hcju/isabelleps huggingface dataset:
apply_chart_restrict_chart: fixes S :: "'c set"   and c :: "('c, 'd) chart" shows "apply_chart (restrict_chart S c) = apply_chart c"
Un_Diff_cancel2: fixes B :: "'c set"   and A :: "'c set" shows "B - A \<union> A = B \<union> A"
exehas_sort_imp_has_sort: fixes sub :: "(String.literal \<times> String.literal) list"   and tcs :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and T :: "typ"   and S :: "String.literal set" assumes "exe_osig_conds (sub, tcs)"   and "exehas_sort (sub, tcs) T S" shows "has_sort (translate_osig (sub, tcs)) T S"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "rpo ground_heads_var gt_sym extf arity_sym arity_var" shows "rpo.gt_same ground_heads_var gt_sym extf a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> rpo.chksubs (rpo.gt ground_heads_var gt_sym extf) t s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (rpo.gt ground_heads_var gt_sym extf) (args t) (args s)))"
monad_alt_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_alt return bind alt" shows "monad_alt (return_writer return) (bind_writer return bind) (alt_writer alt)"
lemMoveByMove: fixes p :: "'a Point"   and u :: "'a Vector"   and v :: "'a Vector" shows "p \<leadsto> u \<leadsto> v = p \<leadsto> (u \<oplus> v)"
deg_r_nonzero: shows "deg R r \<noteq> 0"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) K (N \<prec>' P) [] (\<one>)"
measurable_on_imp_borel_measurable_lebesgue_UNIV: fixes f :: "'a \<Rightarrow> 'b" assumes "f measurable_on UNIV" shows "f \<in> borel_measurable lebesgue"
coprime_crossproduct_int: fixes a :: "int"   and d :: "int"   and b :: "int"   and c :: "int" assumes "coprime a d"   and "coprime b c" shows "(\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar>) = (\<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>)"
supset_bottomI: fixes z :: "'b set"   and P :: "'b set set" assumes "z \<in> P"   and "\<And>y. y \<in> P \<Longrightarrow> order.greater_eq z y" shows "z = dual_order.bottom P"
map_scale_zero_left: fixes t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(0::'b) \<cdot> t = 0"
comp_CLS: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.is_left_adjoint f"   and "B.is_left_adjoint g"   and "f \<star> g \<cong>\<^sub>B h" shows "\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>h\<rbrakk>\<rbrakk>"
supt_ctxt_conv: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "s \<rhd> t = (\<exists>C. C \<noteq> \<box> \<and> s = C\<langle>t\<rangle>)"
pr_in_hom_1: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "\<guillemotleft>\<p>\<^sub>0[a0, a1] : a0 \<otimes> a1 \<rightarrow> a0\<guillemotright>"
assms: shows "P -` {True} \<inter> set l \<noteq> {}"
X: fixes i :: "'l"   and j :: "'k" assumes "(i, j) \<in> B" shows "G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
BoundR_def: fixes x3 :: "subject"   and x1 :: "name"   and x2 :: "pi" shows "x3\<guillemotleft>x1\<guillemotright> \<prec> x2 \<equiv> Late_Semantics.residual.Abs_residual (BoundR_Rep x3 ([x1].x2))"
card_UNIV_integer_def: shows "card_UNIV_class.card_UNIV = Phantom(integer) 0"
scale_one: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "module scale" shows "scale (1::'b) x = x"
rep_eq: fixes x :: "'c upred"   and xa :: "'c upred" shows "\<lbrakk>x \<Rightarrow> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<longrightarrow> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
emax_def: fixes x :: "('a, 'b) IEEE.float itself" shows "emax x = unat (- 1)"
consts_for_unis: fixes cs :: "'a list"   and p :: "('a, 'b) form" assumes "[] \<turnstile> put_unis (length cs) p" shows "[] \<turnstile> consts_for_unis (put_unis (length cs) p) cs"
ev: shows "\<forall>\<^sub>F x in F. f x \<le> g x"
right_neutral: fixes a :: "'b" shows "plus a (zero_class.zero::'b) = a"
indets_def: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets p = \<Union> (keys ` keys p)"
While_transT_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tst :: "'a"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b" assumes "PL.transT tval aval (while tst do c1, s) s'" obtains "\<not> tval tst s" and "s' = s"
borel_adapt_nat_filtration_info: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and w :: "bool stream" assumes "borel_adapt_stoch_proc F X"   and "m \<le> n" shows "X m (pseudo_proj_True n w) = X m w"
lunit_coherence1: fixes a :: "'c" assumes "C.ide a" shows "\<l>\<^sub>1[F a] \<cdot>\<^sub>D (\<psi> \<otimes>\<^sub>D F a) = \<l>\<^sub>D[F a]"
SdlHPPR: shows "SdlHPPB d PP R"
Res: fixes P :: "pi"   and Q :: "pi"   and a :: "name" assumes "(P, Q) \<in> Rel'" shows "(<\<nu>a>P, <\<nu>a>Q) \<in> Rel'"
trace_minus_linear: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "trace (A - B) = trace A - trace B"
RedDynCastFail: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and 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 Cs :: "char list list"   and E :: "char list \<Rightarrow> ty option" assumes "hp s a = \<lfloor>(D, S)\<rfloor>"   and "\<not> P \<turnstile> Path D to C unique"   and "\<not> P \<turnstile> Path last Cs to C unique"   and "C \<notin> set Cs" shows "P,E \<turnstile> \<langle>Cast C (ref (a, Cs)),s\<rangle> \<rightarrow> \<langle>null,s\<rangle>"
iptables_bigstep_defined_if_singleton_rules: fixes rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state" assumes "\<forall>r\<in>set rs. \<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> t" shows "\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
simps_1: fixes P :: "instr list"   and cf :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and cf' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" shows "P \<Turnstile> cf # cf' # cfs = (P \<turnstile> cf \<rightarrow> cf' \<and> P \<Turnstile> cf' # cfs)"
disjoint_projection: fixes X :: "'a set"   and Y :: "'a set"   and l :: "'a list" assumes "X \<inter> Y = {}" shows "l \<upharpoonleft> X \<upharpoonleft> Y = []"
path2_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and n :: "'b"   and ns :: "'b list"   and m :: "'b" shows "graph_path_base.path2 \<alpha>n invar inEdges' g n ns m \<equiv> graph_path_base.path \<alpha>n invar inEdges' g ns \<and> n = hd ns \<and> m = last ns"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>xs. P 0 xs"   and "\<And>n x xs. \<lbrakk>P (Suc n div 2) xs; \<And>xa xb y. \<lbrakk>xa = heapify (Suc n div 2) xs; (xb, y) = xa\<rbrakk> \<Longrightarrow> P (n div 2) y\<rbrakk> \<Longrightarrow> P (Suc n) (x # xs)"   and "\<And>v. P (Suc v) []" shows "P a0 a1"
simps: fixes ant :: "('a, 'b) form multiset"   and suc :: "('a, 'b) form multiset" shows "seq_size ( ant \<Rightarrow>* suc) = size ant + size suc"
cases: fixes a :: "(nat \<times> nat) convnet" assumes "balanced_net a" obtains M :: "nat" where "a = Input M"   | m :: "(nat \<times> nat) convnet" and A :: "nat \<times> nat" where "a = Conv A m" and "balanced_net m"   | m1 :: "(nat \<times> nat) convnet" and m2 :: "(nat \<times> nat) convnet" where "a = Pool m1 m2" and "balanced_net m1" and "balanced_net m2" and "count_weights True m1 = count_weights True m2"
g_scale: fixes c :: "complex"   and x :: "'a" shows "g (c *\<^sub>C x) = c *\<^sub>C g x"
subst_pairs_compose: fixes \<delta> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<I> :: "'d \<Rightarrow> ('a, 'b) Term.term"   and D :: "(('a, 'c) Term.term \<times> ('a, 'c) Term.term) set" shows "D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<delta> \<circ>\<^sub>s \<I> = D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<delta> \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>"
G_pos: fixes x :: "real" assumes "0 < x" shows "0 < G x"
prems_4: shows "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"
jordan_block_constant_bound2: fixes a :: "'b"   and n :: "nat" shows "\<exists>p. norm a < 1 \<longrightarrow> (\<forall>i j k. i < n \<longrightarrow> j < n \<longrightarrow> norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)) \<le> p)"
dterms_psimps_3: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "dterms \<Gamma> (p1 \<oplus> p2) = dterms \<Gamma> p1 \<union> dterms \<Gamma> p2"
index_mat_addrow_mat_3: fixes n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" shows "dim_col (addrow_mat n a k l) = n"
flowderiv_def: fixes x0 :: "'d"   and t :: "real" shows "flowderiv x0 t = comp12 (Dflow x0 t) (blinfun_scaleR_left (ode (flow0 x0 t)))"
nLt_le_conv: fixes x :: "real \<times> real"   and y :: "real \<times> real" shows "(compare x y \<noteq> Lt) = le y x"
nn_integral_measure_spmf': fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf p) f = (\<Sum>\<^sup>+ x\<in>set_spmf p. ennreal (spmf p x) * f x)"
antisym_conv1: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
total_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i. \<exists>j. f (i, j) = top" shows "matrix_stone_relation_algebra.total_var f"
assms: shows "i1 \<notin> set (added_ids [Deliver (i, Expunge e mo i2)] e)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) \<chi>a.A.map (\<lambda>j. D (j, a)) (\<chi> a)"
simps_1: fixes S :: "'b"   and S' :: "'b" shows "(SKIP {S} \<le> SKIP {S'}) = (S \<le> S')"
butlast_take: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
cod_tuple: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "cod (ECC.tuple f g) = ECC.prod (cod f) (cod g)"
assms: shows "\<forall>i. \<exists>j\<ge>i. j \<in> A"
strongest_rhs_is_strongest: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A co B" shows "strongest_rhs F A \<subseteq> B"
gval_fold_gOr_map: fixes l :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gOr l (Bc False)) s = fold (\<or>?) (map (\<lambda>g. gval g s) l) false"
Diff_UNIV: fixes A :: "'a set" shows "A - UNIV = {}"
S_transform_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes \<alpha> :: "'b"   and t\<^sub>\<alpha> :: "('a, 'c, 'b) Tree\<^sub>\<alpha>" shows "S_transform_Tree\<^sub>\<alpha> (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>) = Act\<^sub>\<alpha> (S_action.Act \<alpha>) (S_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
empty_fine_imp_eq: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list" assumes "fine \<delta> (a, b) D"   and "D = []" shows "a = b"
case_3: fixes f1 :: "int \<Rightarrow> int list \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and x31 :: "int"   and x32 :: "int"   and x33 :: "int list" shows "(case NDvd x31 x32 x33 of Le x xa \<Rightarrow> f1 x xa | Dvd x xa xb \<Rightarrow> f2 x xa xb | NDvd x xa xb \<Rightarrow> f3 x xa xb) = f3 x31 x32 x33"
simps: fixes M :: "('e, 'f, 'g) FSM" shows "finite_FSM M = (finite (nodes M) \<and> finite (inputs M) \<and> finite (outputs M))"
Int_empty_left: fixes B :: "'b set" shows "{} \<inter> B = {}"
B: shows "B \<in> sets (stream_space (count_space UNIV))"
prems_1: shows "(tab(x # xs [\<mapsto>] ys)) vn = Some z"
Mul_Propagate_Black_def: fixes n :: "nat" shows "Mul_Propagate_Black n \<equiv> \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>Safe \<or> \<acute>l \<le> \<acute>Queue \<or> \<acute>obc \<subset> Blacks \<acute>M)\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> Blacks \<acute>M \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>Safe \<or> \<acute>l \<le> \<acute>Queue \<or> \<acute>obc \<subset> Blacks \<acute>M) \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>E INV \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>Mul_PBInv \<and> \<acute>ind \<le> length \<acute>E\<rbrace>  DO \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>Mul_PBInv \<and> \<acute>ind < length \<acute>E\<rbrace>  IF \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black THEN \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>Mul_PBInv \<and> \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black \<and> \<acute>ind < length \<acute>E\<rbrace> \<acute>k := snd (\<acute>E ! \<acute>ind);; \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>Safe \<or> \<acute>obc \<subset> Blacks \<acute>M \<or> \<acute>l < \<acute>Queue \<or> (\<forall>i<\<acute>ind. \<not> BtoW (\<acute>E ! i, \<acute>M)) \<and> \<acute>l \<le> \<acute>Queue \<and> \<acute>Mul_Auxk) \<and> \<acute>k < length \<acute>M \<and> \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black \<and> \<acute>ind < length \<acute>E\<rbrace> \<langle>\<acute>M := \<acute>M[\<acute>k := Black],, \<acute>ind := \<acute>ind + 1\<rangle> ELSE \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>Mul_PBInv \<and> \<acute>ind < length \<acute>E\<rbrace> \<langle>IF \<acute>M ! fst (\<acute>E ! \<acute>ind) \<noteq> Black THEN \<acute>ind := \<acute>ind + 1 FI\<rangle> FI OD"
new_element_empty_tag_name: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_tag_name new_element_ptr \<rightarrow>\<^sub>r []"
graph_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "graph (app_iso hom G) = graph G"
inv_eq_1_iff: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "(\<ominus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>) = (x = \<zero>\<^bsub>F\<^esub>)"
chooseNext': fixes g :: "'g" assumes "uninst_code.ssa.redundant (u_g g s1_uses) (p_g g s1_phis) g" shows "uninst_code.chooseNext' (u_g g s1_uses) (p_g g s1_phis) g \<in> dom (Mapping.lookup (p_g g s1_phis g)) \<and> uninst_code.ssa.trivial (u_g g s1_uses) (p_g g s1_phis) g (uninst_code.chooseNext (u_g g s1_uses) (p_g g s1_phis) g)"
Limit_eq_Sup_self: fixes i :: "V" assumes "Limit i" shows "\<Union>\<^sub>\<circ> i = i"
exhaust: fixes y :: "event\<^sub>c" obtains "y = a\<^sub>c"   | "y = b\<^sub>c"   | "y = c\<^sub>c"
restricted_ideal_to_def: fixes B :: "nat set"   and A :: "'a mpoly set" shows "restricted_ideal_to B A = {p \<in> A. vars p \<subseteq> B}"
gs: fixes x :: "real" assumes "x \<in> {a..b}" shows "g x \<in> S"
vsubset_vunion_left: fixes A :: "V"   and B :: "V"   and C :: "V" shows "(A \<union>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> C) = (A \<subseteq>\<^sub>\<circ> C \<and> B \<subseteq>\<^sub>\<circ> C)"
dghm_is_tiny_dghm_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<beta>\<^esub> \<BB>"
\<epsilon>_leg_simps_4: shows "\<epsilon>.dom.leg1 = fg.leg1"
raw_comparator_nat_def: shows "raw_comparator_nat = compare default"
norm_jordan: fixes k :: "nat" shows "\<forall>(n, a)\<in>set n_as. \<forall>e\<in>elements_mat (jordan_block n a ^\<^sub>m k). norm e \<le> c + real k ^ (N - 1)"
S: shows "j < length (map the (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma))"
comp_converter_parallel2'_3: fixes conv1l :: "('i, 'k, 'm, 'o) converter"   and conv1r :: "('j, 'l, 'n, 'p) converter"   and conv2l :: "('m, 'o, 'q, 's) converter"   and conv2r :: "('n, 'p, 'r, 't) converter"   and res :: "('q + 'r, 's + 't) resource" shows "conv1l |\<^sub>= conv1r \<rhd> conv2l |\<^sub>= conv2r \<rhd> res = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r \<rhd> res"
replace_span: fixes q :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "q \<in> pmdl B" shows "pmdl (insert q (B - {p})) \<subseteq> pmdl B"
tendsto_powr': fixes f :: "'b \<Rightarrow> real"   and a :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "a \<noteq> 0 \<or> 0 < b \<and> (\<forall>\<^sub>F x in F. 0 \<le> f x)" shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
reduced_gso_E: fixes \<alpha> :: "'a"   and k :: "nat"   and i :: "nat" assumes "weakly_reduced \<alpha> k"   and "k \<le> length fs"   and "Suc i < k" shows "\<parallel>gso i\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gso (Suc i)\<parallel>\<^sup>2"
closure_cbox: fixes a :: "'a"   and b :: "'a" shows "closure (cbox a b) = cbox a b"
assms_2: shows "0 < d \<or> p $ n \<noteq> 0"
mset_neg_image_zmset: fixes M :: "'a zmultiset"   and f :: "'a \<Rightarrow> 'b" assumes "mset_neg M = {#}" shows "mset_neg (Auxiliary.image_zmset f M) = {#}"
cases: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and a1 :: "('b, 'c, 'a, 'd) seqp"   and a2 :: "('b, 'c, 'a, 'd) seqp" assumes "a1 \<leadsto>\<^bsub>\<Gamma>\<^esub> a2" obtains p1 :: "('b, 'c, 'a, 'd) seqp" and p2 :: "('b, 'c, 'a, 'd) seqp" where "a1 = p1__ \<oplus> p2__" and "a2 = p1__"   | p1 :: "('b, 'c, 'a, 'd) seqp" and p2 :: "('b, 'c, 'a, 'd) seqp" where "a1 = p1 \<oplus> p2" and "a2 = p2"   | pn :: "'a" where "a1 = call(pn)" and "a2 = \<Gamma> pn"
conv: fixes u' :: "'a vec" assumes "u' \<in> carrier_vec n" shows "(- (w \<bullet>c u') / (u' \<bullet>c u') \<cdot>\<^sub>v u') \<bullet>c u = - (w \<bullet>c u') / (u' \<bullet>c u') * (u' \<bullet>c u)"
mat_row_first_nth: fixes i :: "nat"   and n :: "nat"   and A :: "'b mat" assumes "i < n" shows "row (mat_row_first A n) i = row A i"
cval_add_def: fixes u :: "'a \<Rightarrow> 'b"   and d :: "'b" shows "u \<oplus> d = (\<lambda>x. u x + d)"
SSpec_derivable: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ" assumes "G \<rhd> Call : SSpec Call" shows "G \<rhd> c : SSpec c"
eq_class_of_rng_of_frac_def: fixes rel :: "('c \<times> 'd, 'e) eq_object_scheme"   and r :: "'c"   and s :: "'d" shows "r |\<^bsub>rel\<^esub> s \<equiv> {(r', s'). (r', s') \<in> carrier rel \<and> (r, s) .=\<^bsub>rel\<^esub> (r', s')}"
simps_3: fixes uw :: "('c \<times> nat) tree"   and ux :: "'c"   and lva :: "nat"   and uy :: "('c \<times> nat) tree"   and uz :: "'c"   and lvb :: "nat"   and va :: "('c \<times> nat) tree" shows "sngl \<langle>uw, (ux, lva), \<langle>uy, (uz, lvb), va\<rangle>\<rangle> = (lvb < lva)"
rewrite_iter_slow_def: fixes \<phi> :: "'a ltln" shows "rewrite_iter_slow \<phi> \<equiv> iterate (rewrite_syn_imp \<circ> rewrite_modal \<circ> rewrite_X) \<phi> (size \<phi>)"
open_from_initialiD: fixes i :: "'e"   and OI :: "(('e \<Rightarrow> 'f) \<times> 'g) set"   and CI :: "('f \<times> 'g) set"   and \<sigma> :: "'e \<Rightarrow> 'f"   and p :: "'g" assumes "initiali i OI CI"   and "(\<sigma>, p) \<in> OI" shows "\<exists>\<xi>. \<sigma> i = \<xi> \<and> (\<xi>, p) \<in> CI"
eq3: shows "rb_spp_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z) = app_args spp_of (rb_aux (app_args vec_of (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)))"
csin_eq_1: fixes z :: "complex" shows "(sin z = 1) = (\<exists>x. z = complex_of_real (2 * real_of_int x * pi) + complex_of_real pi / 2)"
eq_imp_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs = ys" shows "prefix xs ys"
is_unit_field_poly: fixes p :: "'b poly" shows "is_unit p = (p \<noteq> 0 \<and> Polynomial.degree p = 0)"
lfinite_inf_llist: fixes f :: "nat \<Rightarrow> 'b" shows "linfinite (inf_llist f)"
power_mono_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "less_eq (power.power one times a n) (power.power one times b n) = less_eq a b"
aezdelta0fun_commutes': fixes b1 :: "'a"   and a :: "'b"   and b2 :: "'a" shows "b1 \<delta>\<delta> a * (b2 \<delta>\<delta> (0::'b)) = b2 \<delta>\<delta> (0::'b) * (b1 \<delta>\<delta> a)"
UN_Image: fixes X :: "'d \<Rightarrow> ('c \<times> 'b) set"   and I :: "'d set"   and S :: "'c set" shows "\<Union> (X ` I) `` S = (\<Union>i\<in>I. X i `` S)"
le_numeral_extra_3: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
of_complex_one_dim_iso: fixes \<psi> :: "'c" shows "of_complex (one_dim_iso \<psi>) = one_dim_iso \<psi>"
set_1: fixes x1 :: "'a" shows "set_Literal (Pos x1) = {x1}"
rewr: fixes x :: "int"   and n :: "nat"   and e :: "int" shows "(x \<le> 2 ^ n - e) = (x + e < 2 ^ n + 1)"
sym_group_mult: fixes n :: "nat" shows "(\<otimes>\<^bsub>sym_group n\<^esub>) = (\<circ>)"