list"   and y :: "nat list"   and z :: "nat list" assumes "x pfixLe y"   and "y pfixLe z" shows "x pfixLe z"
unstream_cong: fixes a :: "('b, 'c) Stream"   and b :: "('b, 'd) Stream" assumes "a \<approx> b" shows "unstream\<cdot>a = unstream\<cdot>b"
satC_iff_set: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "Ik.satC \<xi> c = (\<exists>l\<in>set c. Ik.satL \<xi> l)"
linear_sum': fixes S1 :: "'c set"   and S2 :: "'d set"   and scale1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and scale2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and S :: "'f set"   and g :: "'f \<Rightarrow> 'c" assumes "vector_space_pair_on S1 S2 scale1 scale2"   and "\<forall>x. x \<in> S1 \<longrightarrow> f x \<in> S2"   and "\<forall>x. x \<in> S \<longrightarrow> g x \<in> S1"   and "linear_on S1 S2 scale1 scale2 f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
card_QF: fixes k :: "nat"   and j :: "nat"   and i :: "nat" shows "card (QF k (j, i)) = (if j = 0 then \<Sqinter> (DF k i) else del k (if j = k then m - Suc i else i) j)"
times_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A * B \<equiv> mat (dim_row A) (dim_col B) (\<lambda>(i, j). row A i \<bullet> col B j)"
predf: shows "\<forall>la\<in>dom f. pred_cof L ((f(l \<mapsto> t)) la) (f' la) la"
bdd_above_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq {}"
Pred\<^sub>\<alpha>_eq_iff: fixes f1 :: "'d"   and \<phi>1 :: "'b"   and f2 :: "'d"   and \<phi>2 :: "'b" shows "(Pred\<^sub>\<alpha> f1 \<phi>1 = Pred\<^sub>\<alpha> f2 \<phi>2) = (f1 = f2 \<and> \<phi>1 = \<phi>2)"
WT_callee: fixes s :: "'s" assumes "I s" shows "\<I>l \<turnstile>c left s \<surd>"
induct: fixes x1 :: "'a"   and x2 :: "'b list"   and x3 :: "'b list"   and P :: "'b process"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and Pa :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> rel_induct_aux P I D"   and "\<And>u xs ys. \<lbrakk>(u, xs, ys) \<in> rel_induct_aux P I D; Pa u xs ys\<rbrakk> \<Longrightarrow> Pa u ys xs"   and "\<And>u xs ys zs. \<lbrakk>(u, xs, ys) \<in> rel_induct_aux P I D; Pa u xs ys; (u, ys, zs) \<in> rel_induct_aux P I D; Pa u ys zs\<rbrakk> \<Longrightarrow> Pa u xs zs"   and "\<And>u xs ys x. \<lbrakk>(u, xs, ys) \<in> rel_induct_aux P I D; Pa u xs ys; (D x, xs, ys) \<in> rel_induct_aux P I D; Pa (D x) xs ys; x \<in> next_events P xs \<inter> next_events P ys\<rbrakk> \<Longrightarrow> Pa u (xs @ [x]) (ys @ [x])"   and "\<And>u x xs. \<lbrakk>u \<in> range D; (D x, u) \<notin> I; x \<in> next_events P xs\<rbrakk> \<Longrightarrow> Pa u xs (xs @ [x])" shows "Pa x1 x2 x3"
y_le_IH: shows "y < dim_vec (tensors_from_net (insert_weights s (shallow_model' Z M N) (\<lambda>i. w (i + count_weights s (shallow_model' Z M 0)))))"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<times> 'b rose_tree list \<Rightarrow> 'a"   and rose_tree :: "'b rose_tree" shows "P (case rose_tree of Tree x \<Rightarrow> f x) = (\<forall>x. rose_tree = Tree x \<longrightarrow> P (f x))"
fract_poly_eq_0_iff: fixes p :: "'a poly" shows "(map_poly to_fract p = 0) = (p = 0)"
L2_set_le_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "L2_set f A \<le> sum f A"
stop: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" assumes "g s = Done" shows "terminates_onp g s"
base_case_satisfy_equation: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "coprime p q" shows "satisfy_equation p [q] [[], [0]] [[1], [- 1]]"
trm_rep_involutive: fixes t :: "'a trm"   and S :: "'a eclause set" shows "trm_rep (trm_rep t S) S = trm_rep t S"
fst_gb_schema_dummy_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "dickson_grading d"   and "struct_spec sel ap ab compl" shows "dgrad_p_set_le d (fst ` set (fst (gb_schema_dummy data D bs ps))) (args_to_set (gs, bs, ps))"
mtf0: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "mtf x xs ! 0 = x"
exp_Lambert_W: fixes x :: "real" assumes "- exp (- 1) \<le> x"   and "x \<noteq> 0" shows "exp (Lambert_W x) = x / Lambert_W x"
k'': shows "k'' = (GREATEST k''. k'' \<le> n - k - k' \<and> (\<pi> \<guillemotleft> k \<guillemotleft> k') k'' = y)"
bcomp_ccomp: fixes b :: "bexp"   and f :: "bool"   and i :: "int"   and c :: "com"   and P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "bcomp (b, f, i) @ ccomp c @ P \<Turnstile> cfs\<box>"   and "0 \<le> i"   and "\<And>cfs. ccomp c \<Turnstile> cfs\<box> \<Longrightarrow> cpred c (cfs ! 0) (cfs ! (length cfs - 1))" shows "case cfs ! 0 of (pc, s, stk) \<Rightarrow> pc = 0 \<and> (bval b s \<noteq> f \<longrightarrow> (\<exists>k<length cfs. case cfs ! k of (pc', s', stk') \<Rightarrow> pc' = size (bcomp (b, f, i) @ ccomp c) \<and> (c, s) \<Rightarrow> s' \<and> stk' = stk))"
absorb_iff2: fixes b :: "'b word"   and a :: "'b word" shows "(b \<le>s a) = (signed.min a b = b)"
zero_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
set_diff_list: fixes xs :: "'e list"   and ys :: "'e list" shows "set (xs -- ys) = set xs - set ys"
no_subcls_Object: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D" shows "C \<noteq> Object"
A'_def: shows "A' \<equiv> \<lambda>view \<sigma>. D (((a1, a2), \<sigma>, view), \<sigma>, \<sigma> \<oplus> (a1 \<oplus> b1 \<and> a2 \<oplus> b2))"
gs_3: shows "is_idirprod (carrier G) (\<lambda>g. generate G {g}) (set gs)"
multi_member_last: fixes x :: "'b" shows "x \<in># \<LM> x  \<RM>"
False: shows "\<not> (p = 0 \<or> b \<le> a)"
normal_form: fixes x :: "'a itrm" obtains n :: "'a itrm" where "n \<simeq> x" and "n \<in> CF"
that: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "thesis"
lower_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lower (p + q) v = lower p v + lower q v"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "indRelRp a1 a2 = (\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2)"
assms_1: shows "abs_conv_abscissa f < ereal (s \<bullet> (1::'a))"
c''_eq: fixes i :: "nat" shows "fps_to_fls (c'' i) = c' i"
exhaust: fixes y :: "('a, 'b) message" obtains x11 :: "'a" and x12 :: "bool" where "y = <x11, inM x12>"   | x2 :: "bool" where "y = <\<bottom>, outM x2>"   | x31 :: "'a" and x32 :: "'b" where "y = <x31, x32>"
mat_transpose_mat_cnj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_transpose (mat_cnj A) = mat_adj A"
subclsRI: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Repeats D \<in> set Bs" shows "subclsRp P C D"
distinct_clearjunk: fixes al :: "('a \<times> 'b) list" shows "distinct (map fst (AList.clearjunk al))"
function_ring_car_closed: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "a \<in> S"   and "f \<in> carrier (function_ring S R)" shows "f a \<in> carrier R"
marg_dens2_def: shows "marg_dens2 \<equiv> \<lambda>(V, V', \<Gamma>, \<delta>) x y \<rho> v. \<integral>\<^sup>+ \<sigma>. \<delta> (merge V V' (\<sigma>(x := fst (extract_pair v), y := snd (extract_pair v)), \<rho>)) \<partial>state_measure (V - {x, y}) \<Gamma>"
normal_form_zero_iff: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]" shows "(normal_form F p = 0) = (p \<in> ideal F)"
card_partition_on_at_most_size: fixes A :: "'c set"   and k :: "nat" assumes "finite A" shows "card {P. partition_on A P \<and> card P \<le> k} = sum (Stirling (card A)) {..k}"
par_cp_def: fixes P :: "'b com option list"   and s :: "'b" shows "par_cp P s \<equiv> {l. l ! 0 = (P, s) \<and> l \<in> par_cptn}"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b tree\<^sub>0 \<Rightarrow> 'b tree\<^sub>0 \<Rightarrow> 'a"   and tree\<^sub>0 :: "'b tree\<^sub>0" shows "P (case tree\<^sub>0 of ET\<^sub>0 \<Rightarrow> f1 | MKT\<^sub>0 x xa xb \<Rightarrow> f2 x xa xb) = ((tree\<^sub>0 = ET\<^sub>0 \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23. tree\<^sub>0 = MKT\<^sub>0 x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
sliced_path_determ: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and as' :: "'edge list"   and S :: "'node set"   and s :: "'state"   and s' :: "'state" assumes "n -as\<rightarrow>* n'"   and "n -as'\<rightarrow>* n'"   and "slice_edges S as = slice_edges S as'"   and "preds (slice_kinds S as) s"   and "preds (slice_kinds S as') s'"   and "n' \<in> S"   and "\<forall>V\<in>rv S n. state_val s V = state_val s' V" shows "as = as'"
enttI: fixes A :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>A B * true" shows "A \<Longrightarrow>\<^sub>t B"
simps: fixes \<I> :: "('d, 'e) \<I>"   and callee :: "'d \<Rightarrow> ('e \<times> 'f) spmf" shows "\<I> \<turnstile>c callee \<surd> = (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (callee x) \<longrightarrow> xa \<in> responses_\<I> \<I> x)"
bindU_stateT_def: shows "bindU = (\<Lambda> m k. StateT\<cdot>(\<Lambda> s. runStateT\<cdot>m\<cdot>s \<bind> (\<Lambda> (x, s'). runStateT\<cdot>(k\<cdot>x)\<cdot>s')))"
ODE_zero: fixes i :: "'a"   and ODE :: "('b, 'a) ODE"   and I :: "('b, 'c, 'a) interp"   and \<nu> :: "(real, 'a) vec" assumes "Inl i \<notin> BVO ODE"   and "Inr i \<notin> BVO ODE" shows "ODE_sem I ODE \<nu> $ i = 0"
E18: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>\<box>F \<longrightarrow> \<diamond>\<box>F"
constructive_security_aux_obsf: shows "constructive_security_aux_obsf real2 ideal2 sim2 (\<I>_real2_core \<oplus>\<^sub>\<I> \<I>_real2_rest) (\<I>_ideal2_core \<oplus>\<^sub>\<I> \<I>_ideal2_rest) (\<I>_common2_core \<oplus>\<^sub>\<I> \<I>_common2_rest) adv2"
subst_fmla_graph_aux_def: shows "subst_fmla_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_fmla_aux t u i x. x1 = (t EQ u, i, x) \<and> x2 = subst i x t EQ subst i x u) \<or> (\<exists>subst_fmla_aux A B i x. x1 = (A OR B, i, x) \<and> x2 = ((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x) OR ((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (B, i, x) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x)) \<and> p (B, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (B, i, x))) \<or> (\<exists>subst_fmla_aux A i x. x1 = (neg A, i, x) \<and> x2 = neg (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x)) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x))) \<or> (\<exists>subst_fmla_aux j i x A. x1 = (exi j A, i, x) \<and> x2 = exi j (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x)) \<and> atom j \<sharp> (i, x) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fmla_aux) (A, i, x))))"
simps: fixes b :: "nat"   and a :: "C" shows "hbase_extp b a = ((\<exists>n. a = n \<and> hbasep b n) \<or> (\<exists>n m. a = C (n # m) \<and> hbase_extp b n \<and> hbasep b (C m) \<and> (\<forall>x. x \<in> set m \<longrightarrow> evalC b n \<le> evalC b x)))"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> h.cf.E" shows "h.cf (u, v) = 0"
expectation_gpv_const_le: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and fail :: "ennreal"   and c :: "ennreal" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv fail \<I> (\<lambda>_. c) gpv \<le> max c fail"
state_sim_refl: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set" shows "s ~A~ s"
MemFail_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "MemFail ch rs p \<equiv> $Calling ch p \<and> (rs!p)$ = #MemFailure \<and> unchanged (rtrner ch!p)"
dom_char: fixes f :: "'c \<times> 'c" shows "C.VV.dom f = (if C.VV.arr f then C.VxV.dom f else C.VxV.null)"
plus_mult_def: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b" shows "plus_mult id f zer g inver \<equiv> Matrix_Tensor.mult id f \<and> plus_mult_axioms f zer g inver"
x_val: shows "\<tau> \<Turnstile> \<upsilon> x"
mQmpTr3: fixes H :: "'a set"   and N :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N"   and "X \<in> carrier (M /\<^sub>m H)"   and "Y \<in> carrier (M /\<^sub>m H)" shows "(Mp\<^bsub>M  H,N\<^esub>) (mr_cospOp M H X Y) = mr_cospOp M N ((Mp\<^bsub>M  H,N\<^esub>) X) ((Mp\<^bsub>M  H,N\<^esub>) Y)"
solution_of_nat_of_nat': fixes z :: "nat \<times> nat" shows "solution (case z of (a, b) \<Rightarrow> (of_nat a, of_nat b)) = solution z"
comp_ipresIGFreshAll: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('o, 'p, 'q, 'r, 's, 't, 'k, 'm, 'u) model_scheme"   and MOD' :: "('v, 'w, 'q, 'r, 'x, 't, 'l, 'n, 'y) model_scheme"   and h' :: "'l \<Rightarrow> 'z"   and hA' :: "'n \<Rightarrow> 'aa"   and MOD'' :: "('ab, 'ac, 'q, 'ad, 'ae, 't, 'z, 'aa, 'af) model_scheme" assumes "ipresIGWlsAll h hA MOD MOD'"   and "ipresIGFreshAll h hA MOD MOD'"   and "ipresIGFreshAll h' hA' MOD' MOD''" shows "ipresIGFreshAll (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
list_max_Cons: fixes x :: "'t"   and xs :: "'t list" shows "list_max (x # xs) = ord_term_lin.max x (list_max xs)"
mono_demonic: fixes Q :: "'b \<Rightarrow> 'a" shows "mono [: Q :]"
d_mvt: fixes t :: "real"   and y :: "'a" assumes "0 < t"   and "t \<le> d"   and "dist y x < d" shows "s (flow0 y t) < s y"
\<theta>_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
insertCI: fixes a :: "'c"   and B :: "'c set"   and b :: "'c" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> b \<triangleright> B"
simps: fixes M :: "('d, 'e, 'f) FSM" shows "finite_FSM M = (finite (nodes M) \<and> finite (inputs M) \<and> finite (outputs M))"
star: fixes \<alpha> :: "fm"   and p :: "perm"   and V :: "name set"   and F :: "name \<Rightarrow> tm" assumes "ss_fm \<alpha>"   and "atom ` (p \<bullet> V) \<sharp>* V"   and "- p = p"   and "finite V"   and "supp \<alpha> \<subseteq> atom ` V"   and "F = make_F V p" shows "insert \<alpha> (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F)"
T_bij_def: shows "T_bij = {f. \<forall>y. \<exists>!x. f x = y}"
rel_simps_93: shows "(1::'a) \<noteq> (0::'a)"
h_def: shows "h = (\<lambda>p. signof p * g p)"
P_lnth_Suc: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and n :: "nat" assumes "vm_path G P v0" shows "P $ Suc n = ltl P $ n"
fsmult_sum_distrib: fixes r :: "'f"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'v" assumes "r \<in> UNIV"   and "finite A"   and "f ` A \<subseteq> V" shows "r \<sharp>\<cdot> sum f A = (\<Sum>a\<in>A. r \<sharp>\<cdot> f a)"
conv: fixes x :: "'b"   and y :: "'b"   and \<mu> :: "real" assumes "x \<in> C"   and "y \<in> C"   and "0 \<le> \<mu>"   and "\<mu> \<le> 1" shows "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
sublist_cons_exists: fixes h :: "'c"   and l1 :: "'c list"   and l2 :: "'c list" shows "subseq (h # l1) l2 = (\<exists>l2aa l2b. l2 = l2aa @ [h] @ l2b \<and> \<not> ListMem h l2aa \<and> subseq l1 l2b)"
mk_mat_def: fixes f :: "'g \<Rightarrow> 'f \<Rightarrow> 'e" shows "local.mk_mat f = (\<chi>i j. f i j)"
m_closed: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
fundamental_theorem_eq_6: fixes i :: "nat" assumes "3 \<le> i"   and "i \<le> k" shows "ffp (R (n i)) = Polynomial.smult (\<Theta> i) (F i)"
elims: fixes x :: "'a dres"   and y :: "'a" assumes "the_res x = y" obtains xa :: "'a" where "x = dRETURN xa" and "y = xa"   | "x = dSUCCEEDi" and "y = undefined"   | "x = dFAILi" and "y = undefined"
prod_def: fixes f :: "'a"   and g :: "'a" shows "f \<otimes> g \<equiv> \<langle>f \<cdot> \<pp>\<^sub>1[local.dom f, local.dom g], g \<cdot> \<pp>\<^sub>0[local.dom f, local.dom g]\<rangle>"
assms_2: shows "A \<equiv> (P' \<union> Pw' \<union> R' \<union> Rw') \<inter> Ms"
var_rename_inv_def: fixes n :: "nat"   and v :: "'b \<times> nat" shows "var_rename_inv n v \<equiv> Var (fst v, snd v - Suc n)"
wlconf_empty: fixes G :: "prog"   and s :: "st"   and vs :: "'b \<Rightarrow> val option" shows "G,s\<turnstile>vs[\<sim>\<Colon>\<preceq>]Map.empty"
deltaG_metric_completion_def: fixes uu :: "'b metric_completion itself" shows "deltaG uu = deltaG TYPE('b)"
differentiable_imp_piecewise_differentiable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> f differentiable at x within S" shows "f piecewise_differentiable_on S"
ctermsl_stermsl_or_dtermsl: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> ctermsl p" shows "q \<in> stermsl p \<or> (\<exists>p'\<in>dtermsl p. q \<in> ctermsl p')"
assms_3: shows "set_mset (Decreasing_Diagrams.measure r (\<tau>, \<sigma>)) \<subseteq> r \<down>s S"
iwlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "iwlsFSb MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSubstCls MOD \<and> igAbsRen MOD"
is_finite_weak_ranking_def: fixes xs :: "'c set list" shows "is_finite_weak_ranking xs = (is_weak_ranking xs \<and> (\<forall>x\<in>set xs. finite x))"