hcju/isabelleps huggingface dataset:
N_def: shows "N = y\<^sup>2 + p / (2::'a) + m"
reachable_dom_acyclic: fixes n2 :: "nat"   and n1 :: "nat" assumes "reachable n2"   and "dominate n1 n2"   and "dominate n2 n1" shows "n1 = n2"
fst_netgmap_not_none: fixes i :: "nat"   and s :: "'d net_state"   and sr :: "'d \<Rightarrow> 'e \<times> 'f" assumes "i \<in> net_ips s" shows "fst (netgmap sr s) i \<noteq> None"
one_mod_ring_def: shows "1 \<equiv> Abs_mod_ring 1"
ldrop_ltake: fixes n :: "enat"   and m :: "enat"   and xs :: "'a llist" shows "ldrop n (ltake m xs) = ltake (m - n) (ldrop n xs)"
inf_add_K_right: fixes a :: "'a"   and b :: "'a" shows "a \<le> a + b"
prefix_eqI: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "\<And>x. x \<le> n \<Longrightarrow> f x = g x" shows "prefix f n = prefix g n"
Un_iff: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "s2p (A \<union> B) c = (s2p A c \<or> s2p B c)"
cond3: shows "ideal_generated {inv_u * Q $$ (1, 1), inv_u * - Q $$ (0, 1)} = ideal_generated {1::'a}"
s_top_unit_closed: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K " shows "\<one> \<in> H \<diamondop> K"
project_transient_extend_set: fixes C :: "'c set"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "project h C G \<in> transient (project_set h C \<inter> A - B)" shows "G \<in> transient (C \<inter> extend_set h A - extend_set h B)"
simps_2: fixes s :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "statelist' s (a # as) = s # statelist' (state_succ s a) as"
ternary_lift_4: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv = TernaryTrue) = (tv = TernaryFalse)"
simps_7: shows "ACI_nPlus Zero Zero = (if Zero = Zero then Zero else if Zero < Zero then Plus Zero Zero else Plus Zero Zero)"
Sup_eq_MInfty: fixes S :: "ereal set" shows "(Sup S = - \<infinity>) = (S = {} \<or> S = {- \<infinity>})"
scaleR_add_left: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
cnj_add: fixes a :: "'a"   and b :: "'a" shows "cnj_class.cnj (a + b) = cnj_class.cnj a + cnj_class.cnj b"
Glb_in_L: fixes A :: "'a set"   and i :: "'i" assumes "A \<subseteq> L i" shows "Glb i A \<in> L i"
measurable_completion2: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> completion M \<rightarrow>\<^sub>M N"   and "null_sets N \<subseteq> null_sets (distr (completion M) N f)" shows "f \<in> completion M \<rightarrow>\<^sub>M completion N"
a_export'': fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> y) = apd_d x + ad y"
red_wf_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m" shows "wfP (red F)\<inverse>\<inverse>"
tarski_first5_axioms_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_first5_axioms C B \<equiv> (\<forall>q a b c. \<exists>x. B q a x \<and> C a x b c) \<and> (\<forall>a b c d a' b' c' d'. a \<noteq> b \<and> B a b c \<and> B a' b' c' \<and> C a b a' b' \<and> C b c b' c' \<and> C a d a' d' \<and> C b d b' d' \<longrightarrow> C c d c' d')"
cases: fixes x :: "'b up\<^sub>i" obtains t :: "'b tree234" where "x = T\<^sub>i t"   | l :: "'b tree234" and a :: "'b" and r :: "'b tree234" where "x = Up\<^sub>i l a r"
intros_5: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow> Undecided"
apply_rsp': fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "(R1 ===> R2) f g"   and "R1 x y" shows "R2 (f x) (g y)"
sort_by_pri_def: fixes min_pri :: "nat"   and edges :: "'b edge_scheme list" shows "sort_by_pri min_pri edges = foldl (\<lambda>es e. let idx = nat_of_integer \<bar>prio e\<bar> in if min_pri < idx then abort STR ''Invalid priority'' (\<lambda>_. es) else Let (e # es ! idx) (list_update es idx)) (replicate (min_pri + 1) []) edges"
use_iso2: fixes A :: "('d, 'e) gorder_scheme"   and B :: "('f, 'g) gorder_scheme"   and f :: "'d \<Rightarrow> 'f"   and x :: "'d"   and y :: "'d" assumes "isotone A B f"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<sqsubseteq>\<^bsub>A\<^esub> y" shows "f x \<sqsubseteq>\<^bsub>B\<^esub> f y"
cupcake_list_complete0: fixes env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and res :: "'b state \<times> (v list, v) result" assumes "Big_Step_Unclocked_Single.evaluate_list (\<lambda>s e r. Big_Step_Unclocked_Single.evaluate env s e r \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> cupcake_evaluate_single env e (snd r))) s1 es res"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es" shows "\<exists>rs. list_all2_shortcircuit (cupcake_evaluate_single env) es rs \<and> sequence_result rs = snd res"
AE_lim_wf_times: fixes t :: "real"   and s :: "'a" shows "almost_everywhere (K.lim_stream (t, s)) (wf_times t)"
semilat_le_err_OK1: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b err \<Rightarrow> 'b err \<Rightarrow> 'b err"   and z :: "'b" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, Err.le r, f)"   and "OK x \<squnion>\<^bsub>f\<^esub> OK y = OK z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
fdiv_def: fixes m :: "roundmode"   and a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "fdiv m a b = (if is_nan a \<or> is_nan b \<or> is_zero a \<and> is_zero b \<or> is_infinity a \<and> is_infinity b then some_nan else if is_infinity a \<or> is_zero b then if sign a = sign b then \<infinity> else - \<infinity> else if is_infinity b then if sign a = sign b then 0 else - 0 else zerosign (if sign a = sign b then 0 else 1) (IEEE.round m (valof a / valof b)))"
agree1: shows "Vagree \<nu> \<nu>' (FVDiff t1)"
get_mode_def: shows "get_mode \<equiv> l_get_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_mode"
distinct_1: fixes x1 :: "real" shows "real_min_top.R x1 \<noteq> PInfty"
preprm_inv_involution: fixes \<pi> :: "('b \<times> 'b) list" shows "preprm_inv (preprm_inv \<pi>) = \<pi>"
that: fixes i :: "V"   and a :: "V" assumes "i \<in>\<^sub>\<circ> I"   and "a \<in>\<^sub>\<circ> A i"   and "c = \<langle>i, a\<rangle>" shows "thesis"
fsubsetI: fixes A :: "'d fset"   and B :: "'d fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B" shows "A |\<subseteq>| B"
cases: fixes a1 :: "'a"   and a2 :: "'a" assumes "rel_12_23 a1 a2" obtains "a1 = \<^bold>1" and "a2 = \<^bold>2"   | "a1 = \<^bold>2" and "a2 = \<^bold>3"
assms_1: fixes i :: "'i" assumes "i \<in> I" shows "prob_space (M i)"
module_cpx_vec_def: fixes n :: "nat" shows "module_cpx_vec n \<equiv> module_vec TYPE(complex) n"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "MPoly_Type.degree (MPoly x) = (\<lambda>v. Max (insert 0 ((\<lambda>m. lookup m v) ` keys x)))"
flip_altsublist_chain_set: fixes x :: "'w list"   and zs :: "'w list list"   and y :: "'w list" assumes "x \<in> lists S"   and "flip_altsublist_chain (x # zs @ [y])" shows "set y = set x"
getVar_def: fixes v :: "varRef" shows "getVar v = liftVar (\<lambda>gl v idx arg. getVar' gl v idx) v ()"
dis_is_hbl: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set" assumes "discharge_invar c s t u fo lo no f l n" shows "Height_Bounded_Labeling c s t f l"
isArrLenLoc_simps_4: fixes T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "isArrLenLoc (arrLoc T a i) = False"
elims: fixes x :: "('a, 'b) L_action"   and y :: "atom set" assumes "bn x = y" obtains \<alpha> :: "'a" where "x = L_action.Act \<alpha>" and "y = bn \<alpha>"   | uu_ :: "'b" where "x = Eff uu_" and "y = {}"
sum_gp_basic: fixes x :: "'a"   and n :: "nat" shows "((1::'a) - x) * sum ((^) x) {..n} = (1::'a) - x ^ Suc n"
simps_2: fixes n :: "'a"   and l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0" shows "height (MKT\<^sub>0 n l r) = 1 + max (height l) (height r)"
rep_eq: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |-| xa) = fset x - fset xa"
Obj_in_Hom_1: fixes t :: "'a E.term" assumes "E.Obj t" shows "t \<in> E.HHom t t"
a_inv: fixes h :: "'c"   and R :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "h \<in> generate_ring R H" shows "\<ominus>\<^bsub>R\<^esub> h \<in> generate_ring R H"
rel: shows "qtable n (fv_regex r) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < i + length (fst st) \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> Regex.match (Formula.sat \<sigma> V (map the v)) r i j) rel"
crdt_ops_last: fixes xs :: "('a \<times> 'b) list"   and x :: "'a \<times> 'b"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops (xs @ [x]) deps" obtains oid :: "'a" and oper :: "'b" where "x = (oid, oper)" and "crdt_ops xs deps" and "oid \<notin> fst ` set xs" and "\<forall>ref\<in>deps oper. ref \<in> fst ` set xs \<and> ref < oid"
comm_ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "comm_ring_hom hom \<equiv> Ring_Hom.ring_hom hom"
assms_5: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"
i_Equiv_Exec_Stream_Acc_Output_set_eq: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "nat \<Rightarrow> 'h" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec_stable_set A equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"   and "range input \<subseteq> A" shows "i_Exec_Comp_Stream_Acc_Output k1 output_fun1 trans_fun1 (input_fun1 \<circ> input) c1 = i_Exec_Comp_Stream_Acc_Output k2 output_fun2 trans_fun2 (input_fun2 \<circ> input) c2"
continuous_map_real_mult: fixes X :: "'b topology"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "continuous_map X euclideanreal f"   and "continuous_map X euclideanreal g" shows "continuous_map X euclideanreal (\<lambda>x. f x * g x)"
triangle_strict_inequality: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B D"   and "Cong B C B D"   and "\<not> Bet A B C" shows "A C Lt A D"
intT_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP" shows "ProblemIkTpartG_GEModel.intT prot protFw eintT eintP \<sigma> a \<equiv> if ProblemIkTpart.unprot prot protFw \<sigma> then eintT \<sigma> a else eintT \<sigma> a \<and> eintP (Guard \<sigma>) [a]"
streams_mono: fixes s :: "'d stream"   and A :: "'d set"   and B :: "'d set" assumes "s \<in> streams A"   and "A \<subseteq> B" shows "s \<in> streams B"
primeI: fixes p :: "'b"   and G :: "('b, 'c) monoid_scheme" assumes "p \<notin> Units G"   and "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G; p divides\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b\<rbrakk> \<Longrightarrow> p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b" shows "Divisibility.prime G p"
dist_diff_2: fixes a :: "'b"   and b :: "'b" shows "dist (a - b) a = norm b"
eq: shows "((SUP x\<in>R. f x + g x) \<le> Sup (f ` R) + Sup (g ` R)) = (\<forall>x\<in>R. f x + g x \<le> Sup (f ` R) + Sup (g ` R))"
moebius_rotation_comp: fixes \<phi>1 :: "real"   and \<phi>2 :: "real" shows "moebius_comp (moebius_rotation \<phi>1) (moebius_rotation \<phi>2) = moebius_rotation (\<phi>1 + \<phi>2)"
topological_property_of_product_component: fixes P :: "('c \<Rightarrow> 'd) topology \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and Q :: "'d topology \<Rightarrow> bool" assumes "P (product_topology X I)"   and "\<And>z i. \<lbrakk>z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i)); P (product_topology X I); i \<in> I\<rbrakk> \<Longrightarrow> P (subtopology (product_topology X I) (\<Pi>\<^sub>E j\<in>I. if j = i then topspace (X i) else {z j}))"   and "\<And>X X'. X homeomorphic_space X' \<Longrightarrow> P X = Q X'" shows "(\<Pi>\<^sub>E i\<in>I. topspace (X i)) = {} \<or> (\<forall>i\<in>I. Q (X i))"
rep_eq: fixes x :: "'d pmf"   and xa :: "'d \<Rightarrow> 'c pmf" shows "measure_pmf (x \<bind> xa) = measure_pmf x \<bind> measure_pmf \<circ> xa \<circ> id"
lowner_le_transitive: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "positive (f n - f m)"
standard_identity_rules_1: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph" assumes "standard' C G" shows "maintained (reflexivity_rule S_Idt) G"
hyps: assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<exists>y. execute (f x) h = Some (y, h)" shows "execute (fold_map f xs) h = Some (map (\<lambda>x. fst (the (execute (f x) h))) xs, h)"
m_strictly_prefers_def: fixes m :: "'m"   and \<mu> :: "('m \<times> 'w) set" shows "m_strictly_prefers m \<mu> = {w' \<in> Field (Pm m). \<forall>w\<in>{w. (m, w) \<in> \<mu>}. (w, w') \<in> Pm m \<and> (w', w) \<notin> Pm m}"
characteristic_iso_4: fixes f :: "'a" assumes "Left.ide f" shows "Left.iso (Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f]))"
closer_to_chamber1_not_in_rfold_im_chamber_system: fixes F :: "'a set" assumes "order.greater_eq (chamber_distance D F) (chamber_distance C F)" shows "F \<notin> ChamberComplex.chamber_system (opp_fold_A \<turnstile> A)"
measure_UNION_AE: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M"   and "pairwise (\<lambda>i j. AE x in M. x \<notin> F i \<or> x \<notin> F j) I" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) = (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
group_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" assumes "Group.group G" shows "Group.group (subgroup_generated G S)"
quotient_of_nat: fixes i :: "nat" shows "quotient_of (rat_of_nat i) = (int i, 1)"
wordinterval_to_set_alt: fixes r :: "'a wordinterval" shows "wordinterval_to_set r = {x. wordinterval_element x r}"
A: shows "A \<in> carrier_mat (Suc n) (Suc n)"
assms_3: shows "Qn = Set.filter (\<lambda>(A, uu_). A \<subseteq> V \<and> 1 \<le> card A) Q_neg"
awalk_verts_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "u \<in> verts (G \<restriction> S)"   and "set p \<subseteq> arcs (G \<restriction> S)"   and "v \<in> set (H.awalk_verts u p)" shows "v \<in> verts (G \<restriction> S)"
hyps: fixes M :: "('a \<times> bool) list" assumes "\<exists>s. p' @ s = M"   and "e \<in> set (elements p')" shows "prefixToElement e M = prefixToElement e p'"
induct: fixes x1 :: "NFT"   and x2 :: "dB"   and P :: "NFT \<Rightarrow> dB \<Rightarrow> bool" assumes "NFR x1 x2"   and "\<And>ts x xa. \<forall>i<length ts. NFR (xa i) (ts ! i) \<and> P (xa i) (ts ! i) \<Longrightarrow> P (NFT.App ts x xa) (dB.Var x \<degree>\<degree> ts)"   and "\<And>t x. \<lbrakk>NFR x t; P x t\<rbrakk> \<Longrightarrow> P (NFT.Abs t x) (dB.Abs t)" shows "P x1 x2"
emeasure_extend_measure_Pair: fixes M :: "'b measure"   and \<Omega> :: "'b set"   and I :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and G :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and \<mu> :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and \<mu>' :: "'b set \<Rightarrow> ennreal"   and i :: "'c"   and j :: "'d" assumes "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"   and "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"   and "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>"   and "positive (sets M) \<mu>'"   and "countably_additive (sets M) \<mu>'"   and "I i j" shows "emeasure M (G i j) = \<mu> i j"
RAT_fs': shows "map of_int_hom.vec_hom fs' = (map of_int_hom.vec_hom fs)[i := map of_int_hom.vec_hom fs ! (i - 1), i - 1 := map of_int_hom.vec_hom fs ! i]"
strict_trans2: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" assumes "b \<subset># a"   and "c \<subseteq># b" shows "c \<subset># a"
assms_1: shows "A, n \<turnstile> (ps, a) # filter nonempty left @ right"
same_prefix_nil: fixes xs :: "'c list"   and ys :: "'c list" shows "xs @ ys \<le> xs = (ys = [])"
not_exp_less_eq_0_int: fixes n :: "nat" shows "\<not> 2 ^ n \<le> 0"
universe_is_complement_sum: fixes x :: "i" shows "u = x \<oplus> \<midarrow> x"
simps_3: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm" shows "eval i (Neg' p) = (case eval i p of Det True \<Rightarrow> Det False | Det False \<Rightarrow> Det True | Indet x \<Rightarrow> Indet x)"
simps: fixes y :: "name"   and z :: "name"   and w :: "name"   and x :: "tm" assumes "atom y \<sharp> (z, w, x)"   and "atom z \<sharp> (w, x)"   and "atom w \<sharp> x" shows "SentP x = SyntaxN.Ex y (SyntaxN.Ex z (SyntaxN.Ex w (FormP (Var y) AND FormP (Var z) AND FormP (Var w) AND (x EQ Q_Imp (Var y) (Var y) OR x EQ Q_Imp (Var y) (Q_Disj (Var y) (Var z)) OR x EQ Q_Imp (Q_Disj (Var y) (Var y)) (Var y) OR x EQ Q_Imp (Q_Disj (Var y) (Q_Disj (Var z) (Var w))) (Q_Disj (Q_Disj (Var y) (Var z)) (Var w)) OR x EQ Q_Imp (Q_Disj (Var y) (Var z)) (Q_Imp (Q_Imp (Var y) (Var w)) (Q_Disj (Var z) (Var w)))))))"
FinPow_rep_eq: fixes xs :: "'b fset" shows "\<langle>FinPow xs\<rangle>\<^sub>f = {ys. ys |\<subseteq>| xs}"
oseq_nos_incs: fixes i :: "nat"   and ANY :: "state \<Rightarrow> state \<Rightarrow> bool" shows "optoy i \<Turnstile>\<^sub>A (act TT, other ANY {i} \<rightarrow>) onll \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (seqll i (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). nos_inc \<xi> \<xi>'))"
a1: shows "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = None \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = Some yb"
list_of_dlist_inverse: fixes x :: "'d dlist" shows "Abs_dlist (list_of_dlist x) = x"
card_of_Pow_mono: fixes A :: "'c set"   and B :: "'d set" assumes "|A| \<le>o |B|" shows "|Pow A| \<le>o |Pow B|"
assms_3: fixes v :: "'al" assumes "v \<in> A" shows "v \<le>\<^sub>o\<^sub>w u"
1: shows "coreflexive (prim_E w v e * (1::'a) * prim_E w v e\<^sup>T)"
residue_hom_p: fixes m :: "nat"   and n :: "nat"   and p :: "int" assumes "m \<le> n"   and "0 < m"   and "prime p" shows "residue (p ^ m) \<in> ring_hom (residue_ring (p ^ n)) (residue_ring (p ^ m))"
adjust_idx_nth: fixes i :: "nat"   and xs :: "'c list"   and j :: "nat" assumes "i < length xs" shows "remove_nth i xs ! j = xs ! adjust_idx i j"
lprefix_lnthD: fixes xs :: "'a llist"   and ys :: "'a llist"   and n :: "nat" assumes "lprefix xs ys"   and "enat n < llength xs" shows "lnth xs n = lnth ys n"
gval_fold_rev_true: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd (rev G) (Bc True)) s = true" shows "gval (fold gAnd G (Bc True)) s = true"
null_char: shows "D.VV.null = D.VxV.null"
reachable_connected: fixes g :: "'a ugraph"   and r :: "'a" shows "Undirected_Graph.connected (Prim.rg g r)"
unit_simps_2: fixes a :: "'a" assumes "obj a" shows "S.src (UP.unit a) = UP.map\<^sub>0 a"
e2: fixes e :: "nat \<Rightarrow> real" assumes "e \<in> funcset UNIV {- 1..1}" shows "\<exists>e'\<in>funcset UNIV {- 1..1}. pdevs_val e (snd Y) = pdevs_val e' (snd X)"
path_iff_backward: fixes x :: "'a"   and r :: "'a" assumes "is_inj x"   and "is_p_fun x"   and "point r"   and "r ; x \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>" shows "connected x"
bound_on_all_plans_bounds_MPLS_s: fixes P :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> bool"   and f :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> ('c, 'd) fmap \<Rightarrow> nat" assumes "\<forall>PROB as s. finite PROB \<and> P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB s)" shows "\<forall>PROB x s. finite PROB \<and> P PROB \<and> s \<in> valid_states PROB \<longrightarrow> x \<in> MPLS_s PROB s \<longrightarrow> x < f PROB s"
extensional_def: fixes A :: "'b set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
A4_unique: fixes q :: "'p"   and a :: "'p"   and x :: "'p"   and b :: "'p"   and c :: "'p"   and x' :: "'p" assumes "q \<noteq> a"   and "B q a x"   and "a x \<equiv> b c"   and "B q a x'"   and "a x' \<equiv> b c" shows "x = x'"
DERIV_log: fixes x :: "real"   and b :: "real" assumes "0 < x" shows "(log b has_real_derivative 1 / (ln b * x)) (at x)"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and u :: "name" assumes "P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'"   and "x \<sharp> P" shows "P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'[x::=u]"
ext_inject: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a"   and statename' :: "nat option"   and datastate' :: "nat \<Rightarrow>f value option"   and action' :: "String.literal \<times> value list"   and output' :: "value option list"   and more' :: "'a" shows "(\<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = \<lparr>statename = statename', datastate = datastate', action = action', output = output', \<dots> = more'\<rparr>) = (statename = statename' \<and> datastate = datastate' \<and> action = action' \<and> output = output' \<and> more = more')"
deterministic_minsky_UN: fixes M :: "('a \<times> ('a, 'b) Op) set"   and x :: "'a \<times> ('b \<Rightarrow> nat)"   and sa :: "'a"   and vsa :: "'b \<Rightarrow> nat"   and s :: "'a"   and vs :: "'b \<Rightarrow> nat" assumes "deterministic M"   and "(x, sa, vsa) \<in> (step M)\<^sup>*"   and "(x, s, vs) \<in> (step M)\<^sup>*"   and "sa \<notin> fst ` M"   and "s \<notin> fst ` M" shows "(sa, vsa) = (s, vs)"