hcju/isabelleps huggingface dataset:
dual_assume: fixes p :: "'b Assertion" shows "[\<cdot> p ] ^ o = {\<cdot> p }"
currentD_outside_IN: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub>" shows "d_IN f x = 0"
partn_lst_greater_resource: fixes M :: "nat"   and \<alpha> :: "nat list"   and \<gamma> :: "nat"   and N :: "nat" assumes "Ramsey.partn_lst {..<M} \<alpha> \<gamma>"   and "M \<le> N" shows "Ramsey.partn_lst {..<N} \<alpha> \<gamma>"
neq_mems_state_set_neq_len: fixes x :: "('a, 'b) fmap list"   and s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and y :: "('a, 'b) fmap list" assumes "x \<in> state_set (state_list s as)"   and "y \<in> state_set (state_list s as)"   and "x \<noteq> y" shows "length x \<noteq> length y"
sort_rtbl_def: shows "sort_rtbl \<equiv> sort_key routing_rule_sort_key"
I_subst_peps: fixes \<phi> :: "atom fm"   and l :: "nat"   and xs :: "'a list" assumes "nqfree \<phi>" shows "DLO.I (subst\<^sub>+ \<phi> l) xs \<longrightarrow> (\<exists>leps>xs ! l. \<forall>x. xs ! l < x \<and> x \<le> leps \<longrightarrow> DLO.I \<phi> (x # xs))"
simps_4: fixes a :: "floatarith" shows "fold_const_fa (Inverse a) = Inverse (fold_const_fa a)"
permutes_add_one: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "p permutes S"   and "x \<notin> S"   and "y \<in> S" shows "p(x := y, inv p y := x) permutes insert x S"
is_character_data_ptr_def: fixes ptr :: "(_) character_data_ptr" shows "is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of character_data_ptr.Ref x \<Rightarrow> True | character_data_ptr.Ext character_data_ptr \<Rightarrow> False)"
add_diff_cancel_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
ZFfunDomAppCod: fixes f :: "ZF"   and x :: "ZF" assumes "isZFfun f"   and "x |\<in>| |dom|f" shows "f |@| x |\<in>| |cod|f"
closed_nonintDfmla: shows "FV nonintDfmla = {}"
eta: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "wf_term (sig \<Theta>) t"   and "\<turnstile>\<^sub>\<tau> t : \<tau> \<rightarrow> \<tau>'" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (t $ Bv 0)) t"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) R"
map_denote_positive_trace_dim: fixes x1 :: "nat"   and x2a :: "nat \<Rightarrow> complex mat"   and x3a :: "com list"   and x4 :: "complex mat" assumes "well_com (Measure x1 x2a x3a)"   and "x4 \<in> carrier_mat d d"   and "partial_density_operator x4"   and "\<And>x3aaa \<rho>. \<lbrakk>x3aaa \<in> set x3a; well_com x3aaa; \<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (denote x3aaa \<rho>) \<and> trace (denote x3aaa \<rho>) \<le> trace \<rho> \<and> denote x3aaa \<rho> \<in> carrier_mat d d" shows "\<forall>k<x1. positive ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<and> (map denote x3a ! k) (x2a k * x4 * adjoint (x2a k)) \<in> carrier_mat d d \<and> trace ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<le> trace (x2a k * x4 * adjoint (x2a k))"
lborel_has_bochner_integral_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'c"   and x :: "'c"   and t :: "real" assumes "c \<noteq> 0" shows "has_bochner_integral lborel f x = has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a iarray" assumes "\<And>x. Ra x x" shows "rel_iarray Ra x x"
2151: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (x \<squnion> y) = - x"
advantage_multi_def: fixes \<A> :: "unit \<Rightarrow> (unit, 'x + 'x \<times> 'hash, 'hash + unit) gpv" shows "advantage_multi \<A> = spmf (game_multi \<A>) True"
segments_subset_convex_hull_5: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "closed_segment c a \<subseteq> convex hull {a, b, c}"
assms_1: shows "(t, s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
nxtEA1: fixes c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'ND"   and \<gamma> :: "(nat \<Rightarrow> 'ND) \<Rightarrow> nat \<Rightarrow> bool"   and n' :: "nat" assumes "\<exists>i>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<circle>\<^sub>b\<gamma>)"   and "n \<le> n'"   and "\<exists>!i. n \<le> i \<and> latestAct_cond c t n' i" shows "eval c t t' n' \<gamma>"
measure_eqI_generator_eq_countable: fixes E :: "'b set set"   and \<Omega> :: "'b set"   and M :: "'b measure"   and N :: "'b measure"   and A :: "'b set set" assumes "Int_stable E"   and "E \<subseteq> Pow \<Omega>"   and "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"   and "sets M = sigma_sets \<Omega> E"   and "sets N = sigma_sets \<Omega> E"   and "A \<subseteq> E"   and "\<Union> A = \<Omega>"   and "countable A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>" shows "M = N"
MinimalMethodDefs_unique_def: 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 M :: "char list"   and Cs :: "char list list" shows "MinimalMethodDefs_unique P C M Cs = (\<forall>Cs' mthd. MethodDefs' P C M Cs' mthd \<longrightarrow> (leq_path1p P C)\<^sup>*\<^sup>* Cs' Cs \<longrightarrow> Cs' = Cs)"
reflects_iso: fixes f :: "'d"   and a' :: "'d"   and a :: "'d" assumes "\<guillemotleft>f : a' \<rightarrow>\<^sub>D a\<guillemotright>"   and "C.iso (G f)" shows "D.iso f"
well_order_extend: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "well_ordered_set A less_eq"   and "well_ordered_set B less_eq"   and "\<forall>a\<in>A. \<forall>b\<in>B. less_eq a b \<longrightarrow> less_eq b a \<longrightarrow> a = b"   and "\<forall>a\<in>A. \<forall>b\<in>B. less_eq a b" shows "well_ordered_set (A \<union> B) less_eq"
simps_4: fixes m :: "nat"   and k :: "nat" shows "Partition (Suc m) (Suc k) = Partition m k + Partition (m - k) (Suc k)"
prems_3: shows "partial_density_operator \<rho>\<^sub>1"
marg_dens_cexpr_def: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and x :: "nat"   and e :: "cexpr" shows "marg_dens_cexpr \<Gamma> vs x e = map_vars (\<lambda>y. if y = x then 0 else Suc y) (integrate_vars \<Gamma> (filter (\<lambda>y. y \<noteq> x) vs) e)"
Busemann_function_limsup: fixes u :: "nat \<Rightarrow> 'b"   and xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi" shows "limsup (\<lambda>x. ereal (dist x (u x) - dist y (u x))) \<le> ereal (Busemann_function_at xi x y)"
expand_false_propag: fixes n_ns :: "'b node \<times> 'b node set" assumes "false\<^sub>r \<notin> old (fst n_ns) \<and> (\<forall>nd\<in>snd n_ns. false\<^sub>r \<notin> old nd)" shows "expand n_ns \<le> SPEC (\<lambda>nm_nds. \<forall>nd\<in>snd nm_nds. false\<^sub>r \<notin> old nd)"
cases: fixes a :: "fm" assumes "a \<in> special_axioms" obtains A :: "fm" and i :: "name" and x :: "tm" where "a = A(i::=x) IMP SyntaxN.Ex i A"
assms_2: shows "Fun f T \<cdot> \<delta> = Fun g T' \<cdot> \<delta>"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier F; y \<in> carrier F\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>F\<^esub> y = y \<otimes>\<^bsub>F\<^esub> x" shows "Group.comm_monoid F"
primitive_part_smult: fixes a :: "'a"   and p :: "'a poly" shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
rep_eq: fixes x :: "'b list" shows "pdevs_apply (pdevs_of_list x) = (\<lambda>i. if i < length x then x ! i else (0::'b))"
qGood_qTerm_induct: fixes phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; \<And>Y. qGood Y \<and> qSkel Y = qSkel X \<Longrightarrow> phi Y; \<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
inf_top_right: fixes x :: "'d upred" shows "(x \<and> true) = x"
derivative_norm_bounded_axioms: shows "derivative_norm_bounded T X f f' B B'"
rbt_comp_minus: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_minus c = ord.rbt_minus (lt_of_comp c)"
vmatch_welldefined': fixes pat :: "pat"   and v :: "value"   and env :: "(name, value) fmap" assumes "vmatch pat v = Some env"   and "vwelldefined' v" shows "fmpred (\<lambda>_. vwelldefined') env"
k_1: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "F G1 G2 (Suc (k G1 G2)) = 0"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "(h \<cdot>\<^sub>D g) \<cdot>\<^sub>D f = h \<cdot>\<^sub>D g \<cdot>\<^sub>D f"
conSim: shows "weak_reduction_contrasimulation (indRelTEQ TRel) (STCal Source Target)"
interleave_length: fixes x :: "'d list"   and a :: "'d list"   and b :: "'d list" assumes "x \<in> a \<otimes> b" shows "length x = length a + length b"
fold_cons_def: fixes z :: "'b"   and s :: "'s" shows "local.fold_cons z s = fold f (unstream g s) z"
gb_red_def: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "gb_red gs bs ps sps data = (map (\<lambda>h. (h, default)) (gb_red_aux (gs @ bs) sps), snd data)"
has_next_rbt_keys_generator: shows "list.has_next rbt_keys_generator = rbt_has_next"
Tree_wf\<^sub>pI: fixes a :: "('a, 'b, 'c, 'd) Tree"   and b :: "('a, 'b, 'c, 'd) Tree"   and p :: "perm" assumes "(a, b) \<in> Tree_wf" shows "(abs_Tree\<^sub>p (p \<bullet> a), abs_Tree\<^sub>p b) \<in> Tree_wf\<^sub>p"
splits_at_combine_dest: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 a i r b"   and "splits_at a i \<alpha> N \<beta>" shows "b = \<alpha> @ snd r @ \<beta>"
smap_union_def: fixes smap1 :: "'d \<Rightarrow> 'c set"   and smap2 :: "'d \<Rightarrow> 'c set"   and k :: "'d" shows "smap1 \<union>. smap2 k = smap1 k \<union> smap2 k"
handshake_phase_invD: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and m :: "'a" assumes "handshake_phase_inv s" shows "(sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) \<in> hp_step_rel \<and> (sys_hs_pending m s \<longrightarrow> \<not> sys_ghost_hs_in_sync m s)"
mIH_1: fixes ni2 :: "nat"   and n2 :: "nat ifex" assumes "Rmi_g ni2 n2 bdd"   and "fst (snd (pm_pth (dpm bdd) n)) \<noteq> ni2" shows "t \<noteq> n2"
dom_char: shows "V.dom = (\<lambda>\<mu>. if V.arr \<mu> then \<lparr>Chn = span_in_category.apex (\<cdot>) (Dom \<mu>), Dom = Dom \<mu>, Cod = Dom \<mu>\<rparr> else V.null)"
cmult_le_mono: fixes \<kappa>' :: "V"   and \<kappa> :: "V"   and \<mu>' :: "V"   and \<mu> :: "V" assumes "\<kappa>' \<le> \<kappa>"   and "\<mu>' \<le> \<mu>" shows "\<kappa>' \<otimes> \<mu>' \<le> \<kappa> \<otimes> \<mu>"
fin: shows "\<integral>\<^sup>+ x. ennreal (case x of (x, y, z) \<Rightarrow> Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)) \<partial>density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x)) \<noteq> \<infinity>"
phi_in_A: assumes "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"   and "s\<^sub>0 \<in> A" shows "\<tau>s \<in> list (length ins) A"
Vid: shows "set (us[i := us ! i + c \<cdot>\<^sub>v us ! j]) = insert (us ! i + c \<cdot>\<^sub>v us ! j) (set us - {us ! i})"
mem_in_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and a :: "'c" assumes "f \<in> A \<rightarrow> B"   and "a \<in> A" shows "f a \<in> f ` A"
seq_ext: fixes P :: "(char list list \<times> char list exp) prog"   and e'' :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e\<^sub>0 :: "char list exp"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "\<And>e' s'. P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "P \<turnstile> \<langle>e'';; e\<^sub>0,s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
scale_left_commute: fixes a :: "rat"   and b :: "rat"   and x :: "'a" shows "a *R b *R x = b *R a *R x"
conc_epsilon_1: fixes A :: "'a list set" shows "{[]} @@ A = A"
eheight_rbst_reduce: fixes n :: "nat" assumes "1 < n" shows "eheight_rbst n = pmf_of_set {..<n} \<bind> (\<lambda>k. eheight_rbst k \<bind> (\<lambda>h1. eheight_rbst (n - k - 1) \<bind> (\<lambda>h2. return_pmf (2 * max h1 h2))))"
def_hashmap_size_variable_def: fixes uu :: "variable itself" shows "def_hashmap_size uu \<equiv> 10"
total_preorder_on_axioms: shows "total_preorder_on alts Ri'"
elims_2: fixes x :: "'a extN"   and xa :: "'a extN" assumes "x < xa" obtains x :: "'a extN" and y :: "'a extN" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
filter_concat: fixes p :: "'c \<Rightarrow> bool"   and xs :: "'c list list" shows "filter p (concat xs) = concat (map (filter p) xs)"
csi: shows "cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'"
snoc_eq_iff_butlast: fixes xs :: "'d list"   and x :: "'d"   and ys :: "'d list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
sccs_are_sccs: shows "\<forall>scc\<in>sccs s. is_scc (g_E G) scc"
sconjunctive_INF_simp: fixes x :: "'d"   and X :: "'d set"   and S :: "'e \<Rightarrow> 'f"   and Q :: "'d \<Rightarrow> 'e" assumes "x \<in> X"   and "sconjunctive S" shows "S (Inf (Q ` X)) = Inf (S ` Q ` X)"
Hom_FopxC_simp: fixes gf :: "'d \<times> 'c" assumes "DopxC.arr gf" shows "Hom_FopxC.map gf = S.mkArr (HomC.set (F (D.cod (fst gf)), C.dom (snd gf))) (HomC.set (F (D.dom (fst gf)), C.cod (snd gf))) (inC \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>C h \<cdot>\<^sub>C F (fst gf)) \<circ> HomC.\<psi> (F (D.cod (fst gf)), C.dom (snd gf)))"
order_iff: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le>s b) = (a = signed.min a b)"
bij_fn: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "bij f" shows "bij (f ^^ n)"
measurable_comp: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'b \<Rightarrow> 'c"   and L :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "g \<circ> f \<in> M \<rightarrow>\<^sub>M L"
TD': shows "(\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'a}) \<and> (\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a}))"
le_add_diff_inverse2: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and b :: "'k"   and a :: "'k" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
norm_blinfun_bound: fixes b :: "real"   and f :: "'a \<Rightarrow>\<^sub>L 'b" assumes "0 \<le> b"   and "\<And>x. norm (blinfun_apply f x) \<le> b * norm x" shows "norm f \<le> b"
semantics_in_vertices_2: fixes A :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and e :: "'b allegorical_term" assumes "graph A"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>" shows "b \<in> vertices A"
assms: fixes x :: "'a" assumes "x \<in> space M" shows "N x = N' x"
inj_on_image_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "order.greater_eq C A"   and "order.greater_eq C B" shows "(f ` A = f ` B) = (A = B)"
n_ge_1: shows "- fls_subdegree f \<le> int n"
simps_4: fixes c :: "poly"   and s :: "tm" shows "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
x: shows "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>"
openI: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"
wary_cases_apps: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and t :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var t" obtains \<zeta> :: "('a, 'b) hd" and ss :: "('a, 'b) tm list" where "t = apps (Hd \<zeta>) ss" and "\<And>sa. sa \<in> set ss \<Longrightarrow> arity.wary arity_sym arity_var sa" and "enat (length ss) \<le> arity.arity_hd arity_sym arity_var \<zeta>"
ffold_fun_left_comm: fixes x :: "'m"   and z :: "'m"   and A :: "'m fset" shows "funionM x (ffold funionM z A) = ffold funionM (funionM x z) A"
integrable: fixes i :: "'i" assumes "i \<in> I" shows "integrable (M i) (f i)"
distinct_conv_nth: fixes xs :: "'e list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
2: shows "[k] \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> ms = [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms"
chain_LNil: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "\<not> chain R LNil"
inject: fixes a :: "'d"   and b :: "'e"   and a' :: "'d"   and b' :: "'e" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
degree_smult_eq: fixes c :: "'b"   and f :: "'b poly" assumes "c \<noteq> (0::'b)" shows "degree (Polynomial.smult c f) = degree f"
add_mset_add_single: fixes a :: "'f"   and A :: "'f multiset" shows "add_mset a A = A + {#a#}"
lists_eq_set: fixes A :: "'c set" shows "lists A = {xs. set xs \<subseteq> A}"
cases: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" assumes "a1 cd-a2\<rightarrow>\<^sub>d* a3" obtains n :: "'node SDG_node" where "a1 = n__" and "a2 = []" and "a3 = n__" and "valid_SDG_node n__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a1 = n" and "a2 = ns @ [n'']" and "a3 = n'" and "n cd-ns\<rightarrow>\<^sub>d* n''" and "n'' \<longrightarrow>\<^bsub>cd\<^esub> n'"
cases: fixes a :: "'node SDG_node"   and n :: "'node SDG_node" assumes "a \<in> sum_SDG_slice1 n" obtains "a = n" and "valid_SDG_node n"   | n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a = n''__" and "n''__ s\<longrightarrow>\<^bsub>cd\<^esub> n'a__" and "n'a__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a = n''a__" and "n''a__ s-V__\<rightarrow>\<^sub>d\<^sub>d n'b__" and "n'b__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''b__" and "n''b__ s-p__\<rightarrow>\<^bsub>call\<^esub> n'c__" and "n'c__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and V :: "'var" and n' :: "'node SDG_node" where "a = n''c__" and "n''c__ s-pa__:V\<rightarrow>\<^bsub>in\<^esub> n'd__" and "n'd__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" and "n' \<in> sum_SDG_slice1 n"
s1'_in_S: shows "s1' \<in> S\<^bsub>SES\<^esub>"
k: fixes k :: "nat" assumes "n < enat k" shows "m < enat k"
equality: fixes r :: "('c, 'd) pre_map_scheme"   and r' :: "('c, 'd) pre_map_scheme" assumes "edge_rev r = edge_rev r'"   and "edge_succ r = edge_succ r'"   and "pre_map.more r = pre_map.more r'" shows "r = r'"
transfer: shows "IEEE.cr_float (0, - 1, 0) plus_infinity"
scast_eq2: shows "sint (SCAST(32 \<rightarrow> 64) 0x80000001) = sint 0x80000001"
assms_1: shows "algebraic_semidom_class.coprime a b"
eq: shows "\<phi> (2, 1) [^]\<^bsub>S\<^esub> 2 ^ (p - 1) = \<ominus>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub>"
top_le: fixes a :: "'a" assumes "\<top> \<le> a" shows "a = \<top>"
lt_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "B A Lt C D"
get_shadow_root_eq2_h2: fixes ptr' :: "(_) element_ptr" shows "|h2 \<turnstile> get_shadow_root ptr'|\<^sub>r = |h' \<turnstile> get_shadow_root ptr'|\<^sub>r"
valuation_iff_STRIPS: fixes M :: "object atom formula set"   and \<phi> :: "object atom formula" assumes "wm_basic M"   and "is_STRIPS_fmla \<phi>" shows "valuation M \<Turnstile> \<phi> = M \<TTurnstile> \<phi>"
adj_mat_2: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "A * adj_mat A = det A \<cdot>\<^sub>m 1\<^sub>m n"
one_eq_numeral_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(one = numeral.numeral one plus n) = (num.One = n)"
simps_1: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'b" shows "program.history_consistent program_step \<theta> p [] = True"
partitions_of_def: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" shows "partitions_of A B F = univ (\<lambda>f. (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) F"
Res: fixes R :: "pi"   and S :: "pi"   and x :: "name" assumes "(R, S) \<in> Rel'" shows "(<\<nu>x>R, <\<nu>x>S) \<in> Rel'"
sorted_list_of_set_eq_pick: fixes i :: "nat"   and I :: "nat set" assumes "i < length (sorted_list_of_set I)" shows "sorted_list_of_set I ! i = pick I i"
bin_nth_numeral: fixes x :: "int"   and y :: "int"   and n :: "num" assumes "x div 2 = y" shows "bit x (numeral n) = bit y (pred_numeral n)"
assms_1: shows "f maps\<^bsub>C\<^esub> X to Y"
ale_neg: fixes x :: "ant" assumes "0 \<le> x" shows "- x \<le> 0"
span_finiteE: fixes B :: "'b set"   and p :: "'b" assumes "finite B"   and "p \<in> ideal B" obtains q :: "'b \<Rightarrow> 'b" where "p = (\<Sum>b\<in>B. q b * b)"
Rep_fin_string_inverse: fixes x :: "fin_string" shows "Abs_fin_string (Rep_fin_string x) = x"
e_res: shows "ord_resolve_rename S CAs DA AAs As \<sigma> E"