hcju/isabelleps huggingface dataset:
reads_restrict2: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'a"   and \<sigma>' :: "'var \<Rightarrow> 'a" assumes "\<forall>x\<in>local.reads n. \<sigma> x = \<sigma>' x" shows "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n"
homogeneous_set_homogeneous_ideal: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f" shows "homogeneous_set (ideal F)"
flow_to_path_arc: fixes t :: "real"   and t' :: "real"   and x :: "'a" assumes "t \<le> t'"   and "{t..t'} \<subseteq> existence_ivl0 x"   and "\<forall>s\<in>{t<..<t'}. flow0 x s \<noteq> flow0 x t"   and "flow0 x t \<noteq> flow0 x t'" shows "arc (flow_to_path x t t')"
flow_vderiv_on_right: fixes x0 :: "'a" shows "(flow t0 x0 has_vderiv_on (\<lambda>x. f x (flow t0 x0 x))) (existence_ivl t0 x0 \<inter> {t0..})"
raw_subst_wellformed: fixes t :: "'a" shows "abs_pred (\<lambda>t. pre_strong_term_class.wellformed t \<longrightarrow> (\<forall>env. wellformed_env env \<longrightarrow> pre_strong_term_class.wellformed (subst t env))) t"
Mignotte_bound: fixes g :: "int poly"   and k :: "nat" shows "real_of_int \<bar>poly.coeff g k\<bar> \<le> real (degree g choose k) * mahler_measure g"
simps_3: fixes xs :: "nat list"   and uw :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((S, xs, uw) # fs, rv) = (fs, Some (Suc (hd xs)))"
fv_subst_int: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp" assumes "x \<notin> S"   and "y \<notin> S" shows "fv e[y::=x] \<inter> S = fv e \<inter> S"
nnf: fixes i :: "nat"   and x :: "'a" shows "0 \<le> f i x"
subst_subst_compose: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<tau> :: "'d \<Rightarrow> ('a, 'b) Term.term" shows "t \<cdot> \<sigma> \<circ>\<^sub>s \<tau> = t \<cdot> \<sigma> \<cdot> \<tau>"
simps_2: fixes \<sigma> :: "nat \<Rightarrow> val"   and v :: "val" shows "expr_sem \<sigma> (Val v) = return_val v"
coset_eq: shows "\<forall>x\<in>carrier G. H #> x = x <# H"
f_Exec_State_Idle_replicate_NoMsg_gr0_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)"   and "0 < n" shows "output_fun (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg"
gauss2complex_i: shows "gauss2complex \<i>\<^sub>\<int> = \<i>"
indicator_simps_2: fixes x :: "'a"   and S :: "'a set" assumes "x \<notin> S" shows "chi S x = (0::'b)"
distrib_right: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(a +\<^sub>o\<^sub>w b) *\<^sub>o\<^sub>w c = a *\<^sub>o\<^sub>w c +\<^sub>o\<^sub>w b *\<^sub>o\<^sub>w c"
assms: shows "pred_state (\<lambda>x. pred_state P (f x)) m"
swap_cols_rows_block_similar: fixes A :: "'b mat"   and n :: "nat"   and j :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "j < n"   and "i \<le> j" shows "similar_mat (swap_cols_rows_block i j A) A"
elims_3: fixes x :: "'edge list"   and xa :: "'edge list" assumes "\<not> same_level_path_aux x xa" obtains cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "\<not> (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> same_level_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> same_level_path_aux cs' as | _ \<Rightarrow> same_level_path_aux cs as)"
lex_p_def: shows "lex_p = measure (\<lambda>m. m) <*lex*> measure (\<lambda>n. n)"
finprod_cong': fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> int vec"   and f :: "'a \<Rightarrow> int vec" assumes "A = B"   and "g \<in> B \<rightarrow> carrier_vec n"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum V f A = finsum V g B"
floor_le_zero: fixes x :: "'a" shows "(\<lfloor>x\<rfloor> \<le> 0) = (x < (1::'a))"
int_less_real_le: fixes n :: "int"   and m :: "int" shows "(n < m) = (real_of_int n + 1 \<le> real_of_int m)"
dimensionI: fixes K :: "'a set"   and Us :: "'a list"   and E :: "'a set" assumes "subfield K R"   and "local.independent K Us"   and "Span K Us = E" shows "dimension (length Us) K E"
carr2_3: shows "gs1.gso ` {0..<i} \<subseteq> Rn"
HInfiniteD: fixes x :: "'a star"   and r :: "real star" assumes "x \<in> HInfinite"   and "r \<in> \<real>" shows "r < hnorm x"
assms_1: shows "cmod (z - z0) \<noteq> r"
tl_stack_hd_tree_path: fixes v :: "'a" assumes "stack s \<noteq> []"   and "v \<in> set (tl (stack s))" shows "(v, hd (stack s)) \<in> (tree_edges s)\<^sup>+"
memberid_pair_simp1: fixes p :: "'a \<times> 'b" shows "memberid p = memberid (snd p)"
pderiv_const: fixes b :: "'a" assumes "b \<in> carrier R" shows "pderiv (monom P b 0) = \<zero>\<^bsub>P\<^esub>"
ennreal_fact: fixes n :: "nat" shows "ennreal (fact n) = fact n"
N: shows "nfa_is_node (quantify_nfa v N) q"
exec_stop1: fixes \<sigma> :: "'b control_state_scheme" assumes "break_status \<sigma>" shows "exec_stop \<sigma>"
freshEnv_getEnv2: fixes ys :: "'d"   and y :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm set option"   and xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm set" assumes "freshEnv ys y rho"   and "rho xs x = Some X" shows "fresh ys y X"
L: shows "subtopology (Euclidean_space n) S homeomorphic_space Euclidean_space n"
lb3_ub: shows "real_of_float (lapprox_rat prec 1 3) < 1"
measurable_dens_ctxt_measure_component: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and x :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "density_context V V' \<Gamma> \<delta>"   and "x \<in> V" shows "(\<lambda>\<sigma>. \<sigma> x) \<in> dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<rightarrow>\<^sub>M stock_measure (\<Gamma> x)"
apply1: fixes i :: "'a" shows "blinfun_apply invf (blinfun_apply (f' x0) i) = i"
sifum_compositionality: fixes cmds :: "('Var, 'AExp, 'BExp) Stmt list" assumes "list_all com_sifum_secure cmds"   and "no_assumptions_on_termination cmds"   and "\<forall>mem. sound_mode_use (add_initial_modes cmds, mem)" shows "prog_sifum_secure cmds"
minarc_arc: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<noteq> mbot \<longrightarrow> matrix_stone_relation_algebra.arc (minarc\<^sub>M x)"
thyps_2: shows "\<forall>\<tau>\<in>{0..t}. \<tau> \<le> - (ln (Tmin / T) / a)"
choice_iff': fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) = (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
While: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c P,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P While b c (P \<inter> - b),A"
is_none_bind: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option" shows "Option.is_none (Option.bind f g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
simps_2: fixes s :: "'b word"   and e :: "'b word" shows "l2wi [(s, e)] = WordInterval s e"
simps: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and zs' :: "'c"   and z :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "qAbs zs' z X $[[x \<and> y]]_zs = qAbs zs' (z @zs'[x \<and> y]_zs) (X #[[x \<and> y]]_zs)"
embedS_iff: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "Well_order r"   and "embed r r' f" shows "embedS r r' f = (f ` Field r \<subset> Field r')"
sup_commute: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "sup_prod x y = sup_prod y x"
prod_induct3: fixes P :: "'e \<times> 'f \<times> 'g \<Rightarrow> bool"   and x :: "'e \<times> 'f \<times> 'g" assumes "\<And>a b c. P (a, b, c)" shows "P x"
simps_1: fixes x2 :: "'a"   and y2 :: "'a" shows "(ForwardOrEdit x2 = ForwardOrEdit y2) = (x2 = y2)"
mult_rel_carrier: fixes X :: "('c set \<times> 'd) set"   and Y :: "('c set \<times> 'd) set" assumes "X \<in> stfx.carrier_direct_lim"   and "Y \<in> stfx.carrier_direct_lim" shows "stfx.mult_rel X Y \<in> stfx.carrier_direct_lim"
sum_np_eq: fixes C :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "C \<subseteq> PR" shows "sum f {..<np} = sum f C + sum f ({..<np} - C)"
vD_iD_gives_kD_2: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> iD rt" shows "ip \<in> kD rt"
UN_I: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "s2p A a"   and "s2p (B a) b" shows "s2p (\<mu> (\<P> B A)) b"
lexordp_cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less"   and "ord.lexordp less xs ys" obtains y :: "'a" and ys' :: "'a list" where "xs = []" and "ys = y__ # ys'__"   | x :: "'a" and xs' :: "'a list" and y :: "'a" and ys' :: "'a list" where "xs = x__ # xs'__" and "ys = y # ys'a__" and "less x__ y"   | x :: "'a" and xs' :: "'a list" and ys' :: "'a list" where "xs = x # xs'" and "ys = x # ys'" and "ord.lexordp less xs' ys'"
Melem_subst_cls_mset: fixes AA :: "'a literal multiset"   and CC :: "'a literal multiset multiset"   and \<sigma> :: "'s" shows "(AA \<in># CC \<cdot>cm \<sigma>) = (\<exists>BB. BB \<in># CC \<and> AA = BB \<cdot> \<sigma>)"
id: shows "fs[i := fs ! i - 0 \<cdot>\<^sub>v fs ! j] = fs"
zero_le: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
slice1_no_bin: fixes n :: "nat"   and w :: "num" shows "slice1 n (numeral w) = of_bl (takefill False n (bin_to_bl LENGTH('c) (numeral w)))"
zero_leastII: fixes x :: "'a" assumes "(0::'a) \<succ> x" shows "x = (0::'a)"
atrm_imp_trm: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t \<in> Syntax_Arith.atrm var Var num suc pls tms" shows "t \<in> trm"
residuated_pair_multr: fixes x :: "'a" shows "residuated_pair ((\<cdot>) x) ((\<rightarrow>) x)"
simps: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and a :: "('a, 'b, 'c, 'd) converter" shows "plossless_converter \<I> \<I>' a = (\<exists>conv. a = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> plossless_gpv \<I>' (run_converter conv x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> plossless_converter \<I> \<I>' xb))"
that: fixes m :: "int"   and e :: "int" assumes "f = Float m e" shows "thesis"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>J f = n \<and> f \<cdot>\<^sub>J n = n"
sorted_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sort_key less_eq (\<lambda>x. x) xs)"
elims_3: fixes x :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and xa :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "\<not> bisim_helper x xa" obtains c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" where "x = \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>" and "xa = \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" and "\<not> mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
finfun_const_eq_update: fixes b :: "'b"   and f :: "'a \<Rightarrow>f 'b"   and a :: "'a"   and b' :: "'b" shows "((K$ b) = f(a $:= b')) = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> f $ a' = b))"
cnj_of_real: fixes x :: "real" shows "Quaternions.cnj (quat_of_real x) = quat_of_real x"
assms: fixes i :: "'a"   and j :: "'a" assumes "i \<in> I \<and> j \<in> I"   and "i \<noteq> j" shows "A i \<inter> A j = {}"
eff_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and s :: "'b"   and i :: "nat" assumes "PL cval" shows "PL.eff aval tval cval c s i = snd (snd (PL.wt_cont_eff aval tval cval c s i))"
assms: shows "x \<in>\<^sub>\<circ> CZH_Sets_BRelations.vimage (R \<times>\<^sub>\<circ> X)"
reaches1_reaches_iff2: fixes x :: "'a"   and y :: "'a" shows "A.reaches1 x y = (\<exists>z. A.reaches x z \<and> A z y)"
ideD_1: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.VV.arr a"
valid_distinguishing_formula: fixes P' :: "'state" shows "P' \<Turnstile> distinguishing_formula P Q = (\<forall>x\<in>{- p \<bullet> (SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}. P' \<Turnstile> x)"
basis_in_max_indep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> X"   and "x \<in> \<E> - X" shows "\<not> \<E>.indep_in \<E> (insert x X)"
flist_props_1: fixes xs :: "'a fset" shows "sorted (flist xs)"
low_proc_no_input_change_s\<^sub>L\<^sub>0: shows "low_proc_no_input_change s\<^sub>L\<^sub>0"
strategyproofness_1: shows "(pmf (sds R43) b + pmf (sds R43) d < pmf (sds R5) c + pmf (sds R5) a \<or> pmf (sds R43) b + (pmf (sds R43) d + pmf (sds R43) a) < pmf (sds R5) c + (pmf (sds R5) a + pmf (sds R5) d)) \<or> pmf (sds R43) b + pmf (sds R43) d = pmf (sds R5) c + pmf (sds R5) a \<and> pmf (sds R43) b + (pmf (sds R43) d + pmf (sds R43) a) = pmf (sds R5) c + (pmf (sds R5) a + pmf (sds R5) d)"
simps_3: fixes \<sigma> :: "ty" shows "RED (T \<sigma>) = _Collect t (\<forall>k\<in>SRED \<sigma>. SN (t \<star> k))"
bin_split_trunc1: fixes n :: "nat"   and c :: "int"   and a :: "int"   and b :: "int"   and m :: "nat" assumes "bin_split n c = (a, b)" shows "bin_split n (take_bit m c) = (take_bit (m - n) a, take_bit m b)"
h2: shows "h \<turnstile> set_disconnected_nodes owner_document (child # prev_disc_nodes) \<rightarrow>\<^sub>h h2"
OneState_HPT_SA: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and T :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> Source TS"   and "T \<in> Source TS"   and "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs (HA ST)" shows "S = T"
a: shows "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"
steps: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"
OCom_iff_indRelRTPO_is_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_complete (TRel\<^sup>*) \<and> weak_reduction_simulation (TRel\<^sup>+) Target) = weak_reduction_simulation (indRelRTPO TRel) (STCal Source Target)"
illegal_instruction_ASR_def: fixes r :: "word5" shows "illegal_instruction_ASR r \<equiv> False"
DERIV_cmult: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c"   and s :: "'c set"   and c :: "'c" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
lemma_partial_derivatives0: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a" assumes "linear f"   and "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> (0::'b)) (at (0::'a) within S)"   and "\<And>v. v \<noteq> (0::'a) \<Longrightarrow> \<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0::'a} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>" shows "f x = (0::'b)"
cardOneTheElem: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "card A = 1" shows "the_elem (f ` A) = f (the_elem A)"
simps: fixes a :: "('sequent, 'rule, 'link) ctree" shows "cwf a = ((\<exists>l. a = Link l \<and> cwf (pointsTo l)) \<or> (\<exists>r s cts. a = cNode (s, r) cts \<and> r \<in> R \<and> eff r s (seqOf |`| cts) \<and> (\<forall>x. x |\<in>| cts \<longrightarrow> cwf x)))"
dp_consistency_heap_empty_axioms: shows "dp_consistency_heap_empty P update lookup empty"
sqn_if_updated: fixes rip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat" shows "sqn (\<lambda>x. if x = rip then Some v else rt x) ip = (if ip = rip then \<pi>\<^sub>2 v else sqn rt ip)"
assms_1: shows "(a, b) \<in> A\<^sup>*"
simps_1: fixes e :: "'c"   and a :: "'d"   and ts :: "('c, 'd) SkewBinomialTree list" shows "tree_invar (Node e a 0 ts) = (ts = [])"
length_remdups_adj: fixes f :: "real poly \<Rightarrow> real"   and s :: "real" assumes "\<And>q. q \<in> set ps \<Longrightarrow> f q = s * f (q div d)"   and "s \<noteq> 0" shows "length (remdups_adj (filter (\<lambda>x. x \<noteq> 0) (map f ps'))) - 1 = length (remdups_adj (filter (\<lambda>x. x \<noteq> 0) (map f ps))) - 1"
lrev_lrev_ww_eq: shows "lrev = lrev_wrap"
IH: assumes "\<not> (\<exists>i\<in>I. \<exists>j\<in>I. f i = - \<infinity> \<and> f j = \<infinity>)" shows "eexp (sum f I) = (\<Prod>i\<in>I. eexp (f i))"
assms_1: shows "det2 a11 a12 a21 a22 = (0::'a)"
feasible_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "feasible t = (\<forall>P b. bounded_by b P \<and> nneg P \<longrightarrow> bounded_by b (t P) \<and> nneg (t P))"
fls_shift_eq_iff: fixes m :: "int"   and f :: "'b fls"   and g :: "'b fls" shows "(fls_shift m f = fls_shift m g) = (f = g)"
less_eqI: fixes xs :: "nat list"   and ys :: "nat list" assumes "length xs = length ys"   and "\<forall>i<length xs. xs ! i \<le> ys ! i" shows "xs \<le>\<^sub>v ys"
isom_gch_unitsTr1: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat" assumes "Ugp E" shows "\<forall>g h f. Gchain n g \<and> Gchain n h \<and> Gch_bridge n g h f \<longrightarrow> card {i. i \<le> n \<and> g i \<cong> E} = card {i. i \<le> n \<and> h i \<cong> E}"
cases: fixes x :: "ground_action" obtains pre :: "object atom formula" and eff :: "object ast_effect" where "x = Ground_Action pre eff"
prob0_def: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "prob0 n v i \<equiv> \<Sum>j\<in>{k |k. k < 2 ^ n \<and> \<not> select_index n i k}. (cmod (v $$ (j, 0)))\<^sup>2"
image_orthogonal_transformation_ball: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and r :: "real" assumes "orthogonal_transformation f" shows "f ` ball x r = ball (f x) r"
asIMOD_igAbs2: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and s :: "'sort"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and xs :: "'varSort"   and x :: "'var" assumes "sWlsDisj SEM"   and "igWls (asIMOD SEM) s X" shows "igAbs (asIMOD SEM) xs x X = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM s) else undefined)"
Term_quot_tm: fixes e :: "(name, hf) finfun"   and t :: "tm" shows "Term \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e"
max_enat_simps_4: fixes q :: "enat" shows "max q \<infinity> = \<infinity>"
galois_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> y + - z) = (x \<cdot> z \<le> y)"
min_y_yz: shows "minimum j = minimum (j \<squnion> k)"
assms: shows "term_variants_pred P (Fun f T) (Fun h S)"
lsimulation_axioms_def: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme" shows "lsimulation_axioms R A B \<equiv> \<forall>a b. (a, b) \<in> R \<longrightarrow> sa_L A a = sa_L B b"
assms_1: shows "\<A> \<Turnstile> \<Phi>"
bij_betw_permutations: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x) {\<pi>. \<pi> permutes A} {\<pi>. \<pi> permutes B}"
Worder_refl: fixes S :: "('c, 'd) Order_scheme" assumes "Worder S" shows "ord_equiv S S"
eta_subst: fixes s :: "dB"   and t :: "dB"   and u :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "s[u/i] \<rightarrow>\<^sub>\<eta> t[u/i]"
double_equivp: shows "equivp (\<lambda>x y. IEEE.is_nan x \<and> IEEE.is_nan y \<or> x = y)"
inject_3: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f"   and y31 :: "nat"   and y32 :: "'d"   and y33 :: "'d"   and y34 :: "'e"   and y35 :: "'e"   and y36 :: "'f" shows "(Recv x31 x32 x33 x34 x35 x36 = Recv y31 y32 y33 y34 y35 y36) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34 \<and> x35 = y35 \<and> x36 = y36)"
wit_lsingleton: fixes x :: "'a" shows "local.wit (LCons x LNil) = LCons x LNil"
eval_monom_def: shows "eval_monom = (\<lambda>(c, es) basis x. c * prod_list (map2 (\<lambda>b. (powr) (b x)) basis es))"
vcons_eqI: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence_pair xs\<^sub>1 xs\<^sub>2"   and "xs\<^sub>1 = xs\<^sub>2"   and "x\<^sub>1 = x\<^sub>2" shows "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2"
mutual_information_distr: fixes S :: "'b measure"   and T :: "'f measure"   and X :: "'a \<Rightarrow> 'b"   and Px :: "'b \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and Pxy :: "'b \<times> 'f \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>y. y \<in> space T \<Longrightarrow> 0 \<le> Py y"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)" shows "mutual_information b S T X Y = LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
simps: fixes x :: "nat"   and y :: "nat"   and u :: "nat"   and v :: "nat" shows "times_int_raw (x, y) (u, v) = (x * u + y * v, x * v + y * u)"
Afix_comp2join_fresh: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "atom ` domA \<Delta> \<sharp>* \<Gamma>" shows "ABinds \<Delta>\<cdot>(Afix \<Gamma>\<cdot>ae) = ABinds \<Delta>\<cdot>ae"
that_1: assumes "xs = []"   and "y = z" shows "thesis"