hcju/isabelleps huggingface dataset:
wf_chain_append: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "wf_chain \<Gamma> (rs1 @ rs2) = (wf_chain \<Gamma> rs1 \<and> wf_chain \<Gamma> rs2)"
induct: fixes P :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list \<times> 'a rose_tree \<Rightarrow> bool"   and a0 :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list \<times> 'a rose_tree" assumes "\<And>t. P ([], t)"   and "\<And>a l r z t. P (z, Tree (a, l @ t # r)) \<Longrightarrow> P ((a, l, r) # z, t)" shows "P a0"
legacy_Complex_simps_26: fixes x :: "real"   and y :: "real" shows "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
interval_extension1_def: fixes F :: "float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real" shows "interval_extension1 F f = (\<forall>ivl ivl'. F ivl = Some ivl' \<longrightarrow> (\<forall>x. x \<in>\<^sub>r ivl \<longrightarrow> f x \<in>\<^sub>r ivl'))"
tailrec_transform_aux2: fixes m :: "'c nres"   and a :: "'c \<Rightarrow> 'c nres" shows "lfp (\<lambda>x. sup m (x \<bind> a)) \<le> m \<bind> REC (\<lambda>D s. sup (a s \<bind> D) (RETURN s))"
ring_hom_trans: fixes f :: "'g \<Rightarrow> 'h"   and R :: "('g, 'i) ring_scheme"   and S :: "('h, 'j) ring_scheme"   and g :: "'h \<Rightarrow> 'k"   and T :: "('k, 'l) ring_scheme" assumes "f \<in> ring_hom R S"   and "g \<in> ring_hom S T" shows "g \<circ> f \<in> ring_hom R T"
c_types_agree_top1: fixes ts :: "t list" shows "c_types_agree (TopType []) ts"
assms_3: fixes zs :: "('a \<times> 'b) list" assumes "oalist_inv_raw zs" shows "g zs = zs"
flow_in_domain: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "flow t0 x0 t \<in> X"
len_BB: fixes K :: "nat set"   and j :: "nat" assumes "K \<in> \<K> j j"   and "j \<le> j" shows "length (BB j j K) = enum (d j) j"
homotopic_loops_part_circlepath: fixes c :: "complex"   and r :: "real"   and A :: "complex set"   and b1 :: "real"   and a1 :: "real"   and k :: "int"   and b2 :: "real"   and a2 :: "real" assumes "sphere c r \<subseteq> A"   and "0 \<le> r"   and "b1 = a1 + 2 * real_of_int k * pi"   and "b2 = a2 + 2 * real_of_int k * pi" shows "homotopic_loops A (part_circlepath c r a1 b1) (part_circlepath c r a2 b2)"
prems_2: shows "sat_precond_as s1 (a # as)"
sup_mono: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (sup a b) (sup c d)"
fract_poly_0: shows "map_poly to_fract 0 = 0"
f_def: shows "f = (\<lambda>u. u powr (b - 1) / (1 + u) powr (a + b))"
scale_zero_right: fixes a :: "'a" shows "a *a (0::'b) = (0::'b)"
remainder_theorem_exist: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "carrier R \<noteq> {\<zero>}" shows "\<exists>q r. q \<in> carrier P \<and> r \<in> carrier P \<and> f = (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> deg R r = 0"
semiring_norm_92: fixes n :: "num" shows "(1::'a) \<le> numeral n"
abs_ge_self: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq a (abs a)"
trivial_outside: fixes v :: "'a" assumes "v \<notin> \<Union> X" shows "f v = v"
assms_5: shows "e \<in> {real_of_float el..real_of_float eu}"
int: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "P {a..b}"
ext_singleton_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_singleton_axioms ext \<equiv> \<forall>y x gt. y \<noteq> x \<longrightarrow> ext gt [y] [x] = gt y x"
alg_equiv_inv_auto_12: fixes \<Gamma> :: "(name \<times> ty) list"   and p :: "trm"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> App p s \<leftrightarrow> t : T" obtains q :: "trm" and T\<^sub>1 :: "ty" and t :: "trm" where "ta__ = App q t" and "\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1\<rightarrow>T" and "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1"
mk_vec_def: fixes f :: "'b \<Rightarrow> 'a" shows "HMA_Connect.mk_vec f = (\<chi>i. f i)"
substAbs_def: fixes xs :: "'varSort"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x :: "'var" shows "substAbs xs X x \<equiv> psubstAbs (idEnv [x \<leftarrow> X]_xs)"
l10_7: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and Q' :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Reflect Bet Cong P' P A B"   and "Tarski_neutral_dimensionless.Reflect Bet Cong Q' Q A B"   and "P' = Q'" shows "P = Q"
le_measure: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "(M \<le> N) = (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"
RangeI: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
one: fixes p' :: "nat" shows "(- 1) ^ p' * (- 1) ^ p' = 1"
mono_gamma_o: fixes S1 :: "'av st option"   and S2 :: "'av st option" assumes "S1 \<le> S2" shows "\<gamma>\<^sub>o S1 \<subseteq> \<gamma>\<^sub>o S2"
elims: fixes x :: "('fsym, 'a) efsym"   and y :: "'tp list" assumes "GE_arOf x = y" obtains f :: "'fsym" where "x = Oldf f" and "y = arOf f"   | \<sigma> :: "'a" where "x = Wit \<sigma>" and "y = []"
steps_append_single: fixes xs :: "'a list"   and x :: "'a" assumes "local.steps xs"   and "last xs \<rightarrow> x"   and "xs \<noteq> []" shows "local.steps (xs @ [x])"
assms_1: shows "r1\<^sup>*\<^sup>* = r2\<^sup>*\<^sup>*"
bdd_aboveI2: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and M :: "'d multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "subset_mset.bdd_above (f ` A)"
uninst_trivial_simps_4: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and pa :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option" shows "uninst.trivial u (pa(g := p g)) g = uninst.trivial u p g"
list_of_oalist_inverse: fixes x :: "('a, 'b) oalist" shows "Abs_oalist (list_of_oalist x) = x"
sorted_hd_last: fixes l :: "'a list" assumes "sorted l"   and "l \<noteq> []" shows "hd l \<le> last l"
reduced_GB_ideal_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "ideal (punit.reduced_GB F) = ideal F"
disc_2: fixes x2 :: "'i" shows "\<not> isl (Inr x2)"
mult_monom: fixes a :: "'a"   and m :: "nat"   and b :: "'a"   and n :: "nat" shows "monom a m * monom b n = monom (a * b) (m + n)"
linear_continuous_on: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "bounded_linear f" shows "continuous_on s f"
Q_hit'_def: shows "Q_hit' \<equiv> P_k_pre @ y # Q_hit_post"
fromIMor_termFSbMorph: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "termFSbImorph h hA (fromMOD MOD)" shows "termFSbMorph (fromIMor h) (fromIMorAbs hA) MOD"
S_def: shows "S = {norm (blinfun_apply f x) |x. x \<in> sphere (0::'a) 1}"
sup_exists: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "\<exists>s. is_lub L s A"
mod_mult_left_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times (modulo a c) b) c = modulo (times a b) c"
nrows_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "nrows A = CARD('c)"
assms_1: shows "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"
fd: fixes x :: "real" assumes "x \<in> {a..c} - S" shows "f differentiable at x within {a..c}"
is_non_overlapping_def: fixes P :: "'a set set" shows "is_non_overlapping P = (\<forall>X\<in>P. \<forall>Y\<in>P. (X \<inter> Y \<noteq> {}) = (X = Y))"
pn_choose_k_modp_0: fixes p :: "nat"   and k :: "nat"   and n :: "nat" assumes "prime p"   and "1 \<le> k \<and> k \<le> p ^ n - 1"   and "0 < n" shows "(p ^ n choose k) mod p = 0"
gt_compat_arg: fixes s' :: "('s, 'v) tm"   and s :: "('s, 'v) tm"   and t' :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "s' >\<^sub>t s"   and "t' \<ge>\<^sub>t t" shows "App s' t' >\<^sub>t App s t"
only_vertical_division_def: fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "only_vertical_division one_chain two_chain \<equiv> \<exists>\<V> \<H>. finite \<H> \<and> finite \<V> \<and> (\<forall>(k, \<gamma>)\<in>\<V>. \<exists>(k', \<gamma>')\<in>two_chain_vertical_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<and> (common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H> \<or> common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)) \<and> boundary_chain \<H> \<and> one_chain = \<V> \<union> \<H> \<and> (\<forall>(k, \<gamma>)\<in>\<H>. valid_path \<gamma>)"
existence_ivl_empty2: fixes x0 :: "'a" assumes "x0 \<notin> X" shows "existence_ivl0 x0 = {}"
assms_2: shows "\<forall>d. d division_of \<Union> d \<longrightarrow> (\<Sum>k\<in>d. norm (integral k f)) \<le> B"
erb_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<lless>\<^sub>\<K> B) = (\<forall>a. a \<in> A \<longrightarrow> (\<exists>b. b \<in> B \<and> a \<le>\<^sub>\<K> b))"
cInf_eq_Min: fixes X :: "'b set" assumes "finite X"   and "X \<noteq> {}" shows "Inf X = Min X"
sum_msets_chop: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n" shows "sum_list (map mset (chop n xs)) = mset xs"
of_int_eq_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and w :: "int"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus w = ring_1.of_int one plus zero minus z) = (w = z)"
distinct_3: fixes x3 :: "'h" shows "Bottom \<noteq> sum_bot.Right x3"
that_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "x = P"   and "xa = E"   and "xb = []"   and "xc = []" shows "thesis"
CJ_O_O: fixes A :: "c \<Rightarrow> w \<Rightarrow> bool"   and B :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. (\<^bold>O\<langle>B|A\<rangle>) c \<sqsubseteq> (\<lambda>w. ob \<top> ((A \<^bold>\<rightarrow> B) c))"
unfold_congs_3: fixes r :: "'a env_scheme"   and r' :: "'a env_scheme"   and v' :: "lenv"   and f :: "lenv \<Rightarrow> lenv"   and f' :: "lenv \<Rightarrow> lenv" assumes "r = r'"   and "lcl r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "lcl_update f r = lcl_update f' r'"
bconf_INIT: fixes P :: "'a prog"   and sh :: "sheap"   and C :: "char list"   and Cs :: "char list list"   and b' :: "bool"   and e :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (INIT C (Cs,b') \<leftarrow> e,b) \<surd>) = (\<not> b)"
is_AddrE: fixes v :: "'a val" assumes "is_Addr v" obtains a :: "'a" where "v = Addr a"
invalid_assn_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> assn"   and x :: "'d"   and y :: "'e" shows "invalid_assn R x y \<equiv> \<up> (\<exists>h. h \<Turnstile> R x y) * true"
elem_carrier: fixes A :: "'a set" assumes "A \<in> E" obtains "A \<subseteq> carrier" and "A \<noteq> {}"
ipt_iprange_to_interval: fixes ip :: "'b ipt_iprange"   and s :: "'b word"   and e :: "'b word" assumes "ipt_iprange_to_interval ip = (s, e)" shows "{s..e} = ipt_iprange_to_set ip"
simps_8: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Since \<phi> I \<psi>) j = min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)"
assms: fixes v :: "'a set"   and x :: "'a" assumes "openin (top_of_set S) v"   and "x \<in> v" shows "\<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
def_transrec: fixes f :: "V \<Rightarrow> V"   and H :: "(V \<Rightarrow> V) \<Rightarrow> V \<Rightarrow> V"   and a :: "V" assumes "\<And>x. f x \<equiv> transrec H x" shows "f a = H (restrict f (elts a)) a"
nonnegative_def: fixes f :: "'b \<Rightarrow> 'c" shows "nonnegative f = (\<forall>x. (0::'c) \<le> f x)"
assms_5: shows "mset (map snd xs) = mset (map snd ys)"
assms_1: shows "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs)"
swap_side: fixes side1 :: "'a set"   and side2 :: "'a set"   and y :: "'a"   and t :: "real" assumes "side1 \<inter> side2 = {}"   and "open side2"   and "frontier side1 = path_image J"   and "frontier side2 = path_image J"   and "side1 \<union> side2 = - path_image J"   and "y \<in> side2"   and "0 \<le> t"   and "t \<in> existence_ivl0 y"   and "flow0 y t \<in> closure side1" shows "\<exists>T>0. T \<le> t \<and> (\<forall>s\<in>{0..<T}. flow0 y s \<in> side2) \<and> flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
change_conditions: fixes b :: "nat"   and k :: "nat" obtains "change_on_0 b k"   | "change_on_1 b k"   | "change_on_neither b k"
abs_conv_abscissa_minus: fixes f :: "'a fds" shows "abs_conv_abscissa (- f) = abs_conv_abscissa f"
distinct_mset_eq: fixes a :: "'b list"   and b :: "'b list" assumes "distinct a"   and "mset a = mset b" shows "distinct b"
inject: fixes x1 :: "'b list"   and x2 :: "nat"   and y1 :: "'b list"   and y2 :: "nat" shows "(Fundef x1 x2 = Fundef y1 y2) = (x1 = y1 \<and> x2 = y2)"
splus_left_commute: fixes s :: "'a"   and t :: "'a"   and v :: "'t" shows "s \<oplus> (t \<oplus> v) = t \<oplus> (s \<oplus> v)"
right_total_rel_converter: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "right_unique A"   and "left_total A"   and "right_total B"   and "right_total C"   and "right_unique R"   and "left_total R" shows "right_total (rel_converter A B C R)"
rep_eq: shows "(\<le>) = (\<lambda>X Y. Rep_upset Y \<subseteq> Rep_upset X)"
prems: fixes i :: "'i" assumes "i \<in> insert i__ I" shows "integrable (M i) (f i)"
simps_2: fixes b :: "'a"   and sp :: "'b"   and s :: "'c stream" shows "run\<^sub>\<mu> (Put b sp) s = ((b, sp), s)"
get_component_def: shows "Shadow_DOM_Components.get_component \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"
rep_nat_pp_inverse: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat_pp :: "'b \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'b"   and lex_comp' :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and deg' :: "'b \<Rightarrow> nat"   and x :: "'b" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "abs_nat_pp (rep_nat_pp x) = x"
containsUnacceptableEdgeSnd_eq: fixes N :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and v :: "nat"   and "is" :: "nat list" shows "containsUnacceptableEdgeSnd N v is = containsUnacceptableEdgeSnd' N (v # is)"
weak_reduction_correspondence_simulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_correspondence_simulation Rel Cal" shows "weak_reduction_correspondence_simulation (Rel\<^sup>+) Cal"
int_pow_int: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> int n = x [^]\<^bsub>G\<^esub> n"
count_Sup_multiset_nonempty: fixes A :: "'k multiset set"   and x :: "'k" assumes "A \<noteq> {}"   and "subset_mset.bdd_above A" shows "count (Sup A) x = (SUP X\<in>A. count X x)"
hyps_3: assumes "s1 \<approx> s2" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s1' \<approx> s2'"
expand_fps_eq: fixes p :: "'a fps"   and q :: "'a fps" shows "(p = q) = (\<forall>n. fps_nth p n = fps_nth q n)"
lc_invar: fixes xc :: "'xc" assumes "xc \<in> set lc" shows "invar xc"
terminates_wfE: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "terminates g" obtains R :: "('e \<times> 'e) set" where "wf R" and "\<And>s s'. g s = Skip s' \<Longrightarrow> (s', s) \<in> R" and "\<And>s a s'. g s = Yield a s' \<Longrightarrow> (s', s) \<in> R"
nres_simp_internals_2: shows "FAILi = FAIL"
l2p_aux2: fixes list :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "list = []" shows "list2FWpolicy (a # list) = a"
case_1: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x1 :: "nat" shows "(case I x1 of I x \<Rightarrow> f1 x | R x \<Rightarrow> f2 x) = f1 x1"
assms_1: fixes e :: "'a" assumes "e \<in> set_pmf p" shows "pmf p e = pmf q e"
strict_mono_add: fixes k :: "'c" shows "strict_mono (\<lambda>n. n + k)"
closed_diagonal: shows "closed {y. \<exists>x. y = (x, x)}"
ins: shows "insert x H <#> J = (H <#> J) \<union> ({x} <#> J)"
abs_eq: fixes x :: "int" shows "Word.of_int x = Word.Word (take_bit LENGTH('b) x)"
subst_compose_assoc: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'g) Term.term"   and \<tau> :: "'g \<Rightarrow> ('d, 'f) Term.term"   and \<mu> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> \<circ>\<^sub>s \<mu> = \<sigma> \<circ>\<^sub>s (\<tau> \<circ>\<^sub>s \<mu>)"
intros_1: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ"   and v :: "variable" assumes "wf_type \<Sigma> T" shows "wf_term \<Sigma> (Fv v T)"
UnionE: fixes A :: "'d"   and C :: "'d set set" assumes "A \<in> \<Union> C" obtains X :: "'d set" where "A \<in> X" and "X \<in> C"
true_dsij_id: fixes P :: "'b \<Rightarrow> bool" shows "(P \<or> false) = P"
P1: shows "Col P X Y \<and> X Y Perp A B \<and> X Y Perp C D"
updates_def: fixes ks :: "'c list"   and vs :: "'d list" shows "updates ks vs = fold (\<lambda>(x, y). update x y) (zip ks vs)"
le_nofailI: fixes M' :: "'c nres"   and M :: "'c nres" assumes "nofail M' \<Longrightarrow> M \<le> M'" shows "M \<le> M'"
relative_homology_group_Euclidean_complement_step: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and p :: "int"   and k :: "nat" assumes "closedin (Euclidean_space n) S" shows "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - S) \<cong> relative_homology_group (p + int k) (Euclidean_space (n + k)) (topspace (Euclidean_space (n + k)) - S)"
prob_eq_0: fixes A :: "'c set"   and M :: "'c pmf" assumes "A \<in> measure_pmf.events M" shows "(measure_pmf.prob M A = 0) = (AE x in measure_pmf M. x \<notin> A)"
not_acc_down: fixes x :: "'c"   and R :: "('c \<times> 'c) set" assumes "x \<notin> Wellfounded.acc R" obtains z :: "'c" where "(z, x) \<in> R" and "z \<notin> Wellfounded.acc R"
lifting_inv_init_fin_lift_inv: shows "lifting_inv init_fin_final init_fin (init_fin_lift_inv P)"
cases: fixes a1 :: "Lambda_Example.type list"   and a2 :: "dB"   and a3 :: "Lambda_Example.type" assumes "a1 \<turnstile> a2 : a3" obtains env :: "Lambda_Example.type list" and x :: "nat" and T :: "Lambda_Example.type" where "a1 = env__" and "a2 = Var x" and "a3 = T__" and "nth_el1 env__ x T__"   | T :: "Lambda_Example.type" and env :: "Lambda_Example.type list" and t :: "dB" and U :: "Lambda_Example.type" where "a1 = enva__" and "a2 = Abs Ta__ ta__" and "a3 = Ta__ \<Rightarrow> Ua__" and "Ta__ # enva__ \<turnstile> ta__ : Ua__"   | env :: "Lambda_Example.type list" and s :: "dB" and U :: "Lambda_Example.type" and T :: "Lambda_Example.type" and t :: "dB" where "a1 = env" and "a2 = s \<degree> t" and "a3 = U" and "env \<turnstile> s : U \<Rightarrow> T" and "env \<turnstile> t : T"
le_fin_def: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2 \<equiv> \<exists>v\<^sub>1. w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2"
order_subst2: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
imp_graph_rtrancl_skew_sym: fixes l\<^sub>1 :: "lit"   and l\<^sub>2 :: "lit"   and cnf :: "lit uprod set" assumes "(l\<^sub>1, l\<^sub>2) \<in> (imp_graph cnf)\<^sup>*" shows "(negate l\<^sub>2, negate l\<^sub>1) \<in> (imp_graph cnf)\<^sup>*"
nn_cond_exp_def: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "nn_cond_exp M F f = (if f \<in> borel_measurable M \<and> subalgebra M F then RN_deriv (restr_to_subalg M F) (restr_to_subalg (density M f) F) else (\<lambda>_. 0))"
assms_7: shows "(lb, l) \<in> set (identify_blocks A k)"
simps_1: fixes p :: "('a, 'b) form"   and c :: "'a"   and cs :: "'a list" shows "consts_for_unis (Forall p) (c # cs) = consts_for_unis (p[App c []/0]) cs"
cases: fixes a1 :: "cdi sequent list"   and a2 :: "cdi sequent" assumes "(a1, a2) \<in> g3ip" obtains A :: "cdi form" and B :: "cdi form" where "a1 = [ (\<LM> Aa__  \<RM>) \<oplus> Ba__ \<Rightarrow>* Em]" and "a2 = ( \<LM> Aa__ \<and>* Ba__  \<RM> \<Rightarrow>* Em)"   | A :: "cdi form" and B :: "cdi form" where "a1 = [ \<Empt> \<Rightarrow>* Ab__,  \<Empt> \<Rightarrow>* Bb__]" and "a2 = ( \<Empt> \<Rightarrow>* Ab__ \<and>* Bb__)"   | A :: "cdi form" and B :: "cdi form" where "a1 = [ \<LM> Ac__  \<RM> \<Rightarrow>* Em,  \<LM> Bc__  \<RM> \<Rightarrow>* Em]" and "a2 = ( \<LM> Ac__ \<or>* Bc__  \<RM> \<Rightarrow>* Em)"   | A :: "cdi form" and B :: "cdi form" where "a1 = [ \<Empt> \<Rightarrow>* Ad__]" and "a2 = ( \<Empt> \<Rightarrow>* Ad__ \<or>* Bd__)"   | B :: "cdi form" and A :: "cdi form" where "a1 = [ \<Empt> \<Rightarrow>* Be__]" and "a2 = ( \<Empt> \<Rightarrow>* Ae__ \<or>* Be__)"   | A :: "cdi form" and B :: "cdi form" where "a1 = [ \<LM> Af__ \<supset> Bf__  \<RM> \<Rightarrow>* Af__,  \<LM> Bf__  \<RM> \<Rightarrow>* Em]" and "a2 = ( \<LM> Af__ \<supset> Bf__  \<RM> \<Rightarrow>* Em)"   | A :: "cdi form" and B :: "cdi form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* B]" and "a2 = ( \<Empt> \<Rightarrow>* A \<supset> B)"