hcju/isabelleps huggingface dataset:
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D'" shows "P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
rel_generat_same: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and r :: "('d, 'e, 'f) generat" shows "rel_generat A B C r r = ((\<forall>x\<in>generat_pures r. A x x) \<and> (\<forall>out\<in>generat_outs r. B out out) \<and> (\<forall>c\<in>generat_conts r. C c c))"
point_rep_num_inv_non_empty: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and x :: "'b" assumes "incidence_system point_set block_collection"   and "block_collection rep x < int (size block_collection)"   and "block_collection \<noteq> {#}" shows "filter_mset ((\<notin>) x) block_collection \<noteq> {#}"
ring_subset_carrier: fixes x :: "'h"   and A :: "'h set"   and R :: "('h, 'i) partial_object_scheme" assumes "x \<in> A"   and "A \<subseteq> carrier R" shows "x \<in> carrier R"
Gram_Schmidt2_def: fixes xs :: "('a, 'b) vec list" shows "Gram_Schmidt2 xs = Gram_Schmidt (rev xs)"
semiring_norm_74: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m \<le> num.Bit0 n) = (m < n)"
mult_not_zero: fixes times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
spec_is_affine_scheme: shows "affine_scheme {0} ring0.subtraction ring0.subtraction 0 0 cring0.spectrum cring0.is_zariski_open cring0.sheaf_spec cring0.sheaf_spec_morphisms cring0.\<O>b cring0.add_sheaf_spec cring0.mult_sheaf_spec cring0.zero_sheaf_spec cring0.one_sheaf_spec (identity cring0.spectrum) (\<lambda>U. identity (cring0.sheaf_spec U))"
Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "\<Sqinter> A \<le> x"
f2_nth_simps_1: shows "f\<^sup>2 $ 1 = - of_nat 2"
con_K_validity: fixes con_K :: "msg"   and evs :: "event list"   and A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat" assumes "con_K \<in> used evs"   and "con_K = Crypt (priEK TTP) \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K\<rbrace>"   and "evs \<in> zg" shows "Notes TTP \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K, con_K\<rbrace> \<in> set evs"
assms_1: shows "\<forall>x. \<forall>y\<in>X. x \<rightarrow> y \<in> X"
simps_2: fixes uu :: "'b" shows "extT.Val uu + extT.Top = extT.Top"
K_absorb_HK: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K" shows "H \<diamondop>\<^bsub>G\<^esub> K = K"
f: shows "(\<lambda>(x, y). f x y) \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M N"
dist_eq_0_iff: fixes x :: "'a"   and y :: "'a" shows "(dist x y = 0) = (x = y)"
unifless_def: shows "unifless = inv_image (finite_psubset <*lex*> measure size_mset) (\<lambda>x. (vars_mset x, x))"
current_Sup: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "\<And>f. f \<in> Y \<Longrightarrow> current \<Gamma> f"   and "countable (support_flow (\<Squnion> Y))" shows "current \<Gamma> (\<Squnion> Y)"
2: shows "G \<circ> F \<in> A \<rightarrow> C"
mat_oner: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<otimes> \<epsilon> = f"
distinct_1: fixes x1 :: "nat" shows "(_ x1 _) \<noteq> (_Entry_)"
in_notinI: fixes j :: "'a Bonk_Schramm_extension_unfolded"   and i :: "'a Bonk_Schramm_extension_unfolded" assumes "(j, i) \<notin> Bonk_Schramm_extension_unfolded_wo \<or> j = i"   and "i \<in> Field Bonk_Schramm_extension_unfolded_wo"   and "j \<in> Field Bonk_Schramm_extension_unfolded_wo" shows "(i, j) \<in> Bonk_Schramm_extension_unfolded_wo"
is_valid_operator_strips_def: fixes \<Pi> :: "'a strips_problem"   and op :: "'a strips_operator" shows "is_valid_operator_strips \<Pi> op \<equiv> let vs = \<Pi>\<^sub>\<V>; pre = precondition_of op; add = add_effects_of op; del = delete_effects_of op in list_all (\<lambda>v. ListMem v vs) pre \<and> list_all (\<lambda>v. ListMem v vs) add \<and> list_all (\<lambda>v. ListMem v vs) del \<and> list_all (\<lambda>v. \<not> ListMem v del) add \<and> list_all (\<lambda>v. \<not> ListMem v add) del"
WriteNonVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write False a (D, f) A L R W # is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is, \<theta>, x, m(a := f \<theta>), \<D>, \<O>, \<R>, \<S>)"
Diagonalize_Dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (monoidal_language.Dom C t) = monoidal_language.Dom C (monoidal_language.Diagonalize C t)"
cases: fixes x :: "C" obtains "x = C []"   | n :: "C" and ns :: "C list" where "x = C (n # ns)"
comp_proj_i_finite: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and n :: "nat"   and i :: "nat"   and y :: "'a" assumes "borel_adapt_stoch_proc F X" shows "finite (comp_proj_i X n i y)"
ln_realpow: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "ln (x ^ n) = real n * ln x"
N_1: fixes p :: "'a" assumes "p \<in> X" shows "p \<in> N p"
sum_map_hom: fixes M1 :: "'c set"   and M2 :: "'c set" assumes "LinearCombinations.submodule R M1 M"   and "LinearCombinations.submodule R M2 M" shows "mod_hom R (direct_sum (md M1) (md M2)) M (\<lambda>v. fst v \<oplus>\<^bsub>M\<^esub> snd v)"
distinct_5: fixes x11 :: "tm"   and x12 :: "tm"   and x4 :: "fm" shows "x11 IN x12 \<noteq> Neg x4"
adapted_init: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure"   and x :: "nat \<Rightarrow> 'b \<Rightarrow> real" assumes "triv_init_disc_filtr_prob_space M F"   and "borel_adapt_stoch_proc F x" shows "\<exists>c. \<forall>w\<in>space M. x 0 w = c"
compl_eq_compl_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(uminus x = uminus y) = (x = y)"
1: shows "\<Theta>,\<Gamma> - {A} \<turnstile> A \<longmapsto> B"
partList2_partList3_equi: fixes s :: "'b set"   and ts :: "'b set list" shows "set (partList2 s ts) - {{}} = set (partList3 s ts) - {{}}"
yun_factorization_induct: fixes P :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"   and bn :: "'a poly"   and p :: "'a poly"   and cn :: "'a poly" assumes "\<And>bn cn. bn = 1 \<Longrightarrow> P bn cn"   and "\<And>bn cn. \<lbrakk>bn \<noteq> 1; P (bn div gcd bn (cn - pderiv bn)) ((cn - pderiv bn) div gcd bn (cn - pderiv bn))\<rbrakk> \<Longrightarrow> P bn cn"   and "bn = p div gcd p (pderiv p)"   and "cn = pderiv p div gcd p (pderiv p)"   and "monic p" shows "P bn cn"
I_2: fixes x :: "'b"   and i :: "'a" assumes "x \<in> space M"   and "P i x" shows "i \<in> I"
fls_pow_nth_below_subdegree: fixes k :: "int"   and n :: "nat"   and f :: "'a fls" assumes "k < int n * fls_subdegree f" shows "fls_nth (f ^ n) k = (0::'a)"
create_graph_aimpl_def: fixes \<phi> :: "'a ltlr" shows "create_graph_aimpl \<phi> = expand_aimpl (\<lparr>name = expand_new_name expand_init, incoming = {expand_init}, new = {\<phi>}, old = {}, next = {}\<rparr>, {}) \<bind> (\<lambda>(uu_, nds). RETURN nds)"
CutAuxProp_def: fixes H :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "CutAuxProp H c A = (\<forall>G P D. H = insert P D \<longrightarrow> G \<rhd> Call : P \<longrightarrow> G \<subseteq> D \<longrightarrow> D \<rhd> c : A)"
connex_axioms: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "well_related_set A less_eq" shows "connex A less_eq"
OrdP_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm" assumes "H \<turnstile> x EQ x'" shows "H \<turnstile> OrdP x IFF OrdP x'"
exp_half: shows "exp (1 / 2) \<le> 2"
sngr_assn_def: fixes r :: "'b ref"   and x :: "'b" shows "r \<mapsto>\<^sub>r x \<equiv> Abs_assn (sngr_assn_raw r x)"
WriteVolatile: fixes \<O>s :: "nat set list"   and i :: "nat"   and a :: "nat"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> \<O>s ! j = {}"   and "a \<notin> read_only \<S>" shows "\<O>s,i\<turnstile> (Write True a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a"   and TRel :: "('b \<times> 'b) set" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelLT Enc TRel"
WF1: fixes P :: "state \<Rightarrow> bool"   and N :: "state \<times> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" assumes "\<turnstile> $P \<and> N \<longrightarrow> P$ \<or> Q$"   and "\<turnstile> ($P \<and> N) \<and> <A>_v \<longrightarrow> Q$"   and "\<turnstile> $P \<and> N \<longrightarrow> $Enabled (<A>_v)" shows "\<turnstile> \<box>N \<and> WF(A)_v \<longrightarrow> (P \<leadsto> Q)"
UN_absorb: fixes k :: "'d"   and I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "k \<in> I" shows "A k \<union> \<Union> (A ` I) = \<Union> (A ` I)"
denormal_exponent_def: fixes x :: "('a, 'b) IEEE.float itself" shows "denormal_exponent x = 1 - (int LENGTH('b) + int (bias TYPE(('a, 'b) IEEE.float)))"
constant_functor_axioms: shows "constant_functor (\<cdot>\<^sub>J) (\<cdot>) (C.dom f)"
measurable_entropy_density: fixes b :: "real"   and M :: "'b measure"   and N :: "'b measure" shows "entropy_density b M N \<in> borel_measurable M"
assms_2: shows "\<I> \<turnstile>g gpv \<surd>"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "ring_functions R"
inv_one: shows "inv\<^bsub>tetrahedral_group\<^esub> \<one>\<^bsub>tetrahedral_group\<^esub> = \<one>\<^bsub>tetrahedral_group\<^esub>"
res_def: fixes \<eta> :: "nat" shows "CIPHER.res \<eta> = 1\<^sub>C |\<^sub>= CIPHER.enc \<eta> |\<^sub>= CIPHER.dec \<eta> \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> CIPHER.KEY.res \<eta> \<parallel> CIPHER.AUTH.res"
standard_decomp_shift_list: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h1 :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U1 :: "'x set"   and h2 :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U2 :: "'x set"   and x :: "'x" assumes "standard_decomp k ps"   and "(h1, U1) \<in> set ps"   and "(h2, U2) \<in> set ps"   and "poly_deg h1 = poly_deg h2"   and "card U2 \<le> card U1"   and "(h1, U1) \<noteq> (h2, U2)"   and "x \<in> U2" shows "standard_decomp k (shift_list (h2, U2) x ps)"
closed_mono: fixes i :: "nat"   and j :: "nat"   and p :: "('a, 'b) form" assumes "i \<le> j"   and "closed i p" shows "closed j p"
I: shows "t - 2 * deltaG TYPE('a) \<le> Gromov_product_at x (geodesic_segment_param G x t) (geodesic_segment_param H x t)"
clo: fixes i :: "'a" assumes "i \<in> I" shows "closedin X (T i)"
cmp_simps_4: shows "trg rs.cmp = src rs.tab"
oconf_init_obj_lemma: fixes G :: "prog"   and obj :: "obj"   and r :: "oref"   and s :: "st" assumes "\<And>C c. class G C = Some c \<Longrightarrow> unique (DeclConcepts.fields G C)"   and "\<And>C c f fld. \<lbrakk>class G C = Some c; table_of (DeclConcepts.fields G C) f = Some fld\<rbrakk> \<Longrightarrow> is_type G (type fld)"   and "case r of Inl a \<Rightarrow> is_type G (obj_ty obj) | Inr C \<Rightarrow> is_class G C" shows "G,s\<turnstile>obj\<lparr>values := init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"
abs_zero: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
executionally_equivalent_transitivity: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and x :: "(String.literal \<times> value list) list"   and e3 :: "((nat \<times> nat) \<times> transition) fset"   and s3 :: "nat"   and r3 :: "nat \<Rightarrow>f value option" assumes "executionally_equivalent e1 s1 r1 e2 s2 r2 x"   and "executionally_equivalent e2 s2 r2 e3 s3 r3 x" shows "executionally_equivalent e1 s1 r1 e3 s3 r3 x"
undefined_transfer: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "T (Rep undefined) undefined"
comp_in_homI': fixes f :: "'c"   and g :: "'c"   and a :: "'c"   and c :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.dom f = a"   and "C.cod g = c"   and "C.dom g = C.cod f" shows "\<guillemotleft>C g f : a \<rightarrow>\<^sub>C c\<guillemotright>"
igWlsDisj_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igWlsDisj MOD \<equiv> \<forall>s s' X. igWls MOD s X \<and> igWls MOD s' X \<longrightarrow> s = s'"
f'_def: shows "f' = f \<circ> (\<lambda>s. s + complex_of_real (\<sigma> - 1))"
shows_sep_append: fixes xs :: "'a list"   and showsx :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list"   and r :: "char list"   and s :: "char list" assumes "\<And>r s. \<forall>x\<in>set xs. showsx x (r @ s) = showsx x r @ s"   and "\<And>r s. sep (r @ s) = sep r @ s" shows "shows_sep showsx sep xs (r @ s) = shows_sep showsx sep xs r @ s"
distinct_7: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com" shows "Skip \<noteq> Cond x51 x52 x53"
cone_def: fixes s :: "'a set" shows "cone s = (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
listE_set: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set" assumes "xs \<in> list n A" shows "set xs \<subseteq> A"
simps_2: fixes j :: "nat"   and xs :: "nat list"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_i_PQ (j # xs) i bezout (P, A, Q) = (if A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j then diagonal_to_Smith_i_PQ xs i bezout (P, A, Q) else let (p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j); A' = diagonal_step A i j d v; (P', Q') = diagonal_step_PQ A i j bezout in diagonal_to_Smith_i_PQ xs i bezout (P' ** P, A', Q ** Q'))"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
set_tl: fixes xs :: "'b list" shows "set (tl xs) \<subseteq> set xs"
bi_unique_iff: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "bi_unique R = ((\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R x z) \<and> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R z x))"
card_AF_sum: fixes i :: "nat" assumes "i < m" shows "card (AF k i) + (\<Sum>j\<in>{0<..<ka}. card (RF j i)) = enum (DF k i) (ka - 1)"
arith_simps_57: fixes a :: "'a" shows "a - (0::'a) = a"
res_of_natural_eq: fixes n :: "nat" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat> = int n mod m"
init_fin_\<tau>diverge_conv: fixes t :: "'t"   and status :: "status"   and x :: "'x1"   and m :: "'m1" shows "\<tau>trsys.\<tau>diverge (r1.init_fin t) r1.init_fin_\<tau>move ((status, x), m) = (r1.\<tau>diverge t (x, m) \<and> status = Running)"
pred_mono_strong: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e option"   and Pa :: "'e \<Rightarrow> bool" assumes "pred_option P x"   and "\<And>z. \<lbrakk>z \<in> set_option x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_option Pa x"
wt_App_arg_\<delta>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "legal_zpassign A"   and "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (App s t))) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t)) = zhmset_of \<delta>\<^sub>h"
distinct_4: fixes x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Ite x31 x32 x33 \<noteq> VE x1"
is_ground_clss_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and CC :: "'b literal multiset set" shows "substitution_ops.is_ground_clss subst_atm CC = (\<forall>C\<in>CC. substitution_ops.is_ground_cls subst_atm C)"
hyps: fixes s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" assumes "length s + length t < length sa__ + length ta__"   and "(s, t) \<in> greek_less r" shows "(x # s, x # t) \<in> greek_less r"
maximal_execution_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.maximal_execution subst e s s' \<equiv> substitution.execution subst e s s' \<and> (\<nexists>s''. (s', s'') \<in> substitution.steps subst)"
unfold_congs_1: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "m r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "m_update f r = m_update f' r'"
r_zero: fixes x :: "'c" assumes "x \<in> carrier S" shows "x \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub> = x"
arr_pr1_iff: fixes a :: "'a"   and b :: "'a" shows "arr \<pp>\<^sub>1[a, b] = (ide a \<and> ide b)"
swapidseq_inverse_exists: fixes n :: "nat"   and p :: "'c \<Rightarrow> 'c" assumes "swapidseq n p" shows "\<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
joinI: fixes x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<And>l. is_lub L l {x, y} \<Longrightarrow> P l"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "P (x \<squnion> y)"
finsert_not_fempty: fixes a :: "'c"   and A :: "'c fset" shows "finsert a A \<noteq> {||}"
similar_bool: fixes b :: "bool discr" shows "B\<cdot>b \<triangleleft>\<triangleright> CB\<cdot>b"
fst_eucl_of_list_prod: fixes xs :: "real list" shows "fst (eucl_of_list xs) = eucl_of_list (take DIM('a) xs)"
i_f_join_i_shrink_iT_Plus_iT_Div_mod: fixes k :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "finite I"   and "\<forall>x\<in>I. x mod k = 0" shows "f \<longmapsto> k \<Join>\<^sub> (I \<oplus> k - 1) = f \<div> k \<Join>\<^sub> (I \<oslash> k)"
relcomp_UNION_distrib2: fixes r :: "'e \<Rightarrow> ('b \<times> 'd) set"   and I :: "'e set"   and s :: "('d \<times> 'c) set" shows "\<Union> (r ` I) O s = (\<Union>i\<in>I. r i O s)"
ide_in_hom: fixes a :: "'a" shows "Left.ide a = Left.in_hom a a a"
vlrestriction_vsingleton_in: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {\<langle>a, b\<rangle>} \<restriction>\<^sup>l\<^sub>\<circ> A = set {\<langle>a, b\<rangle>}"
total_preorder_of_weak_ranking: fixes xs :: "'b set list"   and A :: "'b set" assumes "\<Union> (set xs) = A"   and "is_weak_ranking xs" shows "total_preorder_on A (of_weak_ranking xs)"
simps_2: fixes a :: "('fsym, 'psym) atm" shows "Ik.wtL (Neg a) = Ik.wtA a"
toProcess_start: fixes sidx :: "nat"   and p :: "proc"   and ss :: "(integer \<times> edge list) iarray"   and a :: "nat"   and n :: "String.literal"   and l :: "(String.literal, nat) assoc_list"   and idx :: "nat"   and start :: "edgeIndex"   and r :: "procArg list \<times> varDecl list" assumes "toProcess sidx p = (ss, a, n, l, idx, start, r)" shows "\<exists>s. start = Index s \<and> s < IArray.length ss"
axioms_2: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "Group.group H"
P_rel_G_zipping: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) G"   and y :: "('j, 'k, 'l, 'm, 'n, 'o, 'i) G"   and L1 :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and L2 :: "'d \<Rightarrow> 'k \<Rightarrow> bool"   and Co1 :: "'e \<Rightarrow> 'p \<Rightarrow> bool"   and Co1' :: "'p \<Rightarrow> 'l \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'q \<Rightarrow> 'm \<Rightarrow> bool"   and Contra1 :: "'g \<Rightarrow> 'r \<Rightarrow> bool"   and Contra1' :: "'r \<Rightarrow> 'n \<Rightarrow> bool"   and Contra2 :: "'h \<Rightarrow> 's \<Rightarrow> bool"   and Contra2' :: "'s \<Rightarrow> 'o \<Rightarrow> bool"   and z :: "('b \<times> 'j, 'd \<times> 'k, 'p, 'q, 'r, 's, 'i) G"   and tytok :: "('b \<times> ('b \<times> 'j) \<times> 'j \<times> 'd \<times> ('d \<times> 'k) \<times> 'k \<times> 't) itself" assumes "P x"   and "P y"   and "rel_G L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y"   and "rel_G (\<lambda>x (x', y). x' = x \<and> L1 x y) (\<lambda>x (x', y). x' = x \<and> L2 x y) Co1 Co2 Contra1 Contra2 x z"   and "rel_G (\<lambda>(x, y') y. y' = y \<and> L1 x y) (\<lambda>(x, y') y. y' = y \<and> L2 x y) Co1' Co2' Contra1' Contra2' z y"   and "rel_S_neg_distr_cond' Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok" shows "P z"
submodule_sum_def: fixes N1 :: "'c set"   and N2 :: "'c set" shows "subspace_sum N1 N2 = (\<lambda>(x, y). x \<oplus>\<^bsub>V\<^esub> y) ` {(x, y). x \<in> N1 \<and> y \<in> N2}"