hcju/isabelleps huggingface dataset:
lookup_correct: fixes m :: "('a, 'b) HashMap.hashmap"   and k :: "'a" assumes "hm.invar m" shows "hm.lookup k m = hm.\<alpha> m k"
ordIso_iff_ordLeq: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" shows "(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
MGT_Calls: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set" shows "\<forall>p\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} Call p {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
simps_2: fixes i :: "natural"   and j :: "natural" shows "random_aux_set (Code_Numeral.Suc i) j = collapse (Random.select_weight [(1, Pair valterm_emptyset), (Code_Numeral.Suc i, random j \<circ>\<rightarrow> (\<lambda>x. random_aux_set i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_insert x s))))])"
has_derivative_vec: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. (f i has_derivative f' i) F" shows "((\<lambda>x. \<chi>i. f i x) has_derivative (\<lambda>h. \<chi>i. f' i h)) F"
kill_short_order_of_graph: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "finite (uverts G)" shows "card (uverts G) - card (short_cycles G k) \<le> card (uverts (kill_short G k))"
abs_of_pos: fixes abs :: "'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less zero a" shows "abs a = a"
1: shows "of_nat n + (1::'a) = of_nat (n + 1)"
transfer: shows "(rel_set pcr_CoCalls ===> pcr_CoCalls) \<Union> coCallsLub"
axioms_3: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexFolding X g"
proj_list_insert_2: fixes i :: "'a"   and d :: "'b"   and D :: "('a strand_label \<times> 'b) list" shows "proj i (List.insert (\<star>, d) D) = List.insert (\<star>, d) (proj i D)"
eval_pos_if_dominant_term_pos: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "basis_wf basis"   and "is_expansion F basis"   and "trimmed F"   and "0 < fst (dominant_term F)" shows "\<forall>\<^sub>F x in at_top. 0 < eval F x"
3: shows "\<forall>x. ad x \<sqinter> x = bot"
ff_def: shows "ff = (\<lambda>s. f (w + s *\<^sub>R (z - w)) \<bullet> rot (x - y))"
ts_tuple_rel_def: fixes ys :: "(nat \<times> 'a option list set) set" shows "ts_tuple_rel ys = {(t, as). \<exists>X. as \<in> X \<and> (t, X) \<in> ys}"
span_induct_alt: fixes x :: "'b"   and S :: "'b set"   and h :: "'b \<Rightarrow> bool" assumes "x \<in> span S"   and "h (0::'b)"   and "\<And>c x y. \<lbrakk>x \<in> S; h y\<rbrakk> \<Longrightarrow> h (c *s x + y)" shows "h x"
Vfrom_succ: fixes A :: "V"   and i :: "V" shows "Vfrom A (ZFC_in_HOL.succ i) = A \<squnion> VPow (Vfrom A i)"
pr_simps_5: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "local.dom \<p>\<^sub>1[a0, a1] = a0 \<otimes> a1"
infinite_finite_induct: fixes P :: "'k set \<Rightarrow> bool"   and A :: "'k set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
exec: shows "(ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>')"
quadratic_sub: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and A :: "atom" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "aEval A (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub var a b c d A) (L[var := sqrt Cv])"
Graph_Invariants_axioms: shows "Graph_Invariants A PA QA"
idiv_infinity_right: fixes n :: "enat" assumes "n \<noteq> \<infinity>" shows "n div \<infinity> = 0"
distinct_removeAll: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs" shows "distinct (removeAll x xs)"
less_imp_not_eq2: fixes x :: "'b word"   and y :: "'b word" assumes "x <s y" shows "(y = x) = False"
simps_6: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "not\<^sub>n (\<phi> or\<^sub>n \<psi>) = not\<^sub>n \<phi> and\<^sub>n not\<^sub>n \<psi>"
finite_count_ne: fixes M :: "'b multiset"   and N :: "'b multiset" shows "finite {x. count M x \<noteq> count N x}"
ideal_prod_r_distr: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set"   and K :: "'c set" assumes "ring R"   and "ideal I R"   and "ideal J R"   and "ideal K R" shows "I \<cdot>\<^bsub>R\<^esub> (J <+>\<^bsub>R\<^esub> K) = I \<cdot>\<^bsub>R\<^esub> J <+>\<^bsub>R\<^esub> I \<cdot>\<^bsub>R\<^esub> K"
Unbal2: shows "ins3 (size t) 0 a t = Unbal2 x31_ x32_ x33_"
invertible_def: fixes u :: "'a" assumes "u \<in> G" shows "sub.invertible u = (\<exists>v\<in>G. u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
is_valid_problem_strips_operator_variable_sets_2: fixes \<Pi> :: "'b strips_problem"   and op :: "'b strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (add_effects_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
ans_d_def: fixes x :: "'a" shows "antidomain_near_semiring.ans_d antirange_op x = ar ar x"
LI_unify_measure_lt: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and f :: "'b"   and T :: "('b, 'a) Term.term list"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list"   and \<theta>\<^sub>1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta>\<^sub>2 :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "Some \<delta> = mgu (Fun f T) t"   and "fv t \<subseteq> fv\<^sub>s\<^sub>t S" shows "((S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta>\<^sub>1), S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S', \<theta>\<^sub>2) \<in> measure\<^sub>s\<^sub>t"
check_eqv_complete: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "lang \<Sigma> n r = lang \<Sigma> n s"   and "WS1S_Equivalence_Checking.wf \<Sigma> n r"   and "WS1S_Equivalence_Checking.wf \<Sigma> n s" shows "slow.check_eqvRE \<Sigma> n r s"
1: shows "distinct (map (\<lambda>i. SOME j. j < length ls \<and> ls ! j = i) ss)"
assert_inf: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "{\<cdot> p \<sqinter> q } = {\<cdot> p } \<sqinter> {\<cdot> q }"
unique_fls_subdegree: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "\<exists>!n. fls_nth f n \<noteq> (0::'a) \<and> (\<forall>m. fls_nth f m \<noteq> (0::'a) \<longrightarrow> n \<le> m)"
unsat_core: shows "set (the (\<U>\<^sub>c (set_unsat I s'))) = set I"
approx_st_eq: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "x \<approx> y" shows "st x = st y"
qAbs_preserves_alpha2: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and X' :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGoodAbs (qAbs xs x X) \<or> qGoodAbs (qAbs xs x X')" shows "qAbs xs x X $= qAbs xs x X'"
dvd_def: fixes b :: "'b"   and a :: "'b" shows "(b dvd a) = (\<exists>k. a = b * k)"
red_single_0_lt: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and h :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single f 0 h t" shows "lt f = t \<oplus> lt h"
lit_ord_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.lit_ord trm_ord = {(x, y). (basic_superposition.mset_lit x, basic_superposition.mset_lit y) \<in> mult trm_ord}"
X1: fixes m :: "nat \<Rightarrow> nat option" assumes "\<forall>x v. m x = Some v \<longrightarrow> x < numVars \<and> v < numVals x" shows "s ++ m \<in> valid_states"
cases: fixes G :: "prog"   and accclass :: "qtname"   and a1 :: "qtname \<times> memberdecl"   and a2 :: "qtname" assumes "G \<turnstile> a1 of a2 accessible_from accclass" obtains membr :: "qtname \<times> memberdecl" and "class" :: "qtname" where "a1 = membr__" and "a2 = class__" and "G \<turnstile> membr__ member_of class__" and "G \<turnstile> Class class__ accessible_in pid accclass" and "G \<turnstile> membr__ in class__ permits_acc_from accclass"   | membr :: "qtname \<times> memberdecl" and "class" :: "qtname" and C :: "qtname" and new :: "mdecl" and old :: "qtname \<times> mdecl" and supr :: "qtname" where "a1 = membr" and "a2 = class" and "G \<turnstile> membr member_of class" and "G \<turnstile> Class class accessible_in pid accclass" and "membr = (C, mdecl new)" and "G \<turnstile> (C, new) overrides\<^sub>S old" and "G\<turnstile>class\<prec>\<^sub>C supr" and "G \<turnstile>Method old of supr accessible_from accclass"
diff_cong: fixes g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> src.carrier \<Longrightarrow> f x = g x" shows "diff k charts charts_eucl g"
induct: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and x1 :: "char list"   and x2 :: "((char list \<times> char list) \<times> ty) list"   and Pa :: "char list \<Rightarrow> ((char list \<times> char list) \<times> ty) list \<Rightarrow> bool" assumes "P \<turnstile> x1 has_fields x2"   and "\<And>C D fs ms FDTs FDTs'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D has_fields FDTs; Pa D FDTs; FDTs' = map (\<lambda>(F, T). ((F, C), T)) fs @ FDTs\<rbrakk> \<Longrightarrow> Pa C FDTs'"   and "\<And>D fs ms FDTs. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs\<rbrakk> \<Longrightarrow> Pa Object FDTs" shows "Pa x1 x2"
isometry_subset_subspace: fixes S :: "'a set"   and T :: "'b set" assumes "subspace S"   and "subspace T"   and "dim S \<le> dim T" obtains f :: "'a \<Rightarrow> 'b" where "linear f" and "f ` S \<subseteq> T" and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x"
max_v_ev_def: shows "local.max_v_ev = (SOME v_ev. eigen_vector A (fst v_ev) (snd v_ev) \<and> cmod (snd v_ev) = HMA_Connect.spectral_radius A)"
alpha_trans: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and X' :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and X'' :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGood X \<or> qGood X' \<or> qGood X''"   and "X #= X'"   and "X' #= X''" shows "X #= X''"
\<phi>_def: shows "\<phi> \<equiv> inv_into A (ordermap A VWF)"
mult_onel: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c" assumes "class.ab_near_semiring_one plus times one" shows "times one x = x"
step: fixes k :: "nat"   and X :: "'b option list set"   and Y :: "'b option list set"   and z :: "'a" assumes "min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j) \<le> k"   and "Suc k \<le> Monitor.progress \<sigma> \<phi> j'"   and "Suc k \<le> Monitor.progress \<sigma> \<psi> j'"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) k \<phi>) X"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) k \<psi>) Y"   and "U k z" shows "U (Suc k) (f X Y (\<tau> \<sigma> k) z)"
abs_eq: fixes x :: "dflt_size word" shows "uint64_of_uint (Abs_uint x) = Abs_uint64 (ucast x)"
assms_1: shows "((\<lambda>x. exp (f x - c * ln (b x))) expands_to MS xs g) (b # basis)"
vseq_implies_hpar_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "I.map \<nu> = I.map \<mu>"
rel_prod_inject: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and b :: "'d"   and c :: "'c"   and d :: "'e" shows "rel_prod R1 R2 (a, b) (c, d) = (R1 a c \<and> R2 b d)"
edges_of_walk_in_E: fixes xs :: "'a list" assumes "walk xs" shows "edges_of_walk xs \<subseteq> E"
index_uminus_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "(- A) $$ (i, j) = - A $$ (i, j)"
mult_strict_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times a c) (times b c)"
1: shows "\<exists>!H'. H' \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H' C D"
RootEx_Root_ran: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" assumes "RootEx F G" shows "Root F G \<notin> \<Union> (ran G)"
IH: fixes gpv :: "('e, 'f, 'a) gpv" shows "interaction_bound' (map_gpv' f g h gpv) = interaction_bound'' gpv"
Un_assoc: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
usubst_lookup_id: fixes x :: "'e \<Longrightarrow> 'f" shows "\<langle>id\<rangle>\<^sub>s x = x"
linear_componentwise: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and j :: "'b" assumes "linear f" shows "f x \<bullet> j = (\<Sum>i\<in>Basis. x \<bullet> i * (f i \<bullet> j))"
frontier_straddle: fixes a :: "'b"   and S :: "'b set" shows "(a \<in> frontier S) = (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
card_cosets_limit: fixes x :: "nat" assumes "euler_witness (int x) p"   and "x \<in> carrier G" shows "2 \<le> card (rcosets\<^bsub>G\<^esub> H)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_locate_b (as, lm) (s, l, r) ires = in_middle (as, lm) (s, l, r) ires"
filled_gap_1_init_tau: fixes f\<^sub>1 :: "nat \<Rightarrow> nat option"   and i :: "nat"   and t :: "nat" assumes "f\<^sub>1 = \<tau> i(gap i t \<mapsto> 1)" shows "f\<^sub>1 \<triangleright> (t + 1) = e_update (block i t) (gap i t) 1"
not_cd_2: shows "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
rawpsubst2_fresh_switch: fixes \<phi> :: "'fmla"   and t :: "'trm"   and s :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "rawpsubst \<phi> [(s, y), (t, x)] = rawpsubst \<phi> [(t, x), (s, y)]"
zimplode_zexplode: fixes z :: "ZF" shows "zimplode (zexplode z) = z"
simps_153: fixes x1 :: "('a, 'b, 'c) val" shows "RID\<^sub>E (VE x1) = RID\<^sub>V x1"
tuple_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "arr \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>"
l3_inv9_derived: fixes implem :: "chan \<Rightarrow> msg" assumes "dhlvl3 implem" shows "l3_inv2 \<inter> dhlvl3.l3_inv3 implem \<inter> l3_inv4 \<subseteq> dhlvl3.l3_inv9 implem"
dpd1: fixes x :: "'a" shows "x \<le> x \<cdot> ar (ar x)"
SUP_pair: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and B :: "'d set"   and A :: "'c set" shows "(SUP i\<in>A. Sup (f i ` B)) = (SUP p\<in>A \<times> B. f (fst p) (snd p))"
2: shows "\<exists>e e' \<phi> \<psi> \<theta> \<nu> \<theta>' \<nu>'. internal_equivalence e e' \<phi> \<psi> \<and> \<guillemotleft>\<theta>' : src w \<star> e' \<Rightarrow> m\<guillemotright> \<and> \<guillemotleft>\<nu> : g \<star> n \<Rightarrow> (g \<star> w) \<star> e'\<guillemotright> \<and> local.iso \<nu> \<and> \<sigma> = (((g \<star> w) \<star> \<theta>') \<cdot> \<a>[g \<star> w, src w, e'] \<cdot> (\<rho> \<star> e')) \<cdot> \<nu> \<and> \<guillemotleft>\<theta> : m \<star> e \<Rightarrow> src w\<guillemotright> \<and> \<guillemotleft>\<nu>' : g \<star> w \<Rightarrow> (g \<star> n) \<star> e\<guillemotright> \<and> local.iso \<nu>' \<and> \<rho> = ((g \<star> w) \<star> \<theta>) \<cdot> \<a>[g \<star> w, m, e] \<cdot> (\<sigma> \<star> e) \<cdot> \<nu>'"
add_zero_right: fixes z :: "complex_homo" shows "z +\<^sub>h 0\<^sub>h = z"
case_resumption_Branch: fixes t :: "'g \<Rightarrow> 'f"   and l :: "'h \<Rightarrow> ('g, 'h, 'i, 'j) resumption \<Rightarrow> 'f"   and br :: "'i \<Rightarrow> ('j \<Rightarrow> ('g, 'h, 'i, 'j) resumption) \<Rightarrow> 'f"   and c :: "'i"   and rs :: "'j \<Rightarrow> ('g, 'h, 'i, 'j) resumption" shows "(case Branch c rs of Terminal x \<Rightarrow> t x | Linear x xa \<Rightarrow> l x xa | Branch x xa \<Rightarrow> br x xa) = br c rs"
inject_1: fixes x11 :: "name"   and x12 :: "ty"   and y11 :: "name"   and y12 :: "ty" shows "(PVar x11 x12 = PVar y11 y12) = (x11 = y11 \<and> x12 = y12)"
step_3_a_similar: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i < j"   and "j < n" shows "similar_mat (step_3_a i j A) A"
uassoc: shows "vars1 \<union> (vars2 \<union> vars3) = vars1 \<union> vars2 \<union> vars3"
acc_induct: fixes a :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<forall>y. (y, x) \<in> r \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
fmdrop_set_add_distrib: fixes A :: "'c set"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmdrop_set A (m ++\<^sub>f n) = fmdrop_set A m ++\<^sub>f fmdrop_set A n"
False: shows "\<not> (p = {} \<or> cbox a b = {})"
span_is_subset: fixes S :: "'e set"   and N :: "'e set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N W" shows "W.module.span S \<subseteq> N"
add_rconjby: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (plus (plus (uminus x) y) x) (plus (plus (uminus x) z) x) = plus (plus (uminus x) (plus y z)) x"
sum_list_replicate_mono: fixes w1 :: "nat"   and w2 :: "nat"   and n :: "nat" assumes "w1 \<le> w2" shows "sum_list (replicate n w1) \<le> sum_list (replicate n w2)"
prime_2: shows "prime (of_nat y + \<i>\<^sub>\<int> * of_nat x)"
a_insert_before_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and child :: "(_) node_ptr option" shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before get_parent get_child_nodes set_child_nodes get_ancestors adopt_node set_disconnected_nodes get_disconnected_nodes get_owner_document ptr node child = Heap_Error_Monad.bind (l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_ensure_pre_insertion_validity get_parent get_child_nodes get_ancestors node ptr child) (\<lambda>_. Heap_Error_Monad.bind (if Some node = child then l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_next_sibling get_parent get_child_nodes node else return child) (\<lambda>reference_child. Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>owner_document. Heap_Error_Monad.bind (adopt_node owner_document node) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (remove1 node disc_nodes)) (\<lambda>_. l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_node get_child_nodes set_child_nodes ptr node reference_child))))))"
vsv_appI: fixes a :: "V"   and b :: "V"   and aa :: "V"   and ba :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> ZFC_in_HOL.set {\<langle>aa, ba\<rangle>}" shows "ZFC_in_HOL.set {\<langle>aa, ba\<rangle>}\<lparr>a\<rparr> = b"
transpose_add: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "(A + B)\<^sup>T = A\<^sup>T + B\<^sup>T"
eval_fm_HF_axP: fixes e :: "(name, hf) finfun"   and x :: "tm" shows "eval_fm e (HF_axP x) = (\<lbrakk>x\<rbrakk>e \<in> HF_ax)"
rep_eq: fixes x :: "'a coreflexive" shows "Rep_coreflexive (x\<^sup>T) = Rep_coreflexive x"
vid_onE: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> vid_on A" obtains "\<exists>a\<in>elts A. x = \<langle>a, a\<rangle>"
simps_2: fixes A :: "'b set"   and As :: "'b set list" shows "listset (A # As) = set_Cons A (listset As)"
irreducible_prod_rI: fixes a :: "'a list"   and b :: "'a list" assumes "irreducible (K [X]) a"   and "b \<in> Units (K [X])"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "irreducible (K [X]) (a \<otimes>\<^bsub>K [X]\<^esub> b)"
assms_5: shows "\<forall>m. A $ m $ mod_type_class.from_nat k = (0::'a)"
sup_bot_right: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
direct_decomp_span_some_basis: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "local.span (\<Union> (some_basis ` set ss)) = A"
markedElementsEmptyImpliesMarkedElementsToEmpty: fixes M :: "('b \<times> bool) list"   and e :: "'b" assumes "markedElements M = []" shows "markedElementsTo e M = []"
coprime_mod_left_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "algebraic_semidom.coprime times one (modulo a b) b = algebraic_semidom.coprime times one a b"