hcju/isabelleps huggingface dataset:
RedCall: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and hU :: "htype"   and P :: "'addr J_prog"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and pns :: "String.literal list"   and body :: "'addr expr"   and D :: "String.literal"   and vs :: "'addr val list"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "typeof_addr (hp s) a = \<lfloor>hU\<rfloor>"   and "P \<turnstile> class_type_of hU sees M: Ts\<rightarrow>T = \<lfloor>(pns, body)\<rfloor> in D"   and "length vs = length pns"   and "length Ts = length pns" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>M(map Val vs),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>blocks (this # pns) (Class D # Ts) (Addr a # vs) body,s\<rangle>"
strict_implies_not_eq: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a \<noteq> b"
vec_to_iarray_column: fixes i :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" shows "vec_to_iarray (column i A) = column_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
doesnt_read_or_modify_vars_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and X :: "'Var set" shows "doesnt_read_or_modify_vars c X = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> (\<forall>x\<in>X. \<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>))"
mult_pos_neg: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times a b) zero"
mlex_prod_def: fixes f :: "'b \<Rightarrow> nat"   and R :: "('b \<times> 'b) set" shows "f <*mlex*> R = inv_image (less_than <*lex*> R) (\<lambda>x. (f x, x))"
curry0_def: fixes f :: "unit \<Rightarrow> 'a" shows "curry0 f = f ()"
scaleR_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "real"   and y :: "'b"   and n :: "nat" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "power.power one times (scaleR x y) n = scaleR (x ^ n) (power.power one times y n)"
assms_5: shows "(Xi, X) \<in> clw_rel (\<langle>S, A\<rangle>invar_rel a)"
induced_is_subgraph_1: fixes V :: "nat set"   and G :: "nat set \<times> nat set set" assumes "V \<subseteq> uverts G" shows "subgraph (induced_subgraph V G) G"
lm140: fixes R :: "('a \<times> 'b) set"   and A :: "'a set"   and B :: "'a set" assumes "runiq R"   and "runiq (R\<inverse>)" shows "R `` A \<inter> R `` B = R `` (A \<inter> B)"
step_op_inl_miss: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and opinl :: "'opinl"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and x :: "'dyn"   and F' :: "'fenv_inca"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IOpInl opinl"   and "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl) = ar"   and "ar \<le> length \<Sigma>"   and "\<not> \<II>\<ss>\<II>\<nn>\<ll> opinl (take ar \<Sigma>)"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x"   and "Finca_add F f (rewrite_fundef_body fd pc (Inca.instr.IOp (\<DD>\<ee>\<II>\<nn>\<ll> opinl))) = F'" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F' H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
inject_6: fixes x8 :: "'b ltl"   and y8 :: "'b ltl" shows "(G x8 = G y8) = (x8 = y8)"
in_segment_1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> closed_segment a b) = (\<exists>u\<ge>0. u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
map_fun_apply: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'd"   and h :: "'f \<Rightarrow> 'g"   and x :: "'e" shows "map_fun f g h x = g (h (f x))"
nat_upto_subs_top_removal': fixes S :: "nat set"   and n :: "nat" assumes "S \<subseteq> {0..Suc n}"   and "Suc n \<notin> S" shows "S \<subseteq> {0..n}"
heap_is_wellformed_disconnected_nodes_distinct: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
list_all2_conv_all_nth: fixes P :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and xs :: "'k list"   and ys :: "'l list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
l_m: fixes a :: "'a" assumes "a \<in> carrier R" shows "-\<^sub>a a \<plusminus> a = \<zero>"
constrains_imp_subset: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> A co A'" shows "A \<subseteq> A'"
low_dim_interior: fixes S :: "'b set" assumes "aff_dim S \<noteq> int DIM('b)" shows "interior S = {}"
visits_deadend: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player" assumes "vm_path G P v0"   and "lset P \<inter> ParityGame.deadends G p \<noteq> {}" shows "ParityGame.winning_path G p** P"
ResChain: fixes P :: "pi"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "<\<nu>a>P \<Longrightarrow>\<^sub>\<tau> <\<nu>a>P'"
assms: shows "((\<lambda>n. extended_Gromov_product_at a (u n) (v n)) \<longlongrightarrow> \<infinity>) F"
rank_iarray_def: fixes A :: "'b iarray iarray" shows "rank_iarray A = (let A' = Gauss_Jordan_iarrays A; nrows = IArray.length A' in card {i. i < nrows \<and> \<not> is_zero_iarray (A' !! i)})"
neg_antitone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
intersect_segment_xline_eq_None: fixes prec :: "nat"   and p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real" assumes "intersect_segment_xline prec (p0, p1) x = None" shows "closed_segment p0 p1 \<inter> {p. fst p = x} = {}"
equiv_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
comp_ide_self: fixes a :: "'a" assumes "Left.ide a" shows "Left.comp a a = a"
pell_valuation_0_iff: fixes z :: "int \<times> int" shows "(pell_valuation z = 0) = (z = (0, 0))"
unit_vec_eq: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n" shows "(unit\<^sub>v n i = unit\<^sub>v n j) = (i = j)"
removeAll_filter_not_eq: fixes x :: "'g" shows "removeAll x = filter ((\<noteq>) x)"
aux4req: fixes k :: "nat"   and req :: "nat \<Rightarrow> reqType list"   and t :: "nat"   and m :: "nat" assumes "\<forall>m\<le>k + 2. req (t + m) \<noteq> [send]"   and "m \<le> k"   and "req (t + 2 + m) = [send]" shows "False"
hfd': fixes x :: "complex"   and n :: "nat" assumes "x \<in> S" shows "(f n has_field_derivative deriv (f n) x) (at x)"
IH2: fixes x :: "'a" assumes "x \<in> carrier G"   and "P x" shows "P (inv x)"
cond_spmf_fst_map_prod: fixes f :: "'h \<Rightarrow> 'i"   and x :: "'h"   and p :: "('h \<times> 'j) spmf"   and g :: "'h \<Rightarrow> 'j \<Rightarrow> 'k" assumes "inj_on f (insert x (fst ` set_spmf p))" shows "cond_spmf_fst (map_spmf (\<lambda>(x, y). (f x, g x y)) p) (f x) = map_spmf (g x) (cond_spmf_fst p x)"
add_strict_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" assumes "a \<subset># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
ldistinct_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "ldistinct (lappend xs ys) = (ldistinct xs \<and> (lfinite xs \<longrightarrow> ldistinct ys \<and> lset xs \<inter> lset ys = {}))"
transition_uniq: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and root' :: "(att \<times> char list, att, nat) env"   and root'' :: "(att \<times> char list, att, nat) env" assumes "root \<midarrow>x\<rightarrow> root'"   and "root \<midarrow>x\<rightarrow> root''" shows "root' = root''"
mult_pos_neg2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times b a) zero"
WT_converter_mono: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and \<I>1' :: "('i, 'j) \<I>"   and \<I>2' :: "('k, 'l) \<I>" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv \<surd>"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1', \<I>2' \<turnstile>\<^sub>C conv \<surd>"
nc_ccomp_closed_alt: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> (y \<sqinter> nc) \<sqinter> nc = x \<parallel> (y \<sqinter> nc)"
has_integral_closure: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "negligible (frontier S)" shows "(f has_integral y) (closure S) = (f has_integral y) S"
valid_expr_list_NormalI: fixes A :: "'b triple set"   and G :: "prog"   and t :: "expr list"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC T C vs s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0; \<lparr>prg = G, cls = accC, lcl = L\<rparr>,empty_dt\<Turnstile>t\<Colon>\<doteq>T; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In3 t\<guillemotright> C; G\<turnstile>s0 \<midarrow>t\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s1; Normal P Y s0 Z\<rbrakk> \<Longrightarrow> Q (In3 vs) s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{Normal P} t\<doteq>\<succ> {Q}}"
n1_type: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> Bp h" shows "n1 t \<in> T (Suc h)"
fun_relI: fixes A :: "('f \<times> 'g) set"   and f :: "'f \<Rightarrow> 'h"   and f' :: "'g \<Rightarrow> 'i"   and B :: "('h \<times> 'i) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
is_partial_magma_2: shows "partial_magma.null (\<circ>) = Span.null"
flush_store_buffer: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and sb :: "'a memref list"   and is\<^sub>s\<^sub>b :: "instr list"   and "is" :: "instr list"   and p :: "'a"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p\<^sub>s\<^sub>b :: "'a"   and \<O>' :: "nat set"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "instrs sb @ is\<^sub>s\<^sub>b = is @ prog_instrs sb"   and "causal_program_history is\<^sub>s\<^sub>b sb"   and "ts ! i = (p, is, \<theta> |` (dom \<theta> - read_tmps sb), x, \<D>, \<O>, \<R>)"   and "p = hd_prog p\<^sub>s\<^sub>b sb"   and "last_prog p\<^sub>s\<^sub>b sb = p\<^sub>s\<^sub>b"   and "reads_consistent True \<O>' m sb"   and "history_consistent \<theta> p sb"   and "\<forall>sop\<in>write_sops sb. valid_sop sop"   and "distinct_read_tmps sb"   and "volatile_reads_consistent m sb" shows "(ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts[i := (p\<^sub>s\<^sub>b, is\<^sub>s\<^sub>b, \<theta>, x, \<D> \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}, acquired True sb \<O>, release sb (dom \<S>) \<R>)], flush sb m, share sb \<S>)"
intros_1: fixes A :: "'b set" shows "LNil \<in> A\<^sup>\<star>"
enc_atom_lang_Arbitrary_Except_False: fixes r :: "nat"   and I :: "('b + 'b set) list"   and p :: "'b"   and n :: "nat"   and a :: "'a" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p \<noteq> p' | Inr P \<Rightarrow> p \<notin> P"   and "length I = n"   and "a \<in> set \<Sigma>" shows "[enc_atom I p a] \<in> lang n (Atom (Arbitrary_Except r False))"
rep_eq: fixes x :: "integer list" shows "explode (literal_of_asciis x) = map (ascii_of \<circ> char_of) x"
simple_cipher'_axioms: shows "simple_cipher' prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen"
in_unit_table: fixes v :: "'a option list"   and n :: "nat" shows "(v \<in> unit_table n) = wf_tuple n {} v"
eq1: shows "\<Gamma>' = \<Gamma>'' \<oplus> A"
numeral_power_eq_of_nat_cancel_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "num"   and n :: "nat"   and y :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (numeral.numeral one plus x) n = semiring_1.of_nat one plus zero y) = (numeral x ^ n = y)"
repr_independence: fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H #>\<^bsub>add_monoid G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "H #>\<^bsub>add_monoid G\<^esub> x = H #>\<^bsub>add_monoid G\<^esub> y"
isCont_o2: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "isCont f a"   and "isCont g (f a)" shows "isCont (\<lambda>x. g (f x)) a"
symmetry_lemma: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"   and "\<And>a b c d. \<lbrakk>a \<noteq> b; c \<noteq> d; a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d\<rbrakk> \<Longrightarrow> P a b c d" shows "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> P a b c d"
compactE: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and \<T> :: "'b set set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "S \<subseteq> \<Union> \<T>"   and "\<And>B. B \<in> \<T> \<Longrightarrow> open B" obtains \<T>' :: "'b set set" where "\<T>' \<subseteq> \<T>" and "finite \<T>'" and "S \<subseteq> \<Union> \<T>'"
covering_space_def: fixes c :: "'a set"   and p :: "'a \<Rightarrow> 'b"   and S :: "'b set" shows "covering_space c p S \<equiv> continuous_on c p \<and> p ` c = S \<and> (\<forall>x\<in>S. \<exists>T. x \<in> T \<and> openin (top_of_set S) T \<and> (\<exists>v. \<Union> v = c \<inter> p -` T \<and> (\<forall>u\<in>v. openin (top_of_set c) u) \<and> disjoint v \<and> (\<forall>u\<in>v. \<exists>q. homeomorphism u T p q)))"
wf_tree: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and v :: "'e"   and p :: "('b, 'a) antecedent"   and t :: "'e"   and pth :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) list" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst"   and "Vertex_Graph.valid_in_port (Abstract_Task.inPorts anyP antecedent) vertices nodeOf (v, p)"   and "Scoped_Graph.terminal_path (Abstract_Task.outPorts anyP antecedent consequent) vertices nodeOf edges (Abstract_Task.hyps antecedent) v t pth" shows "RuleSystem_Defs.wf (ND_Rules_Inst.eff freshenLC renameLCs lconsts subst subst_lconsts (Abstract_Task.natEff_Inst antecedent consequent)) (Cut ## Axiom ## smap NatRule (Abstract_Task.n_rules consequent rules)) (Tasked_Proof_Graph.tree freshenLC renameLCs subst anyP antecedent vertices nodeOf edges vidx inst v p pth)"
simps: fixes a1 :: "sequent list \<times> sequent"   and a2 :: "form" shows "rightPrincipal a1 a2 = (\<exists>C A Ps. a1 = (Ps, C) \<and> a2 = A \<and> C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>))"
tseytin_untran: fixes fresh :: "'a set \<Rightarrow> 'a"   and \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula" assumes "freshstuff_sema fresh"   and "\<A> \<Turnstile> freshstuff.tseytin_tran fresh F" shows "\<A> \<Turnstile> F"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "H.VV.comp g f = (if H.VV.arr f \<and> H.VV.arr g \<and> H.VxV.seq g f then H.VxV.comp g f else H.VxV.null)"
hom_def: fixes CC :: "('e, 'd, 'f) category_scheme"   and A :: "'e"   and B :: "'e" shows "Hom\<^bsub>CC\<^esub> A B = {f \<in> Ar\<^bsub>CC\<^esub>. Dom\<^bsub>CC\<^esub> f = A \<and> Cod\<^bsub>CC\<^esub> f = B}"
tl1_to_tl2_def: fixes s2 :: "'s2"   and sstls1 :: "('s1 \<times> 's1 \<times> 'tl1 \<times> 's1) llist" shows "tl1_to_tl2 s2 sstls1 = unfold_llist (\<lambda>(s2, sstls1). lnull sstls1) (\<lambda>(s2, sstls1). let (s1, s1', tl1, s1'') = lhd sstls1; (s2', tl2, s2'') = SOME (s2', tl2, s2''). s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> s1'' \<approx> s2'' \<and> tl1 \<sim> tl2 in (s2, s2', tl2, s2'')) (\<lambda>(s2, sstls1). let (s1, s1', tl1, s1'') = lhd sstls1; (s2', tl2, s2'') = SOME (s2', tl2, s2''). s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> s1'' \<approx> s2'' \<and> tl1 \<sim> tl2 in (s2'', ltl sstls1)) (s2, sstls1)"
induct: fixes P :: "'b match_expr \<Rightarrow> bool"   and a0 :: "'b match_expr" assumes "P MatchAny"   and "\<And>m. P (Match m)"   and "\<And>m1 m2. \<lbrakk>P m1; \<And>x. x \<in> set (normalize_match m1) \<Longrightarrow> P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "\<And>m1 m2. \<lbrakk>P (MatchNot m1); P (MatchNot m2)\<rbrakk> \<Longrightarrow> P (MatchNot (MatchAnd m1 m2))"   and "\<And>m. P m \<Longrightarrow> P (MatchNot (MatchNot m))"   and "P (MatchNot MatchAny)"   and "\<And>m. P (MatchNot (Match m))" shows "P a0"
assms: fixes n :: "nat"   and x :: "'a"   and \<Sigma>1 :: "'stack"   and x' :: "'b"   and \<Sigma>2 :: "'stack" assumes "rewrite_instr n x \<Sigma>1 = Ok (x', \<Sigma>2)" shows "f x' = x"
simps_8: fixes P :: "('d, 'e, 'f) psi" shows "guarded (!P) = guarded P"
flip_length: fixes i :: "nat"   and xs :: "bool list" shows "length (flip i xs) = length xs"
sum_cong_Suc: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "0 \<notin> A"   and "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)" shows "sum f A = sum g A"
simps_17: fixes x1 :: "('b, 'a) hd" shows "vars (Hd x1) = vars_hd x1"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (F.map\<^sub>0 a) = F.map\<^sub>0 a"
ord: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set xs"   and "b \<in> set xs" shows "a \<le> b \<or> b \<le> a"
poly_deg_monomial: fixes t :: "'d \<Rightarrow>\<^sub>0 'c"   and c :: "'e" shows "poly_deg (monomial c t) = (if c = (0::'e) then 0::'c else deg_pm t)"
assms_1: shows "bijective (e * top) \<and> bijective (e[[top]])"
skip_eval_step: fixes annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>Skip \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>Stop, mds \<oplus> annos, mem\<rangle>"
inclusion: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.arr (fst f) \<and> C.VV.arr (snd f) \<and> src\<^sub>C (fst f) = trg\<^sub>C (fst (snd f))" shows "C.VxVxV.arr f"
path_components_of_eq_empty: fixes X :: "'a topology" shows "(path_components_of X = {}) = (topspace X = {})"
option_while'_THE: fixes r :: "'b"   and sr' :: "'b option"   and C :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b option" assumes "(Some r, sr') \<in> option_while' C B" shows "(THE s. (Some r, s) \<in> option_while' C B) = sr'"
p_zero: shows "ego2.p 0 = ego.q \<delta>"
direct_decomp_span_some_basis: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and ss :: "'c set list" assumes "vector_space scale"   and "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> module.subspace scale s" shows "module.span scale (\<Union> (vector_space.some_basis scale ` set ss)) = A"
ord_resolve_rename_lifting: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and M :: "'a literal multiset set"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "\<And>\<rho> C. is_renaming \<rho> \<Longrightarrow> S (C \<cdot> \<rho>) = S C \<cdot> \<rho>"   and "ord_resolve (S_M S M) CAs DA AAs As \<sigma> E"   and "selection S"   and "{DA} \<union> set CAs \<subseteq> \<G>_Fset M" obtains \<eta>s :: "'s list" and \<eta> :: "'s" and \<eta>2 :: "'s" and CAs0 :: "'a literal multiset list" and DA0 :: "'a literal multiset" and AAs0 :: "'a multiset list" and As0 :: "'a list" and E0 :: "'a literal multiset" and \<tau> :: "'s" where "is_ground_subst \<eta>" and "is_ground_subst_list \<eta>s" and "is_ground_subst \<eta>2" and "ord_resolve_rename S CAs0 DA0 AAs0 As0 \<tau> E0" and "CAs0 \<cdot>\<cdot>cl \<eta>s = CAs" and "DA0 \<cdot> \<eta> = DA" and "E0 \<cdot> \<eta>2 = E" and "{DA0} \<union> set CAs0 \<subseteq> M" and "length CAs0 = length CAs" and "length \<eta>s = length CAs"
subst4: fixes xvec :: "name list"   and Tvec :: "'a list"   and M :: "'a" assumes "length xvec = length Tvec"   and "distinct xvec"   and "xvec \<sharp>* M" shows "M[xvec::=Tvec] = M"
1: shows "v \<notin> subst_domain (\<sigma>1 \<circ>\<^sub>s \<theta>2)"
mfinal_deadlock: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r2.mfinal s" shows "r2.deadlock s"
one_less_inverse: fixes a :: "'a" assumes "(0::'a) < a"   and "a < (1::'a)" shows "(1::'a) < inverse a"
sp_Cons: fixes g :: "('a, 'b) rgraph_scheme"   and v1 :: "'a"   and e :: "'a edge"   and es :: "'a edge list"   and v2 :: "'a" shows "subpath g v1 (e # es) v2 = (src e = v1 \<and> e \<in> edges g \<and> subpath g (tgt e) es v2)"
Max_atMost: fixes n :: "nat" shows "Max {..n} = n"