hcju/isabelleps huggingface dataset:
cases: fixes a :: "('a, 'b, 'c) exp" assumes "final a" obtains v :: "'c val" where "a = Val v"   | aa :: "'c" where "a = Throw aa"
preprm_ext_compose_right: fixes P :: "('b \<times> 'b) list"   and S :: "('b \<times> 'b) list"   and \<pi> :: "('b \<times> 'b) list" assumes "P =p S" shows "preprm_compose P \<pi> =p preprm_compose S \<pi>"
Inf_aform_err_def: fixes p :: "nat"   and Rd :: "(real \<times> real pdevs) \<times> real" shows "Inf_aform_err p Rd = float_of (truncate_down p (Inf_aform' p (fst Rd) - \<bar>snd Rd\<bar>))"
homeomorphism_on_continuous: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphism_on S f" shows "continuous_on S f"
top_on_acom_def: fixes C :: "(char list \<Rightarrow> 'av) option acom"   and X :: "char list set" shows "top_on\<^sub>c C X = (\<forall>a\<in>set (annos C). top_on\<^sub>o a X)"
apx_empty: shows "Approx\<^sub>\<beta> {} = {}"
that: fixes f :: "'a \<Rightarrow> nat" assumes "\<forall>x\<in>S. \<forall>y\<in>S. (f x \<le> f y) = (x \<le> y)"   and "range f \<subseteq> {0..card S - 1}" shows "thesis"
l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
rtranclp_idemp: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
constant_functor_axioms: shows "constant_functor (\<cdot>\<^sub>J) (\<cdot>) S.unity"
blinding_of_prod_hash: fixes boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and rha :: "'c \<Rightarrow> 'd"   and bob :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'e \<Rightarrow> 'f" assumes "boa \<le> vimage2p rha rha (=)"   and "bob \<le> vimage2p rhb rhb (=)" shows "rel_prod boa bob \<le> vimage2p (map_prod rha rhb) (map_prod rha rhb) (=)"
rcs_fixed1: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c"   and h :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "h \<in> H" shows "H \<bullet>\<^bsub>G\<^esub> a = H \<bullet>\<^bsub>G\<^esub> (h \<cdot>\<^bsub>G\<^esub> a)"
take_lemma: fixes x :: "'a Box"   and y :: "'a Box" assumes "\<And>n. Box_take n\<cdot>x = Box_take n\<cdot>y" shows "x = y"
cases_scheme: fixes r :: "'a state_scheme" obtains inpt :: "Proc \<Rightarrow> InputsOrNi" and outpt :: "Proc \<Rightarrow> InputsOrNi" and disk :: "Disk \<Rightarrow> Proc \<Rightarrow> DiskBlock" and dblock :: "Proc \<Rightarrow> DiskBlock" and phase :: "Proc \<Rightarrow> nat" and disksWritten :: "Proc \<Rightarrow> Disk set" and blocksRead :: "Proc \<Rightarrow> Disk \<Rightarrow> BlockProc set" and allInput :: "InputsOrNi set" and chosen :: "InputsOrNi" and more :: "'a" where "r = \<lparr>inpt = inpt, outpt = outpt, disk = disk, dblock = dblock, phase = phase, disksWritten = disksWritten, blocksRead = blocksRead, allInput = allInput, chosen = chosen, \<dots> = more\<rparr>"
concat_conv_foldr: fixes xss :: "'e list list" shows "concat xss = foldr (@) xss []"
lookup_simps_1: fixes e :: "('f, 'g, 'h) env" shows "lookup e [] = Some e"
assms_2: fixes n :: "nat" shows "x n \<in> sfis (u n) M"
simps_2: fixes w :: "nat"   and w\<^sub>b :: "nat"   and b :: "'a"   and w\<^sub>c :: "nat"   and c :: "'a"   and a :: "'a" shows "sibling (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a = (if a = b then c else if a = c then b else a)"
float_distinct_5: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_infinity a \<and> is_normal a)"
substeq: fixes \<phi> :: "fml"   and \<nu> :: "variable \<Rightarrow> real"   and I :: "interp" assumes "usubstappf \<sigma> Denotational_Semantics.allvars \<phi> \<noteq> undeff" shows "(\<nu> \<in> fml_sem I (the (usubstappf \<sigma> Denotational_Semantics.allvars \<phi>))) = (\<nu> \<in> fml_sem (USubst.adjoint \<sigma> I \<omega>) \<phi>)"
abs_e2: shows "\<bar>e2\<bar> \<le> snd (trunc_bound_pdevs p (scaleR_pdevs alpha (snd X)))"
pr_pri_agrk_items: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and x :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Pri_AgrK x \<notin> U" shows "items (insert (Pri_AgrK x) (A \<union> spies evs)) = insert (Pri_AgrK x) (items (A \<union> spies evs))"
locally_ringed_space_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" shows "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> locally_ringed_space_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str"
hyps_5: assumes "list_all is_Delete (unlabel A)" shows "list_all is_Delete (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
distinct_1: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'c"   and x21 :: "'a"   and x22 :: "bool" shows "NewThread x11 x12 x13 \<noteq> ThreadExists x21 x22"
alphabet_size: fixes x :: "'x"   and idx :: "'n" shows "(x \<in> set (alphabet idx)) = (size x = idx)"
to_tree_order_parent_child_rel: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and child_ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to" shows "((ptr, child_ptr) \<in> (parent_child_rel h)\<^sup>*) = (child_ptr \<in> set to)"
intro: fixes charts1 :: "('c, 'd) chart set"   and k :: "enat"   and charts2 :: "('e, 'f) chart set" assumes "c_manifold charts1 k"   and "c_manifold charts2 k" shows "c_manifolds k charts1 charts2"
minus_minus: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> (\<ominus>\<^bsub>G\<^esub> x) = x"
coeff_Var2: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" shows "coeff (Abstract_Linear_Poly.Var x) y = 0"
case_eq_if: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('h, 'g) tllist \<Rightarrow> 'f"   and tllist :: "('h, 'g) tllist" shows "(case tllist of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = (if is_TNil tllist then f1 (terminal tllist) else f2 (thd tllist) (ttl tllist))"
a: fixes n :: "nat"   and x :: "'a" shows "\<bar>(F (n + 1) x - F n (T x)) * indicat_real A x\<bar> \<le> \<bar>f x\<bar>"
linear_def: fixes f :: "'b \<Rightarrow> 'c" shows "linear f \<equiv> Vector_Spaces.linear (*\<^sub>R) (*\<^sub>R) f"
Says_Server_message_form: fixes B :: "agent"   and NA :: "msg"   and X :: "msg"   and NB :: "msg"   and K :: "nat"   and evs :: "event list" assumes "Says Server B \<lbrace>NA, X, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "K \<notin> range shrK \<and> (\<exists>i. NA = Nonce i) \<and> (\<exists>j. NB = Nonce j)"
distr_bind: fixes N :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and K :: "'b measure"   and f :: "'b \<Rightarrow> 'd"   and R :: "'d measure" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}"   and "f \<in> K \<rightarrow>\<^sub>M R" shows "distr (M \<bind> N) R f = M \<bind> (\<lambda>x. distr (N x) R f)"
interleave_size: fixes s :: "'a event list"   and t :: "'a event list"   and C :: "'a event set"   and u :: "'a event list" assumes "s setinterleaves ((t, u), C)" shows "length s = length t + length u - length (filter (\<lambda>x. x \<in> C) t)"
countable_network_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "countable_network \<Delta> \<equiv> (countable \<^bold>E\<^bsub>\<Delta>\<^esub> \<and> source \<Delta> \<noteq> sink \<Delta>) \<and> (\<forall>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<longrightarrow> capacity \<Delta> e = 0) \<and> (\<forall>e. capacity \<Delta> e \<noteq> \<top>)"
simps_159: fixes x7 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Rfork x7) = RID\<^sub>E x7"
properfactor_lless: fixes G :: "('c, 'd) monoid_scheme" shows "properfactor G = (\<sqsubset>\<^bsub>division_rel G\<^esub>)"
fresh_fn: fixes S :: "'a set"   and x :: "'a"   and A :: "'a trm"   and T :: "PreSimplyTyped.type" assumes "finite S" shows "\<exists>y B. y \<notin> S \<and> B = [y \<leftrightarrow> x] \<cdot> A \<and> Fn x T A = Fn y T B"
ih_4: shows "InvariantCFalse (getConflictFlag state') (getM state') (getC state')"
Cast: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and T :: "ty"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:v:. \<lambda>s.. abupd (raise_if (\<not> G,s\<turnstile>v fits T) ClassCast) .; Q\<leftarrow>In1 v}" shows "G,A\<turnstile>{Normal P} Cast T e-\<succ> {Q}"
Seq_iff_app: fixes s :: "hf"   and d :: "hf"   and x :: "hf"   and y :: "hf" assumes "Seq s d"   and "x \<^bold>\<in> d" shows "(\<langle>x, y\<rangle> \<^bold>\<in> s) = (app s x = y)"
triangle: fixes s :: "'v"   and v :: "'v"   and u :: "'v" shows "\<delta> s v \<le> \<delta> s u + w (u, v)"
CBasis_zero: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "zero \<notin> CBasis"
in_gamma_update: fixes s :: "char list \<Rightarrow> int"   and S :: "'av Abs_State.st"   and i :: "int"   and a :: "'av"   and x :: "char list" assumes "s \<in> \<gamma>\<^sub>f S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>\<^sub>f (update S x a)"
T_inorder2: fixes t :: "'b tree"   and xs :: "'b list" shows "T_inorder2 t xs = 2 * size t + 1"
build_rel_def: fixes \<alpha> :: "'b \<Rightarrow> 'd"   and I :: "'b \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
union_vars_poly_alt_def: fixes xs :: "(char list list \<times> int) list"   and \<V> :: "char list set" shows "union_vars_poly xs \<V> = \<V> \<union> vars_llist xs"
List_rec_CONS: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set"   and c :: "'c"   and h :: "('b, unit) Old_Datatype.node set \<Rightarrow> ('b, unit) Old_Datatype.node set \<Rightarrow> 'c \<Rightarrow> 'c" assumes "M \<in> sexp"   and "N \<in> sexp" shows "List_rec (CONS M N) c h = h M N (List_rec N c h)"
frequently_ex: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "frequently P F" shows "\<exists>x. P x"
measurable_prob_algebraD: fixes f :: "'b \<Rightarrow> 'c measure"   and N :: "'b measure"   and M :: "'c measure" assumes "f \<in> N \<rightarrow>\<^sub>M prob_algebra M" shows "f \<in> N \<rightarrow>\<^sub>M subprob_algebra M"
wtE_pickE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "'univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> intT (tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al) \<and> (\<forall>i<length xl. pickE xl al (xl ! i) = al ! i)"
floorlog_leI: fixes x :: "nat"   and b :: "nat"   and w :: "nat" assumes "x < b ^ w"   and "0 \<le> w"   and "1 < b" shows "floorlog b x \<le> w"
MM4: fixes F :: "bool" shows "\<turnstile> \<box>#F = #F"
inter_eq_subsetI: fixes S :: "'e set"   and S' :: "'e set"   and A :: "'e set"   and B :: "'e set" assumes "S \<subseteq> S'"   and "A \<inter> S' = B \<inter> S'" shows "A \<inter> S = B \<inter> S"
var_not_in_Const: fixes var :: "nat"   and x :: "real" shows "var \<notin> vars (Const x)"
simps: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'d"   and a3 :: "('e \<Rightarrow> 'f) set"   and a4 :: "'d"   and a5 :: "('e \<Rightarrow> 'f) set" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>* \<langle>a4, a5\<rangle> = ((\<exists>A l Z. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l \<and> a5 = Z) \<or> (\<exists>A l Z l' Z' l'' Z''. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l'' \<and> a5 = Z'' \<and> A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle> \<and> A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>))"
succ_tran_mono: fixes N :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and v :: "'a" assumes "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E" shows "succ_tran \<lparr>nodes = N, edges = E'\<rparr> v \<subseteq> succ_tran \<lparr>nodes = N, edges = E\<rparr> v"
assms: shows "x \<squnion> lift x u = x \<squnion> lift x v"
integrable_altD_1: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set"   and a :: "'c"   and b :: "'c" assumes "f integrable_on s" shows "(\<lambda>x. if x \<in> s then f x else (0::'d)) integrable_on cbox a b"
prems_2: shows "item_\<beta> x = \<delta> @ \<omega>"
partGen_def: fixes P :: "'a set set" shows "partGen P \<equiv> {gen P I |I. I \<in> P}"
generate_incl: fixes H :: "(nat \<times> nat) set" assumes "H \<subseteq> carrier S'" shows "generate S' H \<subseteq> carrier S'"
f: shows "2\<^sup>2 * 5 * 11 = 220"
eqpoll_imp_lepoll: fixes A :: "'c set"   and B :: "'d set" assumes "A \<approx> B" shows "A \<lesssim> B"
DomainPI: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "r a b" shows "Domainp r a"
insert_3: fixes M :: "('a, 'b) sq_matrix" shows "Square_Matrix.det (upd_cols M T (\<lambda>i. sum (a i) S)) = (\<Sum>f\<in>T \<rightarrow>\<^sub>E S. Square_Matrix.det (upd_cols M T (\<lambda>i. a i (f i))))"
plus_trilean_commutative: fixes a :: "trilean"   and b :: "trilean" shows "a \<or>? b = b \<or>? a"
transp_relcompp_less_eq: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "transp r" shows "r OO r \<le> r"
Mapping_lookup_upd_set: fixes m :: "('b, 'a) mapping"   and f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and a :: "'b" shows "Mapping.lookup (upd_set m f X) a = (if a \<in> X then Some (f a) else Mapping.lookup m a)"
assms: shows "finite (\<Union> (block_nominals ` S))"
map_filter_M_pure: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> ('b, 'c, 'd option) prog"   and h :: "'b" assumes "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_filter_M f xs) h"
is_path_undir_memb: fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_path_undir G v p v'" shows "v \<in> V \<and> v' \<in> V"
elem_carrier: fixes F :: "'b set set"   and T :: "'b set set"   and A :: "'b set" assumes "Topology.filter F T"   and "A \<in> F" obtains "A \<subseteq> carrier\<^bsub>T\<^esub>" and "A \<noteq> {}"
abs_trieP_Lf: fixes t :: "trieP" shows "(abs_trieP t = Lf) = (t = LfP)"
init: shows "init users \<Midarrow>local.bogus\<Rightarrow> root"
eint_ord_simps_4: fixes q :: "eint" shows "(q < \<infinity>) = (q \<noteq> \<infinity>)"
assms: shows "f \<in> borel_measurable (nat_filtration n)"
not_iff_asym: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(\<not> x \<sqsubseteq> y) = (y \<sqsubset> x)"
i_set0_ind_i_set: fixes I :: "nat set" assumes "i_set_indp I" shows "i_set0_indp I"
qItem_versus_item_simps_2: fixes fAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs \<Rightarrow> 'f"   and Aa :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "fAbs respectsP alphaAbsGood"   and "qGoodAbs Aa" shows "Equiv_Relation2.univ fAbs (asAbs Aa) = fAbs Aa"
conforms_init_lvars_lemma: fixes G :: "prog"   and P :: "pname"   and sig :: "sig"   and mh :: "mhead"   and s :: "st"   and pvs :: "val list"   and pTsa :: "ty list"   and lvars :: "(vname \<times> ty) list" assumes "wf_prog G"   and "wf_mhead G P sig mh"   and "list_all2 (conf G s) pvs pTsa"   and "G\<turnstile>pTsa[\<preceq>]parTs sig" shows "G,s\<turnstile>[pars mh [\<mapsto>] pvs][\<sim>\<Colon>\<preceq>]table_of lvars(pars mh [\<mapsto>] parTs sig)"
constructive_security_sim_obsf: shows "constructive_security_sim_obsf real1 ideal1 sim1 (\<I>_real1_core \<oplus>\<^sub>\<I> \<I>_real1_rest) (\<I>_common1_core \<oplus>\<^sub>\<I> \<I>_common1_rest) (absorb \<A> (obsf_converter (fused_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) adv1"
ext: shows "(r \<in> Ax \<or> r \<in> upRules \<or> r \<in> modRules2) \<and> extendRule S r = (Ps,  \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>)"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b msllist \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b msllist" shows "(case MSLCons x21 x22 of MSLNil \<Rightarrow> f1 | MSLCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
MGU_nil: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "MGU Var s t = (s = t)"
metric_uniform_limit_imp_uniform_limit: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and a :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'e"   and b :: "'c \<Rightarrow> 'e" assumes "uniform_limit S f a F"   and "\<forall>\<^sub>F x in F. \<forall>y\<in>S. dist (g x y) (b y) \<le> dist (f x y) (a y)" shows "uniform_limit S g b F"
Finite_Preflow_axioms: shows "Finite_Preflow c s t f"
map_of_zip_nth: fixes xs :: "'a list"   and ys :: "'b list"   and i :: "nat" assumes "length xs = length ys"   and "distinct xs"   and "i < length ys" shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
adds_canc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and u :: "'c"   and t :: "'c" assumes "class.comm_powerprod plus minus zero" shows "plus.adds plus (plus s u) (plus t u) = plus.adds plus s t"
fake_monotone: fixes c :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and bad' :: "agent set"   and IK' :: "msg set"   and chan' :: "chan set" assumes "c \<in> fake bad IK chan"   and "bad \<subseteq> bad'"   and "IK \<subseteq> IK'"   and "chan \<subseteq> chan'" shows "c \<in> fake bad' IK' chan'"
progress_simps_16: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.MatchP I r) j = progress_regex \<sigma> P r j"
N_summable: fixes n :: "int" shows "summable (\<lambda>k. if k = 0 \<or> int k dvd n then 0 else 1 / q ^ k)"
max_def: fixes a :: "'a list"   and b :: "'a list" shows "suffix_order.max a b = (if suffix a b then b else a)"
exhaust: fixes y :: "color" obtains "y = Red"   | "y = Black"
polyadd_different_degreen: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "degreen p m \<noteq> degreen q m"   and "m \<le> min n0 n1" shows "degreen (p +\<^sub>p q) m = max (degreen p m) (degreen q m)"
cod_eqI: fixes b :: "'b"   and f :: "'b" assumes "D.ide b"   and "D.seq b f" shows "D.cod f = b"
MIN_SET_ELIM': fixes P :: "'c set"   and Q :: "'c \<Rightarrow> bool" assumes "finite P"   and "P \<noteq> {}"   and "\<forall>x. (\<forall>y. y \<in> P \<longrightarrow> x \<le> y) \<and> x \<in> P \<longrightarrow> Q x" shows "Q (Min P)"
S2: shows "nat_to_sch n = Comp_op (nat_to_sch (c_fst (c_snd n))) (nat_to_sch (c_snd (c_snd n)))"
group_l_invI: assumes "\<And>x. x \<in> carrier G2 \<Longrightarrow> \<exists>y\<in>carrier G2. y \<otimes>\<^bsub>G2\<^esub> x = \<one>\<^bsub>G2\<^esub>" shows "Group.group G2"
rep_eq: fixes x :: "'a" shows "measure_pmf (return_pmf x) = return (count_space UNIV) x"
apply_self_in_orbit: fixes f :: "'a perm"   and a :: "'a" shows "f \<langle>$\<rangle> a \<in> orbit f a"
domain_eq: shows "Domainp (pcr_restrictedf (=)) = (\<lambda>xs. length xs < 3)"
f_shrink_last_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b list" shows "xs \<up> (n * k) \<div>\<^sub>l k = xs \<div>\<^sub>l k \<up> n"
fresh_swap_exchange1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "good X" shows "fresh xs x2 (X #[x1 \<and> x2]_xs) = fresh xs x1 X"
composition_conseq2t: fixes t :: "fterm"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" shows "t \<cdot>\<^sub>t \<sigma>\<^sub>1 \<cdot>\<^sub>t \<sigma>\<^sub>2 = t \<cdot>\<^sub>t (\<sigma>\<^sub>1 \<cdot> \<sigma>\<^sub>2)"
hmset_of_enat_inject: fixes m :: "enat"   and n :: "enat" shows "(hmset_of_enat m = hmset_of_enat n) = (m = n)"
outgoing_transitions_deterministic: fixes e :: "((nat \<times> nat) \<times> transition) fset" assumes "\<forall>s. \<forall>((s1, s2), t)|\<in>|outgoing_transitions e s. \<forall>((s1', s2'), t')|\<in>|outgoing_transitions e s. s2 \<noteq> s2' \<or> t \<noteq> t' \<longrightarrow> Label t = Label t' \<longrightarrow> \<not> choice t t'" shows "deterministic e"
d_1: fixes p :: "nat"   and x :: "int" shows "root_int p x \<equiv> if p = 0 then [] else if x = 0 then [0] else let e = even p; s = sgn x; x' = \<bar>x\<bar> in if x < 0 \<and> e then [] else case root_int_main p x' of (y, True) \<Rightarrow> if e then [y, - y] else [s * y] | (y, False) \<Rightarrow> []"
intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "AX0p ((F::'b formula) \<^bold>\<rightarrow> (G::'b formula) \<^bold>\<rightarrow> F)"
sys_block_sizes_def: shows "sys2.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>'}"
Bet_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "Bet A B C \<and> Bet C B A"
surjective: fixes r :: "'a State_scheme" shows "r = \<lparr>getSATFlag = getSATFlag r, getF = getF r, getM = getM r, getConflictFlag = getConflictFlag r, getConflictClause = getConflictClause r, getQ = getQ r, getReason = getReason r, getWatch1 = getWatch1 r, getWatch2 = getWatch2 r, getWatchList = getWatchList r, getC = getC r, getCl = getCl r, getCll = getCll r, getCn = getCn r, \<dots> = more r\<rparr>"
eventually_imp_frequently: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "ultrafilter F"   and "frequently P F" shows "eventually P F"
rep_eq: fixes x :: "'c dense"   and xa :: "'c dense" shows "(x \<le> xa) = (Rep_dense x \<le> Rep_dense xa)"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "arr f"   and "local.dom f = a" shows "f \<cdot> a = f"
rcs_subset_elem: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e"   and x :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet>\<^bsub>G\<^esub> a" shows "x \<in> carrier G"
quicksort_vars_def: fixes x :: "nat"   and y :: "nat"   and z :: "char list list" shows "quicksort_vars x y z = quicksort_ref (\<le>) id (x, y, z)"
NegPos_map_append: fixes C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and bs :: "'c negation_type list" shows "NegPos_map C (as @ bs) = NegPos_map C as @ NegPos_map C bs"
integrable_real_indicator: fixes A :: "'d set"   and M :: "'d measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "integrable M (indicat_real A)"