hcju/isabelleps huggingface dataset:
spec_demonic_choice: fixes p :: "'e \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'd"   and p' :: "'e \<Rightarrow> bool"   and r' :: "'e \<Rightarrow> 'd" shows "({. p .} \<circ> [: r :]) \<sqinter> ({. p' .} \<circ> [: r' :]) = {. p \<sqinter> p' .} \<circ> [: r \<squnion> r' :]"
int_pow_diff: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int"   and m :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (n - m) = x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> m)"
bij_rename_imp_bij: fixes h :: "'c \<Rightarrow> 'd" assumes "bij (rename h)" shows "bij h"
cryptinverts_hilf4: fixes p :: "nat"   and q :: "nat"   and m :: "nat"   and x :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> q"   and "m < p * q"   and "x mod (pred p * pred q) = 1" shows "m ^ x mod (p * q) = m"
composite_simps_3: shows "local.dom (\<nu> \<star> \<pi>) = local.dom \<nu> \<star> local.dom \<pi>"
wfE_min': fixes R :: "('b \<times> 'b) set"   and Q :: "'b set" assumes "wf R"   and "Q \<noteq> {}" obtains z :: "'b" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
between_triv1: fixes a :: "'b"   and b :: "'b" shows "between (a, b) a"
solves_ode_on_subset: fixes x :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "real set"   and Y :: "'a set"   and T :: "real set"   and X :: "'a set" assumes "(x solves_ode f) S Y"   and "T \<subseteq> S"   and "Y \<subseteq> X" shows "(x solves_ode f) T X"
intro: fixes r :: "('a \<times> 'a) set" assumes "Well_order r" shows "wo_rel r"
causal_program_history: fixes is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list"   and sb\<^sub>1 :: "'b memref list"   and sb\<^sub>2 :: "'b memref list" assumes "causal_program_history is\<^sub>s\<^sub>b sb"   and "sb = sb\<^sub>1 @ sb\<^sub>2" shows "\<exists>is. instrs sb\<^sub>2 @ is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2"
simps: fixes L :: "atomUni list" shows "generalVS_DNF L = list_disj_Uni (list_conj_Uni (map substNegInfinityUni L) # map (elimVarUni_atom L) L)"
rep_eq: fixes x :: "'b fpxs" shows "fpxs_nth (- x) = (\<lambda>xa. - fpxs_nth x xa)"
base_start_eq: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat" assumes "p \<in> sparsegrid dm lm" shows "start dm = base {0..<dm} p"
ctor_rec_o_map: fixes f :: "('g, 'g F \<times> 'f) F0 \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "ctor_rec_F f \<circ> map_F g = ctor_rec_F (f \<circ> map_pre_F g (map_prod (map_F g) id))"
cCons_not_0_eq: fixes x :: "'c"   and xs :: "'c list" assumes "x \<noteq> (0::'c)" shows "cCons x xs = x # xs"
vs1: shows "vs1 \<equiv> getFrN (map snd txs) (r1 # map fst txs) [] (length txs)"
subst_wellformed: fixes n :: "nat"   and t :: "term"   and env :: "(name, term) fmap" assumes "wellformed' n t"   and "fmpred (\<lambda>_. wellformed) env" shows "wellformed' n (subst t env)"
conforms_RetD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "x = Some (Jump Ret)" shows "locals s Result \<noteq> None"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool" assumes "type_wf = ShadowRootClass.type_wf" shows "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
not_add_less_right: fixes x :: "hf"   and y :: "hf" shows "\<not> x @+ y < x"
filter_sort: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and P :: "'h \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" assumes "class.linorder less_eq less" shows "filter P (linorder.sort_key less_eq f xs) = linorder.sort_key less_eq f (filter P xs)"
Nil1: fixes ys :: "'a list" assumes "as = []"   and "bs = ys" shows "P"
x0_2: fixes a :: "nat"   and b :: "nat" assumes "x0 \<le> a"   and "a < b" shows "norm (sum f {a<..b}) \<le> g a"
sup_idempotent: fixes x :: "'a" shows "- x \<squnion> - x = - x"
simps_2: fixes x :: "'b memref"   and xs :: "'b memref list"   and W :: "nat set" shows "unforwarded_non_volatile_reads (x # xs) W = (case x of Write\<^sub>s\<^sub>b x a xa xb xc xd xe xf \<Rightarrow> unforwarded_non_volatile_reads xs (insert a W) | Read\<^sub>s\<^sub>b volatile a x xa \<Rightarrow> if a \<notin> W \<and> \<not> volatile then insert a (unforwarded_non_volatile_reads xs W) else unforwarded_non_volatile_reads xs W | _ \<Rightarrow> unforwarded_non_volatile_reads xs W)"
f_in_hhom: shows "\<guillemotleft>f : src f \<rightarrow> trg f\<guillemotright>"
ring_tOp_assoc: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<cdot>\<^sub>r\<^bsub>R\<^esub> z = x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<cdot>\<^sub>r\<^bsub>R\<^esub> z)"
f': fixes t :: "real" assumes "0 < t" shows "(f has_vector_derivative vector_derivative f (at t)) (at t)"
assms_1: shows "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'b)"
sparse_row_add_spmat: fixes A :: "(nat \<times> (nat \<times> 'a) list) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list" shows "sparse_row_matrix (add_spmat A B) = sparse_row_matrix A + sparse_row_matrix B"
prefix_code_2: fixes x :: "'b"   and xs :: "'b list" shows "prefix (x # xs) [] = False"
d: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> d) = (\<exists>k l u v. meets k l \<and> meets l p \<and> meets p u \<and> meets u v \<and> meets k q \<and> meets q v)"
sum_div_1: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "source_code M b fi X enc dec"   and "sum f A \<noteq> (0::'e)" shows "(\<Sum>i\<in>A. f i / sum f A) = (1::'e)"
deriv_c2: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>2 c\<^sub>2 I,A"
accept\<^sub>R_simp: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and \<alpha> :: "(('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set) set"   and w :: "nat \<Rightarrow> 'c" shows "accept\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> Fin = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> Inf \<noteq> {})"
simps_1: fixes n :: "nat" shows "wf_word n [] = True"
intro: fixes f :: "'d \<Rightarrow> 'd" assumes "bij f" shows "bijection f"
Arg_real: fixes z :: "complex" assumes "z \<in> \<real>" shows "Arg z = (if 0 \<le> Re z then 0 else pi)"
assms_2: shows "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
diff_fun_def: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'f" shows "diff_fun k charts f \<equiv> diff k charts charts_eucl f"
fplus_assoc: fixes a :: "'c\<cdot>'d"   and b :: "'c\<cdot>'d"   and c :: "'c\<cdot>'d" shows "fplus\<cdot>(fplus\<cdot>a\<cdot>b)\<cdot>c = fplus\<cdot>a\<cdot>(fplus\<cdot>b\<cdot>c)"
ab_group_add_axioms_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.ab_group_add_axioms plus zero minus uminus \<equiv> (\<forall>a. plus (uminus a) a = zero) \<and> (\<forall>a b. minus a b = plus a (uminus b))"
set_rotate1: fixes xs :: "'a list" shows "set (rotate1 xs) = set xs"
accept_def: shows "\<UU>.accept = Alm_all \<UU>.token_succeeds"
seqr_right_one_point: fixes x :: "'b \<Longrightarrow> 'c"   and P :: "('d, 'c) urel"   and v :: "'b"   and Q :: "('c, 'e) urel" assumes "vwb_lens x" shows "P ;; ($x =\<^sub>u \<guillemotleft>v\<guillemotright> \<and> Q) = P\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<acute>\<rbrakk> ;; Q\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<rbrakk>"
closure_sound: fixes as :: "'b list"   and r :: "'b rexp"   and s :: "'b rexp"   and R :: "('b rexp \<times> 'b rexp) set" assumes "Equivalence_Checking.closure as (r, s) = Some ([], R)"   and "atoms r \<union> atoms s \<subseteq> set as" shows "lang r = lang s"
intro: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" assumes "Group G"   and "GroupHom_axioms G T" shows "GroupHom G T"
weak_upper_semilattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "weak_upper_semilattice L \<equiv> weak_partial_order L \<and> weak_upper_semilattice_axioms L"
prems: shows "\<Gamma>\<turnstile> \<langle>strip_guards F (Catch c1 c2),s\<rangle> =n\<Rightarrow> t"
Suc: shows "replicate n (v 0) \<frown> v \<approx> v"
assms: shows "x \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
proj_onto_def: fixes a :: "'b"   and S :: "'b set" shows "proj_onto a S = sum (proj a) S"
conjug_refl: fixes u :: "'b list" shows "u \<sim> u"
diff_add: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "plus (minus b a) a = b"
tree23_of_list_def: fixes as :: "'a list" shows "tree23_of_list as = join_all (leaves as)"
cases: fixes x :: "'a extB" obtains "x = extB.Bot"   | uu_ :: "'a" where "x = extB.Val uu_"
cPar1B: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> C" shows "F C (P \<parallel> !P) (a<\<nu>x> \<prec> (P' \<parallel> !P))"
simps_3: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and v :: "'b"   and va :: "'a" shows "sum_le P Q (Inr v) (Inl va) = False"
incl: fixes h :: "'f"   and H :: "'f set"   and G :: "('f, 'g) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
attach_shadow_root_def: shows "attach_shadow_root \<equiv> l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_attach_shadow_root set_shadow_root set_mode get_tag_name get_shadow_root"
assms_1: fixes c :: "nat"   and n :: "nat" assumes "0 < c" shows "\<exists>m>n. c * g m < f m"
lt_q: shows "lt (rep_list b \<odot> a) = punit.lt (rep_list b) \<oplus> lt a"
emeasure_finite: fixes na :: "nat"   and A :: "(nat \<Rightarrow>\<^sub>F 'a) set" shows "emeasure (P' na) A \<noteq> top"
expand_ord_def: fixes \<phi> :: "'g ltlr" shows "expand_ord \<phi> \<equiv> inv_image (finite_psupset (old_next_limit \<phi>) <*lex*> less_than) (\<lambda>(n, ns). (old_next_pair ` ns, size_set (new n)))"
span_directional_derivative: fixes a :: "'c" shows "span (directional_derivative \<infinity> a ` Basis) = manifold_eucl.tangent_space \<infinity> a"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (x * xa) = prod_fun (lookup x) (lookup xa)"
size_10: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "size (PApp x31 x32) = size x31 + size x32 + Suc 0"
asset_iv_unique: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and SK' :: "nat option \<times> nat set \<times> nat set"   and C' :: "nat"   and B :: "nat"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, Token m (Auth_PriK m) B C' SK') \<in> s"   and "(Asset n, Token n (Auth_PriK n) B C SK) \<in> s" shows "m = n \<and> C' = C \<and> SK' = SK"
domains_def: fixes f :: "'a" shows "domains f \<equiv> {a. ide a \<and> f \<cdot> a \<noteq> null}"
oconf_def: fixes G :: "prog"   and s :: "st"   and obj :: "obj"   and r :: "oref" shows "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r = (G,s\<turnstile>values obj[\<Colon>\<preceq>]var_tys G (tag obj) r \<and> (case r of Inl a \<Rightarrow> is_type G (obj_ty obj) | Inr C \<Rightarrow> True))"
matching_srcs: fixes rs :: "'a simple_rule list"   and c :: "parts_connection"   and d :: "'a word" assumes "\<forall>r\<in>set rs. simple_conn_matches (match_sel r) c" shows "wordinterval_to_set (matching_srcs d rs Empty_WordInterval) = {s. runFw s d c rs = Decision FinalAllow}"
erH_rHomTr2: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g" assumes "PolynRg A B Y"   and "h \<in> rHom S B" shows "erH R S X A B Y h 1\<^sub>r = 1\<^sub>r\<^bsub>A\<^esub>"
iteration_idem: fixes x :: "'a" shows "x\<^sup>\<infinity> \<cdot> x\<^sup>\<infinity> = x\<^sup>\<infinity>"
lfilter_LNil: fixes P :: "'c \<Rightarrow> bool" shows "lfilter P LNil = LNil"
nxtAct_eq: fixes n :: "nat"   and n' :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'"   and "\<parallel>c\<parallel>\<^bsub>t n'\<^esub>"   and "\<forall>n''\<ge>n. n'' < n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>" shows "n' = \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
simps: fixes G :: "graph"   and m :: "nat"   and n :: "nat" shows "path_set G m n = {p. is_path G p \<and> hd p = m \<and> last p = n}"
IrrFinGroupRep_trivialGSubspace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "IrrFinGroupRepresentation G smult 0"
t_design_axioms: shows "t_design \<V> \<B> \<k> \<t> \<Lambda>\<^sub>t"
restrict_def: fixes G :: "('e, 'f) labeled_graph" shows "restrict G = LG {(l, v1, v2). (l, v1, v2) \<in> edges G \<and> v1 \<in> vertices G \<and> v2 \<in> vertices G} (vertices G)"
simps_2: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_a (Suc i) j A = (let aij = A $$ (i, j) in Let (if A $$ (i, i + 1) = (1::'b) \<and> aij \<noteq> (0::'b) then add_col_sub_row (- aij) (Suc i) j A else A) (??.Jordan_Normal_Form_Existence.step_3_a i j))"
coms_basic_inv: fixes c' :: "command"   and c :: "command" assumes "coms_com (Com c') c" obtains "c = c'"
Doubleton_E: fixes a :: "tm"   and c :: "tm"   and b :: "tm"   and d :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (a EQ c) (insert (b EQ d) H) \<turnstile> A"   and "insert (a EQ d) (insert (b EQ c) H) \<turnstile> A" shows "insert (Eats (Eats Zero b) a EQ Eats (Eats Zero d) c) H \<turnstile> A"
simps_3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
Sup_dual_upset_var: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a"   and X :: "'a set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "\<Inter> (ord.upset less_eq ` X) = ord.upset less_eq (Sup X)"
diff1_2: shows "local.rev_curr f \<in> rst.FINFUNC"
less_dom: shows "\<forall>i\<in>dom \<theta>. i < n"
netlistalternativeSeparate: fixes p :: "('b, 'c) Combinators list" shows "net_list_aux p = net_list_aux (separate p)"
Max_wrt_total_preorder: shows "Max_wrt (R i) = {x \<in> alts. \<forall>y\<in>alts. R i y x}"
fps_add_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f + g) n = fps_nth f n + fps_nth g n"
Simulation_Invariant_axioms: shows "Simulation_Invariant A B (\<sim>) PA PB"
fresh_upds_list: fixes a :: "Vars.var"   and S :: "stack_elem list" assumes "atom a \<sharp> S" shows "atom a \<sharp> upds_list S"
in\<M>: fixes C :: "'a set" assumes "C \<in> \<M>" shows "\<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F"
ro_ifex_unique: fixes x :: "'a ifex"   and y :: "'a ifex" assumes "ro_ifex x"   and "ro_ifex y"   and "\<And>ass. val_ifex x ass = val_ifex y ass" shows "x = y"
IH: shows "vars\<^sub>s\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>s\<^sub>t S"
simps_2: fixes l :: "nat bdd"   and r :: "nat bdd"   and i :: "nat"   and T :: "bool list list" shows "check_eq (Branch l r) (Leaf i) T = (check_eq l (Leaf i) T \<and> check_eq r (Leaf i) T)"
prems_3: fixes k :: "nat" assumes "k < Suc n" shows "h k \<in> carrier_mat d d"
phi_def: shows "phi = (\<lambda>x. INF t\<in>{x<..} \<inter> I. (q x - q t) / (x - t))"
mahler_landau_graeffe_approximation_def: fixes kk :: "nat"   and dd :: "nat"   and f :: "int poly" shows "mahler_landau_graeffe_approximation kk dd f = (let no = \<Sum>a\<leftarrow>coeffs f. a * a in root_int_floor kk (int dd * no))"
beta_confluent: shows "confluent (\<rightarrow>\<^sub>\<beta>)"
compl_unique: fixes x :: "'a upred"   and y :: "'a upred" assumes "(x \<and> y) = false"   and "(x \<or> y) = true" shows "(\<not> x) = y"
id_sum_mod_2: assumes "f = id" shows "(f 0 + f 1) mod 2 = 1"
ANTISYM: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "antisym r"
inducts: fixes x :: "'state"   and P :: "'state \<Rightarrow> bool" assumes "reachNT x"   and "P istate"   and "\<And>trn. \<lbrakk>reachNT (srcOf trn); P (srcOf trn); step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn); \<not> T trn\<rbrakk> \<Longrightarrow> P (tgtOf trn)" shows "P x"
exhaust: fixes y :: "'a acom" obtains x1 :: "'a" where "y = SKIP {x1}"   | x21 :: "char list" and x22 :: "aexp" and x23 :: "'a" where "y = x21 ::= x22 {x23}"   | x31 :: "'a acom" and x32 :: "'a acom" where "y = x31;; x32"   | x41 :: "bexp" and x42 :: "'a" and x43 :: "'a acom" and x44 :: "'a" and x45 :: "'a acom" and x46 :: "'a" where "y = IF x41 THEN {x42} x43 ELSE {x44} x45 {x46}"   | x51 :: "'a" and x52 :: "bexp" and x53 :: "'a" and x54 :: "'a acom" and x55 :: "'a" where "y = {x51} WHILE x52 DO {x53} x54 {x55}"
PPro: fixes r :: "'a pfm"   and x :: "char list" shows "\<turnstile>\<^sub>! ([r]\<^sub>! Pro\<^sub>! x \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! Pro\<^sub>! x)"
assms_5: shows "bounded_hashcode_nat (length l) x = bounded_hashcode_nat (length l) k"
if_distrib: fixes f :: "'d \<Rightarrow> 'c"   and c :: "bool"   and x :: "'d"   and y :: "'d" shows "f (if c then x else y) = (if c then f x else f y)"
sum_nonneg: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "{#} \<subseteq># comm_monoid_add.sum (+) {#} f A"
intros_1: fixes a :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "a \<in> set G" shows "G \<turnstile> a"
tab\<^sub>1_simps_4: fixes r :: "'a" assumes "ide r" shows "trg (tab\<^sub>1 r) = trg r"
simps_1: fixes r :: "'rule" shows "inPorts' (Rule r) = antecedent r"
shiftl_shiftr_id: fixes n :: "nat"   and x :: "'b word" assumes "n < LENGTH('b)"   and "x < 2 ^ (LENGTH('b) - n)" shows "x << n >> n = x"
set_insert: fixes x :: "'d"   and A :: "'d set" assumes "x \<in> A" obtains B :: "'d set" where "A = insert x B" and "x \<notin> B"
is_maxchopper_chop: fixes P :: "'c list \<Rightarrow> bool"   and splitf :: "'c list \<Rightarrow> 'c list \<times> 'c list" assumes "is_maxsplitter P splitf" shows "is_maxchopper P (chop splitf)"
scount_eq_0I: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "alw (\<lambda>xs. \<not> P xs) \<omega>" shows "scount P \<omega> = 0"
matrix_to_iarray_cofactor: fixes A :: "(('c, 'd) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "cofactor A i j = cofactor_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
assms_3: shows "g \<circ> f \<in> Group.iso G K"
inj_on_one_iff: shows "inj_on (\<lambda>f. sum' f I) (carrier (sum_group I (\<lambda>i. free_Abelian_group (S i)))) = (\<forall>x. x \<in> carrier (sum_group I (\<lambda>i. free_Abelian_group (S i))) \<longrightarrow> sum' x I = \<one>\<^bsub>free_Abelian_group (\<Union> (S ` I))\<^esub> \<longrightarrow> x = \<one>\<^bsub>sum_group I (\<lambda>i. free_Abelian_group (S i))\<^esub>)"
mset_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "mset (map f xs) = image_mset f (mset xs)"
ff: fixes w :: "'a" assumes "w \<in> A" shows "ff w \<in> R \<and> ff w ` S \<subseteq> {0..1} \<and> (\<forall>x\<in>S \<inter> Vf w. ff w x < e / real (card subA)) \<and> (\<forall>x\<in>S \<inter> B. 1 - e / real (card subA) < ff w x)"
square_mod_lower: fixes m :: "nat"   and q :: "nat"   and a :: "nat" assumes "0 < m"   and "q\<^sup>2 mod m = a" shows "\<exists>q'<m. q'\<^sup>2 mod m = a"
strict_trans1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "b < c" shows "a < c"
assms_3: shows "path \<sigma> k \<in> dom local.att"
in_restricted_global_in_updated_global_2: fixes l :: "'a"   and s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" assumes "l \<in> LID\<^sub>G (s(r := None))" shows "l \<in> LID\<^sub>G (s(r \<mapsto> ls))"
tau_steps_add: fixes x :: "('idx, 'pred, 'act) formula"   and m :: "nat"   and n :: "nat" shows "tau_steps (tau_steps x m) n = tau_steps x (m + n)"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case INTEG of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f3"
assoc_in_hom: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.in_hom \<a>[a, b, c] ((a \<otimes> b) \<otimes> c) (a \<otimes> b \<otimes> c)"
match: shows "mC_ZOC ZObis c d s s I0 P F"
hQ: shows "reciprocal_poly p P \<circ>\<^sub>p [:1, 1:] \<noteq> 0"
subst_Var_simp1: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "good Y"   and "ys \<noteq> xs \<or> y \<noteq> x" shows "Var xs x #[Y / y]_ys = Var xs x"
le_list_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> x" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs"
numsubst0_I: fixes b :: "real"   and bs :: "real list"   and a :: "MIR.num"   and t :: "MIR.num" shows "Inum (b # bs) (numsubst0 a t) = Inum (Inum (b # bs) a # bs) t"
paths_substitute_substitute'': fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "paths (substitute f T t) = (\<Union>xs\<in>paths t. Collect (substitute'' f T xs))"
maddux_3_21_pp: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "sup (uminus (uminus x)) (inf (uminus x) y) = sup (uminus (uminus x)) y"