hcju/isabelleps huggingface dataset:
prefs_wf: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'c" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "finite_total_preorder_on alts (R i)"
wf_dom_w2: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w" shows "\<forall>x\<in>set w. x < n"
P_eq_1: fixes i :: "nat" shows "events i = sets (Pi\<^sub>M (J i) (\<lambda>i. borel))"
b_2: shows "\<forall>x\<in>S. norm x \<le> b"
ZObisT_Sym: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01T t" shows "t \<approx>01T s"
invar'_def: fixes m :: "'d\<cdot>('e, 'f) writerT" shows "invar' m = (fmapWT\<cdot>ID\<cdot>m = m)"
expansion_factor_2: fixes p :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "cart2_append1 p v* cltn2_rep J = expansion_factor p J *\<^sub>R cart2_append1 (apply_cltn2 p J)"
connect_eq_resource_cong: fixes \<I> :: "('y, 'z) \<I>"   and distinguisher :: "('y, 'z) distinguisher"   and res :: "('y, 'z) resource"   and res' :: "('y, 'z) resource" assumes "\<I> \<turnstile>g distinguisher \<surd>"   and "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>res res \<surd>" shows "connect distinguisher res = connect distinguisher res'"
Suc_minus1: fixes x :: "nat" shows "Suc (x - Suc 0) = (if x = 0 then Suc 0 else x)"
measurable_cong': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "\<And>w. w \<in> space M =simp=> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> M \<rightarrow>\<^sub>M M')"
lappend_ltakeWhile_ldropWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lappend (ltakeWhile P xs) (ldropWhile P xs) = xs"
inf_RETURN_RES_2: fixes X :: "'a set"   and x :: "'a" shows "inf (RES X) (RETURN x) = (if x \<in> X then RETURN x else SUCCEED)"
K_closed: fixes s :: "'s" assumes "s \<in> S" shows "\<Union> (set_pmf ` K s) \<subseteq> S"
that: fixes n' :: "'node"   and as :: "'edge list" assumes "n' controls\<^sub>s n via as" shows "thesis"
mconn_equiv: fixes E :: "('b \<times> 'b) set" shows "equiv UNIV (mconn E)"
ipurge_tr_rev_t_equiv: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_rev_t I D u xs = ipurge_tr_rev I D u xs"
Mprefix_refines_Mndet_D: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "mndet A P \<sqsubseteq>\<^sub>D Mprefix A P"
dom_comp: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.dom C (C g f) = partial_magma.dom C f"
isSome_eq: fixes x :: "'a option" shows "isSome x = (\<exists>y. x = Some y)"
lin_indpt_list_def: fixes n :: "nat"   and fs :: "'b vec list" shows "cof_vec_space.lin_indpt_list n fs = (set fs \<subseteq> carrier_vec n \<and> distinct fs \<and> \<not> module.lin_dep class_ring (module_vec TYPE('b) n) (set fs))"
72139: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - x \<sqinter> - y"
gs: fixes j :: "nat" assumes "j < m" shows "fs.gs.gso j \<in> Rn"
col_suma__col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C B C A D E F" shows "Tarski_neutral_dimensionless.Col Bet A B C"
random_member_None: fixes ss :: "'a fset" shows "(random_member ss = None) = (ss = {||})"
dag_setof_exD: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and n :: "ref" assumes "Dag p low high t"   and "n \<in> set_of t" shows "\<exists>nt. Dag n low high nt"
of_nat_max: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "nat"   and y :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "semiring_1.of_nat one plus zero (max x y) = ord.max less_eq (semiring_1.of_nat one plus zero x) (semiring_1.of_nat one plus zero y)"
next_subseqs1_set_def: fixes tail :: "'a iarray"   and prevs :: "(nat \<times> 'b) list" shows "next_subseqs1_set tail prevs \<equiv> \<Union> {next_subseqs2_set tail v [0..<i] |v i. (i, v) \<in> set prevs}"
Or_None: fixes e2 :: "'a option"   and e1 :: "'a option" shows "(e1 OR e2 = None) = (e1 = None \<and> e2 = None)"
ord_iso_the_inv: fixes f :: "'a \<Rightarrow> 'b" assumes "ord_iso f" shows "mono (the_inv f)"
is_const_def: fixes t :: "'a" shows "is_const t = (unconst t \<noteq> None)"
finfun_update_const_code: fixes b :: "'b"   and a :: "'a"   and b' :: "'b" shows "finfun_update (finfun_const b) a b' = (if b = b' then finfun_const b else finfun_update_code (finfun_const b) a b')"
simps_4: fixes v :: "name"   and env :: "(name, term) fmap" shows "subst (Const v) env = Const v"
Liminf_obtain': fixes F :: "'b filter"   and u :: "'b \<Rightarrow> 'a"   and c :: "'a"   and P :: "'b \<Rightarrow> bool" assumes "Liminf F u < c"   and "eventually P F" shows "\<exists>n. P n \<and> u n < c"
iso_tuple_cons_def: fixes isom :: "('c, 'a, 'b) tuple_isomorphism" shows "Record.iso_tuple_cons isom = curry (Record.abst isom)"
type_less_eq_x_Tuple_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> = Tuple \<pi>"   and "subtuple (\<le>) \<pi> \<xi>" shows "\<tau> \<le> Tuple \<xi>"
WriteInner_enabled: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and l :: "Locs"   and ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and v :: "Vals" assumes "basevars (mm!l, rtrner ch!p, rs!p)" shows "\<turnstile> Calling ch p \<and> arg<ch!p> = #(write l v) \<longrightarrow> Enabled WriteInner ch mm rs p l v"
restrictionVsIntersection: fixes f :: "'a \<Rightarrow> 'b"   and X2 :: "'a set"   and X1 :: "'a set" shows "{(x, f x) |x. x \<in> X2} || X1 = {(x, f x) |x. x \<in> X2 \<inter> X1}"
gOpInj_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gOpInj MOD \<equiv> \<forall>delta delta1 inp' binp' inp binp inp1' binp1' inp1 binp1. wlsInp delta inp' \<and> wlsBinp delta binp' \<and> gWlsInp MOD delta inp \<and> gWlsBinp MOD delta binp \<and> wlsInp delta1 inp1' \<and> wlsBinp delta1 binp1' \<and> gWlsInp MOD delta1 inp1 \<and> gWlsBinp MOD delta1 binp1 \<and> stOf delta = stOf delta1 \<and> gOp MOD delta inp' inp binp' binp = gOp MOD delta1 inp1' inp1 binp1' binp1 \<longrightarrow> delta = delta1 \<and> inp = inp1 \<and> binp = binp1"
enum_ex_UNIV: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "enum_ex P = Bex UNIV P"
concS_def: fixes S1 :: "'a rexp set"   and S2 :: "'a rexp set" shows "concS S1 S2 = {Times a b |a b. a \<in> S1 \<and> b \<in> S2}"
ex: shows "class P C = \<lfloor>a\<rfloor>"
fst_brw_\<alpha>: fixes s :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" shows "fst (brw_\<alpha> s) = dom (fst s)"
galleryD_chamber: fixes xs :: "'a set list"   and x :: "'a set" assumes "domain.gallery xs"   and "x \<in> set xs" shows "domain.chamber x"
root_unity_def: fixes n :: "nat" shows "root_unity n = monom (1::'b) n - 1"
integrable_neg_iff: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "((\<lambda>x. - f x) integrable_on S) = (f integrable_on S)"
zone_set_def: fixes Z :: "('d \<Rightarrow> 'e) set"   and r :: "'d list" shows "Z\<^bsub>r \<rightarrow> 0\<^esub> = {[r\<rightarrow>(0::'e)]u |u. u \<in> Z}"
simps_6: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'a"   and x2 :: "nat set" shows "(case PubK x2 of PriK x \<Rightarrow> f1 x | PubK x \<Rightarrow> f2 x) = f2 x2"
G_face_cycle_set_a': shows "G.face_cycle_set a' = {a'}"
simps: fixes H :: "msg \<Rightarrow> bool"   and a :: "msg" shows "synthp H a = ((\<exists>X. a = X \<and> H X) \<or> (\<exists>A. a = Messages.Agent A) \<or> (\<exists>n. a = Number n) \<or> (\<exists>n. a = NonceA n) \<or> (\<exists>n. a = epubKA n) \<or> (\<exists>n. a = epriKA n) \<or> (\<exists>X. a = Hash X \<and> synthp H X) \<or> (\<exists>X Y. a = \<langle>X, Y\<rangle> \<and> synthp H X \<and> synthp H Y) \<or> (\<exists>X Y. a = Enc X Y \<and> synthp H X \<and> synthp H Y) \<or> (\<exists>X Y. a = Aenc X Y \<and> synthp H X \<and> synthp H Y) \<or> (\<exists>X Y. a = Sign X Y \<and> synthp H X \<and> synthp H Y) \<or> (\<exists>X Y. a = Exp X Y \<and> synthp H X \<and> synthp H Y))"
init_gc_def: fixes prog :: "program" shows "init_gc prog \<equiv> \<lparr>global_config.processes = mset (map init_pc (program.processes prog)), state = \<lparr>global_state.variables = init_valuation (global_vars prog)\<rparr>\<rparr>"
assms_1: shows "homotopic_with_canon (\<lambda>x. True) S S id r"
s_has_real_derivative: fixes t :: "real" assumes "0 \<le> t"   and "v / a \<le> 0"   and "a \<noteq> 0" shows "(s has_real_derivative s' t) (at t within {0..})"
length_find_base_vectors: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "length (find_base_vectors A) = card (set (find_base_vectors A))"
apply_def: fixes f :: "integer \<Rightarrow> ('b \<Rightarrow> 'a) ??.Quickcheck_Narrowing.narrowing_cons"   and a :: "integer \<Rightarrow> 'b ??.Quickcheck_Narrowing.narrowing_cons"   and d :: "integer" shows "apply f a d = (if 0 < d then case f d of Narrowing_cons (Narrowing_sum_of_products ps) cfs \<Rightarrow> case a (d - 1) of Narrowing_cons ta cas \<Rightarrow> let shallow = non_empty ta in Let (if shallow then map (\<lambda>cf xa. case xa of x # xs \<Rightarrow> cf xs (conv cas x)) cfs else []) (Narrowing_cons (Narrowing_sum_of_products (if shallow then map ((#) ta) ps else []))) else Narrowing_cons (Narrowing_sum_of_products []) [])"
intro: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_term_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "term_powerprod pair_of_term term_of_pair"   and "ordered_powerprod ord ord_strict"   and "class.linorder ord_term ord_term_strict"   and "ordered_term_axioms pair_of_term term_of_pair ord ord_term" shows "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
INF_top_conv_1: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf (B ` A) = top) = (\<forall>x\<in>A. B x = top)"
bigthetaI_tendsto_norm: fixes c :: "real"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "c \<noteq> 0"   and "((\<lambda>x. norm (f x / g x)) \<longlongrightarrow> c) F" shows "f \<in> \<Theta>[F](g)"
assms_1: shows "P \<leadsto><weakBisimulation> Q"
mset_insert: fixes q :: "'b tree"   and x :: "'b" assumes "braun q \<and> heap q" shows "mset_tree (insert x q) = mset_tree q + {#x#}"
K_statefun_def: fixes c :: "'c"   and x :: "'d" shows "K_statefun c x \<equiv> c"
poly_eval_focus: fixes a :: "'c \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_eval a (focus X p) = poly_subst (\<lambda>x. if x \<in> X then a x else monomial (1::'d) (monomial 1 x)) p"
codomain_wall_compose: fixes xs :: "wall"   and ys :: "wall" shows "codomain_wall (xs \<circ> ys) = codomain_wall ys"
nn_cond_exp_mono: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "AE x in M. f x \<le> g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_cond_exp M F f x \<le> nn_cond_exp M F g x"
monad_writer_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and tell :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_writer return bind tell" shows "monad_writer (return_option return) (bind_option return bind) (tell_option tell)"
subst_freshen_conclusions: 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 pf :: "'b"   and s :: "'c"   and a :: "nat" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "pf \<in> set conclusions" shows "subst s (Abstract_Formulas.freshen freshenLC renameLCs a pf) = pf"
Smith_2x2_JNF_def: shows "Smith_2x2_JNF = (\<lambda>A'. let (P', S', Q') = Smith_2x2 (Mod_Type_Connect.to_hma\<^sub>m A') in (Mod_Type_Connect.from_hma\<^sub>m P', Mod_Type_Connect.from_hma\<^sub>m S', Mod_Type_Connect.from_hma\<^sub>m Q'))"
nprv_exuE_uni: fixes F :: "'fmla set"   and x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm"   and \<psi> :: "'fmla" assumes "nprv F (exu x \<phi>)"   and "nprv F (subst \<phi> t1 x)"   and "nprv F (subst \<phi> t2 x)"   and "nprv (insert (eql t1 t2) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "nprv F \<psi>"
rel_fun_comp2': fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and R2 :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and h :: "'k \<Rightarrow> 'l"   and S' :: "'i \<Rightarrow> 'l \<Rightarrow> bool" assumes "(R1 ===> S1) f g"   and "(R2 ===> S2) g h"   and "\<And>a b. (S1 OO S2) a b \<Longrightarrow> S' a b" shows "(R1 OO R2 ===> S') f h"
numer_cf7_mono: fixes y :: "real"   and x :: "real" assumes "y \<le> x" shows "numer_cf7 y \<le> numer_cf7 x"
simps_6: fixes r :: "'b rexp" shows "\<guillemotleft>Star r\<guillemotright> = Star \<guillemotleft>r\<guillemotright>"
simps_2: fixes at :: "('b, 'c) atm" shows "nvL (Neg at) = {}"
AIF_leI: fixes \<Phi> :: "bool"   and S :: "'c nres"   and S' :: "'c nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> S \<le> S'" shows "(if \<Phi> then S else FAIL) \<le> S'"
linear_iff_module_hom: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "linear s1 s2 f = module_hom s1 s2 f"
bind_resumption_is_Done: fixes f :: "('f, 'g, 'h) Resumption.resumption"   and g :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done f" shows "f \<bind> g = (if Resumption.resumption.result f = None then ABORT else g (the (Resumption.resumption.result f)))"
simps_1: fixes get_proc :: "'h \<Rightarrow> 'g"   and Main :: "'g"   and n :: "'h" shows "lift_get_proc get_proc Main (Node n) = get_proc n"
poly_cond_degree_gt_1: fixes p :: "int poly"   and x :: "'a" assumes "poly_cond p"   and "1 < degree p"   and "ipoly p x = (0::'a)" shows "x \<notin> \<rat>"
join_left: fixes x :: "'a"   and y :: "'a" shows "x \<sqsubseteq> x \<squnion> y"
not_eq_order_implies_strict: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b"   and "codomain.greater_eq b a" shows "codomain.greater b a"
or_int_def: fixes k :: "int"   and l :: "int" shows "k OR l = NOT (NOT k AND NOT l)"
satisfiableAppendTautologicalFormula: fixes F :: "Literal list list"   and F' :: "Literal list list" assumes "satisfiable F"   and "\<forall>c. c el F' \<longrightarrow> clauseTautology c" shows "satisfiable (F @ F')"
modulo_nat_def: fixes m :: "nat"   and n :: "nat" shows "m mod n = m - m div n * n"
left_invertible_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "(\<exists>B. B ** Finite_Cartesian_Product.transpose A = mat (1::'a)) = (\<exists>B. A ** B = mat (1::'a))"
exhaust_sel: fixes resumption :: "('f, 'g, 'h) Resumption.resumption" obtains "resumption = Resumption.resumption.Done (Resumption.resumption.result resumption)"   | "resumption = Resumption.resumption.Pause (Resumption.resumption.output resumption) (Resumption.resumption.resume resumption)"
existence_ivl_subset: fixes x0 :: "'a" shows "existence_ivl t0 x0 \<subseteq> T"
exists_fset_of_list: fixes S :: "'b fset" shows "\<exists>xs. fset_of_list xs = S"
sgnx_def: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" shows "sgnx f F = (SOME c. (f has_sgnx c) F)"
c: shows "(c, s) \<rightarrow>* (SKIP, s')"
if_is_pr2: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and q1 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and q2 :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p \<in> PrimRec2"   and "q1 \<in> PrimRec2"   and "q2 \<in> PrimRec2" shows "(\<lambda>x y. if p x y = 0 then q1 x y else q2 x y) \<in> PrimRec2"
tauStepChainFresh: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and x :: "name" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "x \<sharp> P" shows "x \<sharp> P'"
from_nat_0: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs 0 = zero"
map_append: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'d list" shows "map f (xs @ ys) = map f xs @ map f ys"
r_coset_consistent: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and I :: "'c set"   and h :: "'c" shows "I #>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> h = I #>\<^bsub>G\<^esub> h"
u_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "y \<le> local.t_stop" shows "u x \<le> u y"
new_Addr_def: fixes h :: "loc \<Rightarrow> obj option" shows "new_Addr h = (if \<forall>a. h a \<noteq> None then None else Some (SOME a. h a = None))"
may_join_cond_action_oks: fixes cas :: "'b conditional_action list"   and s :: "('c, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'c \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "\<And>t'. Join t' \<in> set cas \<Longrightarrow> \<not> r2.not_final_thread s t' \<and> t \<noteq> t'" shows "r2.cond_action_oks s t cas"
0: shows "monom (UP R) d 1 \<in> carrier (UP R)"
Cons: fixes x :: "'a"   and xs :: "'a list" shows "prod_list (x # xs) = x * prod_list xs"
length_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'d list" shows "length (permute_list f xs) = length xs"
HConj_def: fixes A :: "hfm"   and B :: "hfm" shows "A HAND B \<equiv> HNeg (A HIMP HNeg B)"
f1: assumes "rk {R, a} = 1" shows "rk {P, Q, R, A'} = 2"
mult_1_left: fixes a :: "'a" shows "top \<sqinter> a = a"
Peirce: fixes H :: "fm set"   and A :: "fm" assumes "H \<turnstile> Neg A IMP A" shows "H \<turnstile> A"
nonmeets4: fixes u :: "'b"   and r :: "'b" shows "\<not> (u \<parallel> r \<and> (\<exists>p. r \<parallel> p \<and> p \<parallel> u))"
irreflexive_axioms: shows "irreflexive A (\<sqsubset>)"
complex_inner_1_right: fixes x :: "complex" shows "cinner x 1 = cnj x"
x_c_glb_iff: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<cdot> 1\<^sub>\<pi> \<le> x \<cdot> 1\<^sub>\<pi> \<and> z \<cdot> 1\<^sub>\<pi> \<le> y \<cdot> 1\<^sub>\<pi>) = (z \<cdot> 1\<^sub>\<pi> \<le> x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>))"
reflexive_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "reflexive x"   and "reflexive y" shows "reflexive (x * y)"
xk'_1: shows "(x', k') \<in> p1 \<union> p2"
istopology_pullback_topology: fixes T :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f -` U \<inter> A)"
assms_2: shows "a \<noteq> \<infinity>\<^sub>h"
n_value_x_2: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "ant" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "n \<le> n_val K v x"   and "0 \<le> n" shows "x \<in> vp K v\<^bsup> Vr K v n\<^esup>"
record_simps_2: fixes f :: "nat \<Rightarrow> nat"   and x :: "'b state" shows "refs (update_clock f x) = refs x"
x_prop: shows "r < x \<and> x < min (min y1 y2) y3"
TComp_var_instance_closed_has_Fun: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and x :: "('fun, 'atom) Term.term \<times> nat"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and g :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "is_TComp_var_instance_closed \<Gamma> M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"   and "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)"   and "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"   and "t \<sqsubseteq> \<delta> x"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "\<Gamma> t = Fun g T"   and "is_Fun t" shows "\<exists>m\<in>set M. \<exists>\<theta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>) \<and> t = m \<cdot> \<theta> \<and> is_Fun m"
in_pair_trans: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" assumes "p \<in>\<^sub>p A"   and "A \<subseteq> B" shows "p \<in>\<^sub>p B"
intro: fixes SWB :: "('a, 'b) calculusWithBarbs"   and Source :: "'a processCalculus"   and TWB :: "('c, 'b) calculusWithBarbs"   and Target :: "'c processCalculus" assumes "Calculus SWB = Source"   and "Calculus TWB = Target" shows "encoding_wrt_barbs Source Target SWB TWB"
right_inverse_matrix_def: fixes A :: "'b matrix"   and X :: "'b matrix" shows "right_inverse_matrix A X \<equiv> A * X = one_matrix (max (nrows A) (ncols X)) \<and> nrows X \<le> ncols A"
tauChainEqvt: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> p \<bullet> P'"