hcju/isabelleps huggingface dataset:
left_right_simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and l :: "'b"   and r :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.right unapp (app l r) = r"
vrrestriction_atE1: fixes Aa :: "V"   and A :: "V"   and a :: "V"   and b :: "V" assumes "((r \<restriction>\<^sub>\<circ> Aa) \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> (r \<restriction>\<^sub>\<circ> Aa) -`\<^sub>\<circ> A" obtains "(r \<restriction>\<^sub>\<circ> Aa)\<lparr>a\<rparr> = b"
P_def: shows "P \<equiv> \<lambda>(out, head', tail', prevs'). head' = head \<and> tail' = tail"
minusinf_repeats: fixes p :: "fm"   and d :: "int"   and a :: "real"   and bs :: "real list"   and x :: "int"   and k :: "int" assumes "d_\<delta> p d"   and "iszlfm p (a # bs)" shows "Ifm (real_of_int (x - k * d) # bs) (minusinf p) = Ifm (real_of_int x # bs) (minusinf p)"
assms_1: shows "\<guillemotleft>f : \<I> \<otimes> a \<rightarrow> a\<guillemotright>"
D_T: fixes s :: "'b event list"   and P :: "'b process" assumes "s \<in> D P" shows "s \<in> T P"
make\<Pi>\<^sub>1_inverse: fixes y :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" assumes "y \<in> UNIV" shows "eval\<Pi>\<^sub>1 (make\<Pi>\<^sub>1 y) = y"
vid_on_iff: fixes a :: "V"   and b :: "V"   and A :: "V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vid_on A) = (a = b \<and> a \<in>\<^sub>\<circ> A)"
conv_plus_commute: fixes x :: "'a" shows "id (x \<sqinter> top) = id x \<sqinter> top"
rep_eq: fixes x :: "'a"   and xa :: "'a set['b]" shows "bmember x xa = (x \<in> set_bset xa)"
visits_and_not_subsumes: fixes s1 :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and t1 :: "transition"   and t2 :: "transition" assumes "\<exists>c1 c2 t. obtains s1 c1 e1 0 <> t \<and> obtains s2 c2 e2 0 <> t \<and> \<not> subsumes t1 c2 t2" shows "\<not> directly_subsumes e1 e2 s1 s2 t1 t2"
sifum_refinement_eg1_eg2_axioms: shows "sifum_refinement_eg1_eg2"
E_ss_cfinvE: shows "E \<subseteq> cf.E \<union> cf.E\<inverse>"
strict_trans1: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b"   and "b \<subset># c" shows "a \<subset># c"
\<I>_1: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
filtrated_prob_space_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" shows "filtrated_prob_space M F \<equiv> prob_space M \<and> filtrated_prob_space_axioms M F"
obeh_pi_translation: fixes pi :: "'a \<Rightarrow> 'a"   and S :: "('b, 'a) spec" shows "map pi ` obeh S = obeh (S\<lparr>obs := pi \<circ> obs S\<rparr>)"
weak_partial_order_top_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_partial_order_top_axioms L \<equiv> \<exists>x. greatest L x (carrier L)"
ennreal_add_diff_cancel_left: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> \<top>" shows "y + x - y = x"
addrow_mat_def: fixes n :: "nat"   and a :: "'b"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l = mat n n (\<lambda>(i, j). (if k = i \<and> l = j then (+) a else id) (if i = j then 1::'b else (0::'b)))"
simps: fixes domain :: "'a list"   and codom :: "'a"   and \<Sigma> :: "'a list" shows "Subx.sp_gen_pop_push (domain, codom) \<Sigma> = (let ar = length domain in if ar \<le> length \<Sigma> \<and> take ar \<Sigma> = domain then Ok (codom # drop ar \<Sigma>) else Error ())"
set_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "bheight l \<le> bheight r" shows "Tree2.set_tree (joinL l x r) = Tree2.set_tree l \<union> {x} \<union> Tree2.set_tree r"
coeff_map_poly_hom: fixes p :: "int poly"   and i :: "nat" shows "poly.coeff (map_poly of_int_mod_ring p) i = of_int_mod_ring (poly.coeff p i)"
rev_nth_last: fixes xs :: "'b list"   and x :: "'b" assumes "xs !. 0 = Some x" shows "last xs = x"
rbt_map_entry_rbt_greater: fixes a :: "'a"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "(a \<guillemotleft>| rbt_map_entry k f t) = (a \<guillemotleft>| t)"
eSubst_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and ys :: "'varSort"   and Y :: "'a"   and s :: "'sort"   and X :: "'a"   and y :: "'var" assumes "igWls MOD (asSort ys) Y"   and "igWls MOD s X" shows "eSubst MOD ys (OK Y) y (OK X) = OK (igSubst MOD ys Y y X)"
assms_2: shows "t_covering_design V B (int k) t \<Lambda>\<^sub>t"
min_product_set_infinite: fixes X :: "'b fset set set" assumes "infinite X" shows "\<Otimes>\<^sub>m X = {{||}}"
Abs: fixes t :: "lam"   and x :: "name" assumes "NF t" shows "NF (Lam [x].t)"
poly_of: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and p :: "('d, 'c) tpoly" shows "eval_poly \<alpha> (poly_of p) = eval_tpoly \<alpha> p"
k_def: shows "k = (LEAST k. \<not> b ((c ^^ k) s))"
vec_smult_distr_left: fixes as :: "'b list"   and bs :: "'b list"   and \<alpha> :: "'b" assumes "length as = length bs" shows "vec_smult \<alpha> (vec_plus as bs) = vec_plus (vec_smult \<alpha> as) (vec_smult \<alpha> bs)"
xfer_reachable_aux: fixes c :: "'c" assumes "C.reachable c" shows "\<exists>a. R a c \<and> A.reachable a"
elims: fixes x :: "'g + 'h"   and y :: "'g +\<^sub>\<bottom> 'h" assumes "sumbot x = y" obtains xa :: "'g" where "x = Inl xa__" and "y = sum_bot.Left xa__"   | xa :: "'h" where "x = Inr xa" and "y = sum_bot.Right xa"
left_inv_of_unitary_transpose: fixes U :: "complex Matrix.mat" assumes "unitary U" shows "U\<^sup>t\<^sup>\<dagger> * U\<^sup>t = 1\<^sub>m (dim_row U)"
from_list_correct'_2: fixes l :: "('b \<times> 'b \<times> nat) list"   and u :: "'b"   and v :: "'b"   and d :: "nat" assumes "valid_graph_rep l" shows "((u, v, d) \<in> set l) = (\<alpha> (from_list l) (u, v) = enat d)"
mC_C_partCompat_eff: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "mC_C theta c d s t P F" shows "partCompat P indis (eff c s)"
asset_i_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and S :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey"   and "(Asset n, Crypt (Auth_ShaKey n) (PriKey S)) \<in> s" shows "PriKey S \<notin> spied s"
that: fixes k :: "nat \<Rightarrow> nat" assumes "strict_mono k"   and "\<And>x. x \<in> A \<Longrightarrow> convergent (\<lambda>n. f (k n) x)" shows "thesis"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright> \<equiv> C.arr \<mu> \<and> src\<^sub>C \<mu> = a \<and> trg\<^sub>C \<mu> = b"
deleteMin_mset: fixes q :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q"   and "queue_to_multiset q \<noteq> {#}" shows "queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}"
frechet_linear: fixes I :: "('a, 'b, 'c) interp"   and \<theta> :: "('a, 'c) trm"   and v :: "(real, 'c) vec" assumes "is_interp I"   and "dfree \<theta>" shows "bounded_linear (frechet I \<theta> v)"
esing_bot: fixes x :: "'a" shows "esing x\<cdot>\<bottom> = \<bottom>"
group_l_invI: assumes "\<And>x. x \<in> carrier C \<Longrightarrow> \<exists>y\<in>carrier C. y \<otimes>\<^bsub>C\<^esub> x = \<one>\<^bsub>C\<^esub>" shows "Group.group C"
fpxs_shift_0_left: fixes f :: "'b fpxs" shows "fpxs_shift 0 f = f"
fsingleton_iff: fixes b :: "'c"   and a :: "'c" shows "(b |\<in>| {|a|}) = (b = a)"
prems_4: shows "i < dim_row (diag_block_mat (A # As))"
step_3_c_inv: fixes A :: "'a mat"   and k :: "nat"   and lb :: "nat"   and l :: "nat"   and bs :: "(nat \<times> nat) list"   and x :: "'a" assumes "A \<in> carrier_mat n n"   and "k < n"   and "(lb, l) \<in> set (identify_blocks A k)"   and "local.inv_upto local.jb A k"   and "local.inv_from local.uppert A k"   and "local.inv_at local.one_zero A k"   and "local.ev_block n A"   and "set bs \<subseteq> set (identify_blocks A k)"   and "\<And>be. \<lbrakk>be \<notin> snd ` set bs; be \<notin> {l, k}; be < n\<rbrakk> \<Longrightarrow> A $$ (be, k) = (0::'a)"   and "\<And>bb be. (bb, be) \<in> set bs \<Longrightarrow> be - bb \<le> l - lb"   and "x = A $$ (l, k)"   and "x \<noteq> (0::'a)" shows "local.inv_all local.uppert (step_3_c x l k bs A) \<and> local.same_diag A (step_3_c x l k bs A) \<and> local.same_upto k A (step_3_c x l k bs A) \<and> local.inv_at (local.single_non_zero l k x) (step_3_c x l k bs A) k"
ss_2: shows "MP_Rel S (of_int_poly (Mp S))"
reduced_labelling_Suc: fixes n :: "nat"   and x :: "nat \<Rightarrow> nat" assumes "reduced (Suc n) x \<noteq> Suc n" shows "reduced (Suc n) x = reduced n x"
Union_components: fixes u :: "'a set" shows "\<Union> (components u) = u"
rep_props_1: fixes r :: "'a" assumes "ide r" shows "\<guillemotleft>rep r : tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>* \<Rightarrow> r\<guillemotright>"
ProofStepInduct_y_neq_new_last_axioms_def: fixes v1 :: "'c"   and P_new :: "'c list"   and y :: "'c" shows "ProofStepInduct_y_neq_new_last_axioms v1 P_new y \<equiv> y \<noteq> v1 \<and> y \<noteq> DisjointPathsPlusOne.new_last P_new"
invariant_1D_1: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "root_cond plr (real_of_1 plr)"
snth_szip: fixes s1 :: "'b stream"   and s2 :: "'c stream"   and n :: "nat" shows "(s1 ||| s2) !! n = (s1 !! n, s2 !! n)"
modal_trans: fixes \<phi> :: "i \<Rightarrow> bool"   and \<psi> :: "i \<Rightarrow> bool"   and \<chi> :: "i \<Rightarrow> bool" assumes "\<lfloor>\<phi> \<^bold>\<rightarrow> \<psi>\<rfloor> \<and> \<lfloor>\<psi> \<^bold>\<rightarrow> \<chi>\<rfloor>" shows "\<lfloor>\<phi> \<^bold>\<rightarrow> \<chi>\<rfloor>"
vec_hom_smult: fixes ev :: "'a"   and v :: "'a vec" shows "vec\<^sub>h (ev \<cdot>\<^sub>v v) = hom ev \<cdot>\<^sub>v vec\<^sub>h v"
simps: fixes a1 :: "'a set"   and a2 :: "'a list" shows "local.independent a1 a2 = ((\<exists>K. a1 = K \<and> a2 = []) \<or> (\<exists>u K Us. a1 = K \<and> a2 = u # Us \<and> u \<in> carrier R \<and> u \<notin> Span K Us \<and> local.independent K Us))"
bind_eq_Some_conv: fixes f :: "'d option"   and g :: "'d \<Rightarrow> 'c option"   and x :: "'c" shows "(Option.bind f g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
binop_def: fixes A :: "V"   and f :: "V" shows "binop A f \<equiv> nop A (2\<^sub>\<nat>) f"
count_greater_zero_iff: fixes M :: "'d multiset"   and x :: "'d" shows "(0 < count M x) = (x \<in># M)"
MGF: fixes c :: "com" assumes "state_not_singleton"   and "WT_bodies"   and "WT c" shows "{}|-{=}.c.{->}"
bounded_iff: fixes b :: "'c"   and c :: "'c"   and a :: "'c" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
same_fstI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y' :: "'b"   and y :: "'b"   and R :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "P x"   and "(y', y) \<in> R x" shows "((x, y'), x, y) \<in> same_fst P R"
Bolzano: fixes a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> real \<Rightarrow> bool" assumes "a \<le> b"   and "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b" shows "P a b"
genby_uminus_genset_subset: fixes S :: "'a set" shows "order.greater_eq \<langle>S\<rangle> (uminus ` S)"
nfoldli_upt_rule: fixes lb :: "nat"   and ub :: "nat"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a nres"   and P :: "'a \<Rightarrow> bool" assumes "lb \<le> ub"   and "\<sigma>0 = sum f {l..<lb}"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i < ub; \<sigma> = sum f {l..<i}; c \<sigma>\<rbrakk> \<Longrightarrow> f i \<sigma> \<le> SPEC (\<lambda>s. s = sum f {l..<i + 1})"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i \<le> ub; \<sigma> = sum f {l..<i}; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma>. \<lbrakk>\<sigma> = sum f {l..<ub}; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "nfoldli [lb..<ub] c f \<sigma>0 \<le> SPEC P"
not_weak: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(\<not> x \<sqsubseteq> y) = (y \<sqsubset> x)"
rep_eq: shows "lookup 0 = (\<lambda>k. 0::'r)"
Pi_mono: fixes A :: "'e set"   and B :: "'e \<Rightarrow> 'f set"   and C :: "'e \<Rightarrow> 'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi A B \<subseteq> Pi A C"
that: fixes r :: "V"   and A :: "V" assumes "x = \<langle>\<langle>r, A\<rangle>, r \<restriction>\<^sub>\<circ> A\<rangle>"   and "r \<in>\<^sub>\<circ> R"   and "A \<in>\<^sub>\<circ> X" shows "thesis"
minus_of_nat: fixes x :: "real"   and n :: "nat" shows "fds_perzeta (x - real n) = fds_perzeta x"
dens': fixes M :: "real measure"   and \<delta> :: "real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel \<delta>" shows "has_density (distr M borel f) lborel (g \<delta>)"
cis_divide: fixes a :: "real"   and b :: "real" shows "cis a / cis b = cis (a - b)"
iT_Plus_cut_less: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>< t = I \<down>< (t - k) \<oplus> k"
cont: shows "continuous_on U (\<lambda>x. if x \<in> S' then g x else h x)"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsA at" shows "\<xi>1 x = \<xi>2 x"
sym: fixes s :: "'b"   and t :: "'b" assumes "s = t" shows "t = s"
bind_cong_AE: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and B :: "'b measure"   and g :: "'a \<Rightarrow> 'b measure" assumes "M = N"   and "f \<in> N \<rightarrow>\<^sub>M subprob_algebra B"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra B"   and "AE x in N. f x = g x" shows "M \<bind> f = N \<bind> g"
OclIncludes_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Bag('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->includes\<^sub>B\<^sub>a\<^sub>g(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then \<lfloor>\<lfloor>0 < \<lceil>\<lceil>Rep_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> (y \<tau>)\<rfloor>\<rfloor> else \<bottom>)"
join_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "compatible x y"   and "compatible x (y \<squnion> z)"   and "compatible y z" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
foo3a: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators list"   and b :: "((int \<times> int) set set, int) Combinators"   and c :: "((int \<times> int) set set, int) Combinators list"   and r :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x (a @ [b] @ c) = \<lfloor>b\<rfloor>"   and "r \<in> set c"   and "b \<notin> set c" shows "x \<notin> dom (C r)"
strictly_preserves_\<iota>: shows "E.map \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C = \<iota>\<^sub>D"
focus_superset: fixes A :: "('b \<Rightarrow>\<^sub>0 nat) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "finite A"   and "keys p \<subseteq> A" shows "focus X p = (\<Sum>t\<in>A. monomial (monomial (lookup p t) (except t X)) (except t (- X)))"
Rep_seqauto: fixes x :: "('a, 'b, 'c) seqauto" shows "Rep_seqauto x \<in> seqauto"
ord_equiv_isom: fixes E :: "('e, 'f) Order_scheme" assumes "Order E"   and "ord_equiv D E" shows "\<exists>f. ord_isom D E f"
assms_2: shows "safe_distance_2r < s\<^sub>o - s\<^sub>e"
normal_form_zero: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "normal_form F 0 = 0"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<CC>"
arrE: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VxV.arr f" obtains "arr (fst f) \<and> arr (snd f)"
pair_imageI: fixes a :: "'x"   and b :: "'y"   and A :: "('x \<times> 'y) set"   and f :: "'x \<Rightarrow> 'y \<Rightarrow> 'z" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
indRelRSTPO_impl_SRel_and_TRel_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelRSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred SRel (\<lambda>a. hasBarb a SWB)"
usubstappp_fst_mon: fixes U :: "variable set"   and V :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game" assumes "U \<subseteq> V" shows "fst (usubstappp \<sigma> U \<alpha>) \<subseteq> fst (usubstappp \<sigma> V \<alpha>)"
no_matching_Goto: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, ti\<rangle> \<Rightarrow> t"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "thesis"
rep_eq: fixes x :: "'c fset"   and xa :: "'d" shows "fmlookup [ x |=> xa ] = make_map x xa"
\<psi>_def: shows "\<psi> \<equiv> subst \<chi> \<langle>\<chi>\<rangle> xx"
intro: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_cring R S h"   and "UP_cring R" shows "UP_pre_univ_prop R S h"
multiplicative_function_def: fixes f :: "nat \<Rightarrow> 'b" shows "multiplicative_function f \<equiv> f 0 = (0::'b) \<and> f 1 = (1::'b) \<and> (\<forall>a b. 1 < a \<longrightarrow> 1 < b \<longrightarrow> coprime a b \<longrightarrow> f (a * b) = f a * f b)"
cbeta2_beta2: shows "(b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>) O (b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>) \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
iso_locally_ringed_spaces_axioms_def: fixes X :: "'c set"   and is_open\<^sub>X :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>X :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and Y :: "'e set"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'f set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and d :: "'f"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'f \<Rightarrow> 'd" shows "iso_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> homeomorphism X is_open\<^sub>X Y is_open\<^sub>Y f \<and> iso_sheaves_of_rings Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y (im_sheaf.im_sheaf X \<O>\<^sub>X f) (im_sheaf.im_sheaf_morphisms X \<rho>\<^sub>X f) b (im_sheaf.add_im_sheaf X add_str\<^sub>X f) (im_sheaf.mult_im_sheaf X mult_str\<^sub>X f) (im_sheaf.zero_im_sheaf X zero_str\<^sub>X f) (im_sheaf.one_im_sheaf X one_str\<^sub>X f) \<phi>\<^sub>f"
igAbsRenSTR_imp_igAbsCongUSTR: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igAbsRenSTR MOD" shows "igAbsCongUSTR MOD"
all_final_exceptD: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and Ts :: "'b set"   and t :: "'b" assumes "all_final_except s Ts"   and "not_final_thread s t" shows "t \<in> Ts"
valuate_scaleRat: fixes c :: "rat"   and lp :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "c *R lp \<lbrace> v \<rbrace> = c *R lp \<lbrace> v \<rbrace>"
Sigma_insert: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b set" shows "Sigma (insert x A) f = Pair x ` f x \<union> Sigma A f"
fBallI: fixes A :: "'j fset"   and P :: "'j \<Rightarrow> bool" assumes "\<And>x. x |\<in>| A \<Longrightarrow> P x" shows "fBall A P"
in_convex_hull_exchange: fixes a :: "'a"   and S :: "'a set"   and x :: "'a" assumes "a \<in> convex hull S"   and "x \<in> convex hull S" obtains b :: "'a" where "b \<in> S" and "x \<in> convex hull insert a (S - {b})"
steps: shows "\<Gamma>\<turnstile> ([c], [], s) \<rightarrow>\<^sup>* (c' # cs', css', t)"
cnf_of_encode_all_operator_effects_subset_cnf_of_encode_problem: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t) \<subseteq> cnf (\<Phi> \<Pi> t)"
tendsto_iff: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" shows "(f \<longlongrightarrow> l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
low_deg_terms_zero1: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and j :: "nat" assumes "pol_coeff S c" shows "polyn_expr R X (fst c + j) (ext_cf S j c) = X^\<^bsup>R j\<^esup> \<cdot>\<^sub>r polyn_expr R X (fst c) c"
linears_appendD_3: fixes xs :: "term list"   and ys :: "term list" assumes "linears (xs @ ys)" shows "fdisjnt (freess xs) (freess ys)"
switchAux: fixes y :: "NominalSequents.var"   and Xs :: "NominalSequents.var list"   and x :: "NominalSequents.var" assumes "y \<sharp> Xs" shows "[y;x]Xs = [(y, x)] \<bullet> Xs"
assms: shows "sys \<Turnstile>\<^bsub>pre\<^esub> I"
range_csingle: shows "range csingle \<subseteq> {A. A \<noteq> {}\<^sub>c}"
invertible: fixes u :: "'a set" assumes "u \<in> Partition" shows "quotient.invertible u"
wt_var_eq: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "Term.var"   and U :: "tys" shows "(E,dt\<Turnstile>In2 t\<Colon>U) = (\<exists>T. U = Inl T \<and> E,dt\<Turnstile>t\<Colon>=T)"
llength_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "llength (xs @\<^sub>l ys) = llength xs + llength ys"
assms: shows "pEquivalence_Checking.check_eqv r s"
apex_def: shows "\<mu>\<nu>.cod.apex \<equiv> C.dom \<mu>\<nu>.cod.leg0"
unique_favorites: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_unique_favorites agents alts R"   and "i \<in> agents" shows "favorites R i = {favorite R i}"
simps_15: fixes v :: "'b rexp" shows "le_rexp (Star v) One = False"
Range_mono: fixes r :: "('c \<times> 'd) set"   and s :: "('c \<times> 'd) set" assumes "r \<subseteq> s" shows "Range r \<subseteq> Range s"