hcju/isabelleps huggingface dataset:
Posix_Plus2: fixes s :: "'b list"   and r2 :: "'b rexp"   and v :: "'b val"   and r1 :: "'b rexp" assumes "s \<in> r2 \<rightarrow> v"   and "s \<notin> lang r1" shows "s \<in> Plus r1 r2 \<rightarrow> val.Right v"
proj_Snoc: fixes qs :: "'a list"   and S :: "'a set"   and as :: "'a list"   and a :: "'a" assumes "Lxy qs S = as @ [a]"   and "a \<in> S" obtains pre :: "'a list" and suf :: "'a list" where "qs = pre @ [a] @ suf" and "\<And>x. x \<in> S \<Longrightarrow> x \<notin> set suf" and "Lxy pre S = as"
extend_to_S: fixes p :: "proj2"   and q :: "proj2" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S" shows "\<exists>r\<in>S. B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r"
read_only_unowned: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S> = {}"
order_impl_l: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a l\<rightarrow> b = (1::'a)"
g_def: shows "g = (\<lambda>x. gv x + ereal (real_cond_exp M Invariants (u 1) x))"
cscc_outer_invar_def: fixes G :: "('a, 'b) graph_rec_scheme" assumes "fr_graph G" shows "fr_graph.cscc_outer_invar G \<equiv> \<lambda>it (l, D). cscc_outer_invar_loc G it l D"
cong_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "cong a c b = (b dvd a - c)"
apex_def: shows "\<mu>.apex \<equiv> C.dom \<mu>.leg0"
liftrT_assoc_Some: fixes fs :: "(char list \<times> POPLmarkRecord.type) list"   and l :: "char list"   and T :: "POPLmarkRecord.type"   and n :: "nat"   and k :: "nat" assumes "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>T\<rfloor>" shows "\<up>\<^sub>r\<^sub>\<tau> n k fs\<langle>l\<rangle>\<^sub>? = \<lfloor>\<up>\<^sub>\<tau> n k T\<rfloor>"
simps: fixes a1 :: "'d list"   and a2 :: "'e list"   and r :: "('d \<times> 'e) set" shows "((a1, a2) \<in> listrel r) = (a1 = [] \<and> a2 = [] \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> (x, y) \<in> r \<and> (xs, ys) \<in> listrel r))"
c_x_prop: fixes x :: "'a" shows "1\<^sub>\<pi> \<cdot> x = 1\<^sub>\<pi>"
set_return_spmf: fixes x :: "'c" shows "set_spmf (return_spmf x) = {x}"
except_Int_keys: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and U :: "'c set" shows "except p (U \<inter> keys p) = except p U"
simps_7: fixes p :: "'e" shows "atom_com (Call p) = False"
transitive: fixes Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel :: "(pi \<times> pi) set"   and Rel'' :: "(pi \<times> pi) set"   and P :: "pi" assumes "Q \<leadsto>\<^sup>^<Rel'> R"   and "eqvt Rel"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel''> R"
leof_ASSERT_rule: fixes \<Phi> :: "bool"   and m :: "'b nres"   and m' :: "'b nres" assumes "\<Phi> \<Longrightarrow> m \<le>\<^sub>n m'" shows "m \<le>\<^sub>n ASSERT \<Phi> \<bind> (\<lambda>_. m')"
assms_2: shows "xs \<noteq> ys \<or> x \<notin> {y1, y2}"
lemma_3_7_i: fixes F :: "'a set" assumes "F \<in> filters" shows "F = {a. cong_r F a (1::'a)}"
le_greater_neq_conv: fixes n :: "'a"   and a :: "'a" shows "(n \<le> a) = (\<forall>x>a. n \<noteq> x)"
Rep_nat_int: fixes x :: "nat_int" shows "Rep_nat_int x \<in> {S. \<exists>m n. {m..n} = S}"
space_point_measure: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" shows "space (point_measure A f) = A"
strengthen_pre: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<forall>s. P s \<le> P' s"   and "\<turnstile>\<^sub>2 {P} c {Q}" shows "\<turnstile>\<^sub>2 {P'} c {Q}"
\<R>_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   | "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
assms_1: shows "\<not> \<gamma> (Trans s a ou s')"
restrictP_commute: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" shows "r |` A |` B = r |` B |` A"
rep_nat_term_inj: fixes x :: "'a"   and y :: "'a" assumes "rep_nat_term x = rep_nat_term y" shows "x = y"
c_assoc_value_is_pr: shows "c_assoc_value \<in> PrimRec2"
m\<tau>diverge_simulation1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "\<tau>trsys.\<tau>diverge (multithreaded_base.redT final1 r1 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move1) s1" shows "\<tau>trsys.\<tau>diverge (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2) s2"
interchange: fixes \<nu> :: "'d"   and \<mu> :: "'d"   and \<tau> :: "'d"   and \<sigma> :: "'d" assumes "D.seq \<nu> \<mu>"   and "D.seq \<tau> \<sigma>" shows "\<nu> \<cdot>\<^sub>D \<mu> \<star>\<^sub>D \<tau> \<cdot>\<^sub>D \<sigma> = (\<nu> \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<sigma>)"
that: fixes pre :: "'a list"   and suf :: "'a list" assumes "qs = pre @ [a] @ suf"   and "\<And>x. x \<in> S \<Longrightarrow> x \<notin> set pre"   and "Lxy suf S = as" shows "thesis"
set_child_nodes_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> Shadow_DOM.set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
indspace_basis: fixes vs :: "'v list"   and hs :: "'g list" assumes "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) vs"   and "Supgroup.is_rcoset_replist G hs" shows "set (concat (negHorbit_list hs induced_vector vs)) \<subseteq> indV \<and> indV = R_scalar_mult.RSpan UNIV (\<currency>\<currency>) (concat (negHorbit_list hs induced_vector vs)) \<and> R_scalar_mult.R_lin_independent UNIV (\<currency>\<currency>) (concat (negHorbit_list hs induced_vector vs))"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map d)"
obtain_function: fixes A :: "'c set"   and M :: "'d multiset" assumes "finite A"   and "size M = card A" shows "\<exists>f. image_mset f (mset_set A) = M"
card_le_if_inj_on_rel: fixes B :: "'c set"   and A :: "'d set"   and r :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite B"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b \<in> B \<and> r a b"   and "\<And>a1 a2 b. \<lbrakk>a1 \<in> A; a2 \<in> A; b \<in> B; r a1 b; r a2 b\<rbrakk> \<Longrightarrow> a1 = a2" shows "card A \<le> card B"
not_zeroD: fixes x :: "'c"   and u :: "'c \<Rightarrow> real" assumes "\<not> zero x ([u]\<^sub>\<R>)"   and "u \<in> V"   and "x \<in> \<X>" shows "0 < u x"
eval_permutation_of_list_2: fixes x :: "'a"   and x' :: "'a"   and y :: "'a"   and xs :: "('a \<times> 'a) list" assumes "x = x'" shows "permutation_of_list ((x', y) # xs) x = y"
Smith_1x2': shows "\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2' A)"
induct: fixes x1 :: "'b formula multiset"   and x2 :: "'b formula multiset"   and x3 :: "nat"   and P :: "'b formula multiset \<Rightarrow> 'b formula multiset \<Rightarrow> nat \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2 \<down> x3"   and "\<And>\<Gamma> \<Delta> n. \<bottom> \<in># \<Gamma> \<Longrightarrow> P \<Gamma> \<Delta> (Suc n)"   and "\<And>k \<Gamma> \<Delta> n. \<lbrakk>Atom k \<in># \<Gamma>; Atom k \<in># \<Delta>\<rbrakk> \<Longrightarrow> P \<Gamma> \<Delta> (Suc n)"   and "\<And>\<Gamma> F \<Delta> n. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta> \<down> n; P \<Gamma> (F, \<Delta>) n\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> F, \<Gamma>) \<Delta> (Suc n)"   and "\<And>F \<Gamma> \<Delta> n. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta> \<down> n; P (F, \<Gamma>) \<Delta> n\<rbrakk> \<Longrightarrow> P \<Gamma> (\<^bold>\<not> F, \<Delta>) (Suc n)"   and "\<And>F G \<Gamma> \<Delta> n. \<lbrakk>F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n; P (F, G, \<Gamma>) \<Delta> n\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G, \<Gamma>) \<Delta> (Suc n)"   and "\<And>\<Gamma> F \<Delta> n G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta> \<down> n; P \<Gamma> (F, \<Delta>) n; \<Gamma> \<Rightarrow> G, \<Delta> \<down> n; P \<Gamma> (G, \<Delta>) n\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<and> G, \<Delta>) (Suc n)"   and "\<And>F \<Gamma> \<Delta> n G. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta> \<down> n; P (F, \<Gamma>) \<Delta> n; G, \<Gamma> \<Rightarrow> \<Delta> \<down> n; P (G, \<Gamma>) \<Delta> n\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G, \<Gamma>) \<Delta> (Suc n)"   and "\<And>\<Gamma> F G \<Delta> n. \<lbrakk>\<Gamma> \<Rightarrow> F, G, \<Delta> \<down> n; P \<Gamma> (F, G, \<Delta>) n\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<or> G, \<Delta>) (Suc n)"   and "\<And>\<Gamma> F \<Delta> n G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta> \<down> n; P \<Gamma> (F, \<Delta>) n; G, \<Gamma> \<Rightarrow> \<Delta> \<down> n; P (G, \<Gamma>) \<Delta> n\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G, \<Gamma>) \<Delta> (Suc n)"   and "\<And>F \<Gamma> G \<Delta> n. \<lbrakk>F, \<Gamma> \<Rightarrow> G, \<Delta> \<down> n; P (F, \<Gamma>) (G, \<Delta>) n\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<rightarrow> G, \<Delta>) (Suc n)" shows "P x1 x2 x3"
perm_bij: fixes pi :: "(name \<times> name) list"   and x :: "'c"   and y :: "'c" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
strong_reduction_simulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal" shows "strong_reduction_simulation (Rel\<^sup>*) Cal"
elims: fixes x :: "'b"   and xa :: "nat"   and y :: "nat option" assumes "oldest_token x xa = y" obtains q :: "'b" and n :: "nat" where "x = q" and "xa = n" and "y = (if configuration q n \<noteq> {} then Some (Min (configuration q n)) else None)"
repU_def: fixes w :: "32 Word.word"   and r :: "real" shows "w \<equiv>\<^sub>U r \<equiv> \<exists>r'\<ge>r. w \<equiv>\<^sub>E r'"
modelClause_def: fixes valuation :: "Literal list"   and clause :: "Literal list" shows "model valuation clause \<equiv> consistent valuation \<and> clauseTrue clause valuation"
plus_oracle_split: fixes P :: "(('a + 'b) \<times> 'c) spmf \<Rightarrow> bool"   and left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b \<times> 'c) spmf"   and s :: "'c"   and lr :: "'d + 'e" shows "P ((left \<oplus>\<^sub>O right) s lr) = ((\<forall>x. lr = Inl x \<longrightarrow> P (map_spmf (apfst Inl) (left s x))) \<and> (\<forall>y. lr = Inr y \<longrightarrow> P (map_spmf (apfst Inr) (right s y))))"
ex_subseq: fixes s :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono s" shows "\<exists>r'. strict_mono r' \<and> P n (s \<circ> r')"
0: shows "({dual\<^sub>s\<^sub>t \<A>}, []) \<Rightarrow>\<^sup>\<bullet>\<^sup>* ({}, \<A>)"
bal_log_def: fixes t :: "'b tree" shows "bal_log t = (int (height t) \<le> \<lceil>c * log 2 (real (size1 t))\<rceil>)"
index_row_1: fixes i :: "nat"   and A :: "'d mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Matrix.row A i $v j = A $$ (i, j)"
singleCombinatorsStart: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" assumes "singleCombinators (x # xs)" shows "singleCombinators [x]"
reachable_step: fixes a :: "'a"   and b :: "'a" assumes "reachable a"   and "a \<rightarrow> b" shows "reachable b"
sdle: fixes T :: "'a set"   and v :: "'a" assumes "T \<in> \<C>"   and "v \<in> T" shows "setdist {\<V> T} S \<le> 2 * setdist {v} S"
foo: shows "h' \<turnstile> map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes_h3) \<rightarrow>\<^sub>r [cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr] # foo"
longest_common_prefix_max_prefix: fixes ps :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "prefix ps xs"   and "prefix ps ys" shows "prefix ps (longest_common_prefix xs ys)"
face_cycle_succ_in_arcsI: fixes a :: "'b" assumes "a \<in> arcs H" shows "H.face_cycle_succ a \<in> arcs H"
lotteries_on_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "lotteries_on A \<subseteq> lotteries_on B"
closed_def: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
normalize_quotE: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" obtains d :: "'a" where "a = fst (normalize_quot (a, b)) * d" and "b = snd (normalize_quot (a, b)) * d" and "d dvd a" and "d dvd b" and "d \<noteq> (0::'a)"
plossless_converter_1: shows "plossless_converter \<I> \<I>' conv"
reachable_redef: shows "reachable (set (map (\<lambda>\<nu>. (q\<^sub>0, \<nu>, \<delta> q\<^sub>0 \<nu>)) \<Sigma>)) = reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"
col_def: shows "col \<equiv> map_fun to_fun id (\<lambda>M j. \<chi>i. M i j)"
not_integrable_integral: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'c set" assumes "\<not> f integrable_on i" shows "integral i f = (0::'d)"
devBC_def: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb" shows "Blockchain.devBC active cmp bc mining t n nid n' \<equiv> if \<exists>nid'. (n', nid') \<in> Blockchain.his active cmp bc mining t n nid then Some (THE nid'. (n', nid') \<in> Blockchain.his active cmp bc mining t n nid) else None"
prems_5: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
A: shows "Mod_Type_Connect.from_hma\<^sub>m A \<in> carrier_mat CARD('m) CARD('n)"
InitNonObject\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "nat exp prog"   and D :: "char list"   and r :: "fdecl list \<times> nat exp mdecl list"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "nat exp"   and h :: "heap"   and l :: "val list"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C \<noteq> Object"   and "class P C = \<lfloor>(D, r)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (D # C # Cs,False) \<leftarrow> e,(h, l, sh')\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
elapsed_0D: fixes c :: "'c"   and u :: "'c \<Rightarrow> 'b"   and u' :: "'c \<Rightarrow> 'b" assumes "c \<in> \<X>"   and "elapsed u u' \<le> (0::'b)" shows "u' c - u c \<le> (0::'b)"
P1Q1: shows "P1 \<approx>\<cdot> Q1"
less_eq_set_def: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<le> (\<lambda>x. x \<in> B))"
delta_conv_steps: fixes A :: "'a \<times> ('b \<Rightarrow> 'a \<Rightarrow> 'a set) \<times> ('a \<Rightarrow> bool)"   and w :: "'b list"   and p :: "'a" shows "delta A w p = {q. (p, q) \<in> steps A w}"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and p :: "'psym"   and Tl :: "'fsym trm list" shows "Ik.satA \<xi> (Pr p Tl) = Ik_intP p (map (Ik.int \<xi>) Tl)"
pbeta_Call: fixes t :: "sterm"   and t' :: "sterm"   and u :: "sterm"   and u' :: "sterm"   and l :: "Label" assumes "t => t'"   and "u => u'"   and "lc t"   and "lc u" shows "Call t l u => Call t' l u'"
to_bl_eq: fixes w :: "'b word" shows "to_bl w = bin_to_bl LENGTH('b) (uint w)"
converse_rtranclp_induct_sbh_steps: fixes r :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'b \<times> 'c \<times> 'd \<Rightarrow> bool"   and ts :: "'b"   and m :: "'c"   and \<S> :: "'d"   and ts' :: "'b"   and m' :: "'c"   and \<S>' :: "'d"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ts, m, \<S>) (ts', m', \<S>')"   and "P ts' m' \<S>'"   and "\<And>a aa b ab ac ba. \<lbrakk>r (a, aa, b) (ab, ac, ba); r\<^sup>*\<^sup>* (ab, ac, ba) (ts', m', \<S>'); P ab ac ba\<rbrakk> \<Longrightarrow> P a aa b" shows "P ts m \<S>"
Ico_subset_finite: fixes i :: "nat set"   and a :: "nat"   and b :: "nat" assumes "i \<subseteq> {a..<b}" shows "finite i"
monad_fail_stateT': assumes "monad_fail Monad_Overloading.return Monad_Overloading.bind fail" shows "monad_fail Monad_Overloading.return Monad_Overloading.bind fail"
isos_compose: fixes f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "CC.iso f"   and "CC.iso f'"   and "CC.seq f' f" shows "CC.iso (CC.comp f' f)"
assms_4: shows "C = resolvent_upon P1 P2 A"
rtc_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z + x ; y \<le> y \<longrightarrow> x\<^sup>\<star> ; z \<le> y"
\<a>_Nil: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "finite X"   and "ps\<^sub>+ = []" shows "\<a> ps = 0"
antisym: fixes b :: "'b list"   and a :: "'b list" assumes "??.Sublist.prefix b a"   and "??.Sublist.prefix a b" shows "a = b"
quotient_of_Fract: fixes a :: "int"   and b :: "int" shows "quotient_of (Rat.Fract a b) = Rat.normalize (a, b)"
nilTrans_1: fixes \<Psi> :: "'b"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> \<zero> \<longmapsto> Rs" shows "False"
perp_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "B A Perp D C"
eq_adm: fixes f :: "'b \<Rightarrow> Value"   and g :: "'b \<Rightarrow> Value"   and a :: "Arity" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. eq a (f x) (g x))"
add_mult_less_add_mult: fixes x :: "V"   and y :: "V"   and \<beta> :: "V"   and \<mu> :: "V"   and \<alpha> :: "V"   and \<nu> :: "V" assumes "x < y"   and "x \<in> elts \<beta>"   and "y \<in> elts \<beta>"   and "\<mu> \<in> elts \<alpha>"   and "\<nu> \<in> elts \<alpha>"   and "Ord \<alpha>"   and "Ord \<beta>" shows "\<alpha> * x + \<mu> < \<alpha> * y + \<nu>"
not_arr_null: shows "\<not> Left_a.arr Left_a.null"
cupcake_match_resultE: fixes cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and e :: "exp0"   and p :: "pat"   and env' :: "(char list \<times> v) list" assumes "cupcake_match_result cenv v0 pes err_v = Rval (e, p, env')" obtains init :: "(pat \<times> exp0) list" and rest :: "(pat \<times> exp0) list" where "pes = init @ (p, e) # rest" and "distinct (pat_bindings p [])" and "list_all (\<lambda>(p, e). cupcake_pmatch cenv p v0 [] = No_match \<and> distinct (pat_bindings p [])) init" and "cupcake_pmatch cenv p v0 [] = Match env'"
scaleR_le_cancel_left: fixes c :: "real"   and a :: "'a"   and b :: "'a" shows "(c *\<^sub>R a \<le> c *\<^sub>R b) = ((0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a))"
select_convs_2: fixes init :: "'d set"   and trans :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f" shows "automaton.trans \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = trans"
IFind_1: fixes y :: "'a ifex" assumes "ro_ifex xb1"   and "ro_ifex y"   and "\<And>ass. val_ifex xb1 ass = val_ifex y ass" shows "xb1 = y"
surjective: fixes r :: "'a bw_point_scheme" shows "r = \<lparr>cx = cx r, cy = cy r, black = black r, \<dots> = bw_point.more r\<rparr>"
path_entry_append_conv: fixes v :: "nat"   and l :: "nat list"   and w :: "nat" shows "path_entry (g_E G) (v # l) w = (path_entry (g_E G) l v \<and> (v, w) \<in> g_E G)"
zero_min: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq zero x"
LT: fixes x :: "'c"   and y :: "'c" shows "(cmp x y = LT) = (x < y)"
ordinal_neq_0: fixes x :: "ordinal" shows "(x \<noteq> 0) = (0 < x)"
inc_Fx: fixes x :: "'a" shows "incseq (\<lambda>n. F n x)"
mn_loop_correct: fixes B :: "abc_inst list"   and arity :: "nat"   and fap :: "abc_inst list"   and ft :: "nat"   and fft :: "nat"   and xs :: "nat list"   and x :: "nat"   and far :: "nat"   and f :: "recf"   and anything :: "nat list" assumes "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"   and "ft = max (Suc arity) fft"   and "length xs = arity"   and "\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"   and "\<forall>i\<le>x. 0 < rec_exec f (xs @ [i])"   and "rec_ci f = (fap, far, fft)"   and "far = Suc arity" shows "\<exists>stp. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
dist_ell2_def: fixes x :: "'a ell2"   and y :: "'a ell2" shows "dist x y = norm (x - y)"
fv_supp: fixes e :: "'a" shows "supp e = atom ` fv e"