hcju/isabelleps huggingface dataset:
bind_split: fixes P :: "'b option \<Rightarrow> bool"   and m :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = \<lfloor>v\<rfloor> \<longrightarrow> P (f v)))"
le_lappend: fixes r :: "'b llist"   and s :: "'b llist" shows "r \<le> r @@ s"
accepts_def: fixes A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "NAe.accepts A w = (\<exists>q. (start A, q) \<in> NAe.steps A w \<and> fin A q)"
case_prod_conv: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
flow_initial_time_if: fixes t0 :: "real"   and x0 :: "('b, 'sz) vec" shows "ll.flow t0 x0 t0 = (if t0 \<in> UNIV \<and> x0 \<in> UNIV then x0 else 0)"
ffold_def: shows "ffold \<equiv> map_fun id (map_fun id (map_fun fset id)) Finite_Set.fold"
f_def: shows "f = (\<lambda>m. poly.coeff (mpoly_to_poly v p) (lookup m v) when keys m \<subseteq> {v})"
AbGroup_nth_subset_inner_dirsum: fixes Gs :: "'a set list"   and n :: "nat" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs" shows "Gs ! n \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"
gr\<I>_2: shows "ground (subst_range (rm_vars B \<I>))"
a_def: fixes x :: "nat \<Rightarrow>\<^sub>0 nat" shows "a x \<equiv> MPoly_Type.coeff p x"
kronecker_factorization_rat_main_def: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and bnd :: "nat"   and p :: "rat poly" shows "kronecker_factorization_rat_main df dp bnd p \<equiv> map_option of_int_poly (kronecker_factorization_main df dp bnd (snd (rat_to_normalized_int_poly p)))"
cases: fixes a1 :: "ccs"   and a2 :: "ccs" assumes "a1 \<sim> a2" obtains P :: "ccs" and Q :: "ccs" where "a1 = P" and "a2 = Q" and "P \<leadsto>[bisim] Q" and "Q \<sim> P"
Atm_BisT: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>T Atm atm"
exec_gpv2_def: shows "exec_gpv2 \<equiv> exec_gpv"
ln_lower_7_def: shows "ln_lower_7 \<equiv> \<lambda>x. - ln_upper_7 (inverse x)"
cycle_rotate_len_arcs_successive: fixes xs :: "nat list"   and i :: "nat"   and j :: "nat"   and P :: "nat \<times> nat \<Rightarrow> bool"   and a :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "1 < length xs"   and "(i, j) \<in> arcs' xs"   and "successive P (arcs a a xs)"   and "\<not> P (a, hd xs) \<or> \<not> P (last xs, a)" shows "\<exists>ys zs. len M a a xs = len M i i (j # ys @ a # zs) \<and> set (arcs a a xs) = set (arcs i i (j # ys @ a # zs)) \<and> xs = zs @ i # j # ys \<and> successive P (arcs i i (j # ys @ a # zs))"
hyp_edge_from_valid_out_port: fixes "is" :: "nat list"   and c :: "'form"   and n :: "nat"   and s :: "'subst"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" assumes "is \<in> it_paths (it' c)"   and "c \<in> set conclusions"   and "tree_at (it' c) is = HNode n s ants" shows "valid_out_port (hyp_edge_from c is n s)"
simps_1: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" shows "non_volatile_owned_or_read_only pending_write \<S> \<O> [] = True"
set_cut_same: fixes x :: "'b"   and w :: "'b list" shows "set (cut_same x (w @- sconst x)) \<subseteq> set w"
KBasic2_7: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<box>\<psi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) in v]"
S_def: shows "S \<equiv> {p. conic_sgn p = 0}"
le_fin_def: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action 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"
not_is_Done_conv_Pause: fixes r :: "('f, 'g, 'h) Resumption.resumption" shows "(\<not> Resumption.resumption.is_Done r) = (\<exists>out c. r = Resumption.resumption.Pause out c)"
assms_3: shows "x \<in> set xs \<union> set ys"
evalF_subF_eq: fixes M :: "model"   and A :: "formula" shows "\<forall>phi theta. evalF M phi (subF theta A) = evalF M (phi \<circ> theta) A"
cast_cast_below1: fixes A :: "'a defl"   and B :: "'a defl"   and x :: "'a" assumes "A \<sqsubseteq> B" shows "cast\<cdot>A\<cdot>(cast\<cdot>B\<cdot>x) = cast\<cdot>A\<cdot>x"
diffs_sin_coeff: shows "diffs sin_coeff = cos_coeff"
subgroup_nonempty: fixes G :: "('f, 'g) monoid_scheme" shows "\<not> subgroup {} G"
cancel_cancel: fixes a :: "bool \<times> 'c"   and b :: "bool \<times> 'c"   and c :: "bool \<times> 'c" assumes "canceling a b"   and "canceling b c" shows "a = c"
generated_ring_Int: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> G.generated_ring"   and "b \<in> G.generated_ring" shows "a \<inter> b \<in> G.generated_ring"
SOME_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "(SOME i. i \<in> CBasis) \<in> CBasis"
code_ref_list: fixes us :: "'a list list" assumes "us \<in> lists \<C>" shows "Ref \<C> us = us"
hs_isEmpty_def: shows "hs_isEmpty \<equiv> hm_isEmpty"
DiffD1: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A - B" shows "c \<in> A"
mult_one_right: fixes z :: "complex_homo" shows "z *\<^sub>h 1\<^sub>h = z"
atms_less_eq_imp_lit_less_eq_neg: fixes C :: "'a literal multiset"   and A :: "'a"   and L :: "'a literal" assumes "\<And>B. B \<in> atms_of C \<Longrightarrow> B \<le> A"   and "L \<in># C" shows "L \<le> Neg A"
T_invT: fixes h :: "'h" assumes "h \<in> H" shows "T (invT h) = h"
semiring_simprules_11: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "semiring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
ostep_invariant_def': fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool" shows "(A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. \<forall>a s'. (s, a, s') \<in> automaton.trans A \<and> S (fst s) (fst s') a \<longrightarrow> P (s, a, s'))"
S_def: shows "S = {x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> a \<le> x \<and> x \<le> c}"
vsv_vimageI2': fixes b :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V"   and a :: "V" assumes "b = VLambda A f\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f)" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (VLambda A f)"
mset_right_cancel_elem: fixes a :: "'a"   and A :: "'a multiset"   and b :: "'a" assumes "a \<in># A + {#b#}"   and "a \<noteq> b" shows "a \<in># A"
lookup_pair_tl: fixes xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "tc.oalist_inv_raw xs" shows "lookup_pair_tc (tl xs) k = (if \<forall>k'\<in>fst ` set xs. tc.le k k' then 0::'b else lookup_pair_tc xs k)"
param_curry: fixes Ra :: "('c \<times> 'f) set"   and Rb :: "('d \<times> 'g) set"   and Rc :: "('e \<times> 'h) set" shows "(curry, curry) \<in> (Ra \<times>\<^sub>r Rb \<rightarrow> Rc) \<rightarrow> Ra \<rightarrow> Rb \<rightarrow> Rc"
OrdinaryObjectsPossiblyConcreteAxiom: shows "\<forall>x. \<exists>v. ConcreteInWorld x v"
incl: fixes h :: "'b"   and H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
A_Exists_2: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<exists>!x. \<phi> x) in v]"
finite_comm_group_adjoin_axioms: shows "finite_comm_group_adjoin G {\<one>} x"
f_bounded: fixes t :: "real"   and x :: "'a" assumes "t \<in> T"   and "x \<in> X" shows "norm (f t x) \<le> B"
assms_9: shows "\<forall>i\<in>I. \<forall>j\<in>I. Xin i \<inter> Xout j = {}"
subtype_refl: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Gamma> \<turnstile> T <: T"
height_split_min: fixes t :: "'b tree23"   and x :: "'b"   and t' :: "'b upD" assumes "split_min t = (x, t')"   and "0 < height t"   and "complete t" shows "hD t' = height t"
eq: shows "adjoint U1 * (adjoint U2 * P * U2) * U1 = adjoint (U2 * U1) * P * (U2 * U1)"
Per_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C" shows "Per A B C \<and> Per C B A"
omit_branch_mem_dual: fixes i :: "'c"   and xs :: "(nat \<times> nat) set"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm" assumes "p at i in' set (omit_branch xs branch)" shows "p at i in' set branch"
analz_insert_freshK1: fixes evs :: "event list"   and K :: "nat"   and SesKey :: "nat" assumes "evs \<in> kerbIV_gets"   and "K \<in> authKeys evs \<union> range shrK"   and "SesKey \<notin> range shrK" shows "(Key K \<in> analz (insert (Key SesKey) (knows Spy evs))) = (K = SesKey \<or> Key K \<in> analz (knows Spy evs))"
Assign: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> s(x := aval a s)"
simps_25: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "'c" shows "map_tpoly f1 f2 (PVar x1) = PVar (f1 x1)"
simps_8: fixes f :: "'h \<Rightarrow> 'g" shows "map f [] = []"
inf_V_def: fixes x :: "V"   and y :: "V" shows "x \<sqinter> y \<equiv> set (elts x \<inter> elts y)"
swap_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" shows "X #[x \<and> y]_xs = asTerm (pick X #[[x \<and> y]]_xs)"
ideD_2: fixes a :: "'d" assumes "D.ide a" shows "D.dom a = a"
disjoint_image_iff: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and I :: "'b set" shows "(A \<subseteq> - f ` I) = (\<forall>i\<in>I. f i \<notin> A)"
strict_order_iff: fixes b :: "'d"   and a :: "'d" shows "(b < a) = (a = sup a b \<and> a \<noteq> b)"
gcd_proj2_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and m :: "'b"   and n :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd m n = normalize n) = dvd.dvd times n m"
distribute_alpha_over_sum: fixes a :: "real"   and p :: "'b pmf"   and f :: "'b \<Rightarrow> real"   and T :: "'b pmf" shows "(\<Sum>i\<in>set_pmf T. a * pmf p i * f i) = a * (\<Sum>i\<in>set_pmf T. pmf p i * f i)"
cover: shows "{a--b} \<subseteq> (\<Union>x\<in>X. ball x (d' x))"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) G"
bound_on_all_plans_bounds_MPLS_finite: fixes P :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> bool"   and f :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> finite PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB x. P PROB \<and> finite PROB \<longrightarrow> x \<in> MPLS PROB \<longrightarrow> x < f PROB"
ksup_comm: fixes f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" shows "f \<squnion> g = g \<squnion> f"
tagged_division_ofD_2: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "x \<in> K"
value_written_cong: fixes P :: "'a prog"   and P' :: "'a prog"   and a :: "nat"   and a' :: "nat"   and E :: "('c \<times> ('b, 'c) obs_event action) llist"   and E' :: "('c \<times> ('b, 'c) obs_event action) llist" assumes "P = P'"   and "a = a'"   and "action_obs E a' = action_obs E' a'" shows "value_written P E a = value_written P' E' a'"
s_x_c: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<cdot> 1\<^sub>\<pi> = (0::'a)"
exhaust_sel: fixes tree :: "'b tree" obtains "tree = Node (root tree) (left tree) (right tree)"
enumBase_length: fixes ls :: "nat list"   and nmax :: "nat" assumes "ls \<in> set (enumBase nmax)" shows "|ls| = 1"
dvd_triv_left: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times a b)"
fin: fixes x :: "real" shows "finite (S \<inter> f -` {x})"
ZObisT_matchC_ZOC_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01T t" shows "matchC_ZOC ZObisT t s"
simps_4: fixes l1 :: "value"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (L l1) (Minus v va) = True"
Quotient_rep_abs_fold_unmap: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'b"   and Rep' :: "'c \<Rightarrow> 'b" assumes "Quotient R Abs Rep T"   and "x' \<equiv> Abs x"   and "R x x"   and "Rep x' \<equiv> Rep' x'" shows "R (Rep' x') x"
get_ancestors_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok Shadow_DOM.get_ancestors ptr" shows "ptr |\<in>| object_ptr_kinds h"
axioms_4: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" assumes "Limit.cone J C D a \<chi>" shows "constant_functor J C a"
emeasure_bind_prob_algebra: fixes A :: "'d measure"   and N :: "'d measure"   and B :: "'d \<Rightarrow> 'e measure"   and L :: "'e measure"   and X :: "'e set" assumes "A \<in> space (prob_algebra N)"   and "B \<in> N \<rightarrow>\<^sub>M prob_algebra L"   and "X \<in> sets L" shows "emeasure (A \<bind> B) X = \<integral>\<^sup>+ x. emeasure (B x) X \<partial>A"
singleton_eqpoll: fixes x :: "'a"   and y :: "'b" shows "{x} \<approx> {y}"
iLast_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P \<equiv> P (iprev t0 I)"
snth_0: fixes a :: "'c"   and w :: "'c stream" shows "(a ## w) !! 0 = a"
comm_root: fixes x :: "'b list"   and y :: "'b list" shows "(x \<cdot> y = y \<cdot> x) = (\<exists>t. x \<in> t* \<and> y \<in> t*)"
is_extensional: fixes f :: "'b" assumes "\<not> B.arr f" shows "F f = A.null"
sup_idem: fixes x :: "'b multiset" shows "x \<union># x = x"
H_k_ket_zero_k: fixes k :: "nat" assumes "k < n" shows "H_k k *\<^sub>v ket_zero_k k = ket_plus_k k"
next_plane0_nonfinals: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)" shows "nonFinals g \<noteq> []"
mult_less_0_iff: fixes a :: "'a"   and b :: "'a" shows "(a * b < (0::'a)) = ((0::'a) < a \<and> b < (0::'a) \<or> a < (0::'a) \<and> (0::'a) < b)"
arr_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>" shows "partial_magma.arr S (set_category_given_img.mkIde S img \<AA> A) = (A \<subseteq> Collect (category.terminal S) \<and> |A| <o \<AA>)"
inject: fixes x1 :: "'c extended"   and x2 :: "'c"   and x3 :: "'c extended"   and x4 :: "'d"   and y1 :: "'c extended"   and y2 :: "'c"   and y3 :: "'c extended"   and y4 :: "'d" shows "(InsertMessage x1 x2 x3 x4 = InsertMessage y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
frontierI: fixes x :: "'b"   and T :: "'b set set"   and s :: "'b set" assumes "x \<in> closure\<^bsub>T\<^esub> s"   and "x \<in> core\<^bsub>T\<^esub> s \<Longrightarrow> False" shows "x \<in> frontier\<^bsub>T\<^esub> s"
is_locally_frac_cong: fixes U :: "'a set set"   and f :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and g :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "\<And>x. x \<in> U \<Longrightarrow> f x = g x" shows "is_locally_frac f U = is_locally_frac g U"
edge_succ_permutes: shows "edge_succ M permutes arcs G"
arrE: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" obtains "bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Dom F)" and "bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Dom F)" and "bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Cod F)" and "bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Cod F)" and "bicategorical_language.Src src\<^sub>B (concrete_category.Dom F) = bicategorical_language.Src src\<^sub>B (concrete_category.Cod F)" and "bicategorical_language.Trg trg\<^sub>B (concrete_category.Dom F) = bicategorical_language.Trg trg\<^sub>B (concrete_category.Cod F)" and "partial_magma.in_hom V\<^sub>B (concrete_category.Map F) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Dom F)) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Cod F))" and "F \<noteq> concrete_category.Null"
rangeE: fixes b :: "'f"   and f :: "'g \<Rightarrow> 'f" assumes "b \<in> range f" obtains x :: "'g" where "b = f x"
strongAssign: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and a :: "aexp"   and v :: "int"   and x :: "char list"   and ds :: "int"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "symeval P a v"   and "Frame P ($ 1 \<and>* x \<hookrightarrow> ds) F"   and "\<And>s. (x \<hookrightarrow> v \<and>* F) s \<Longrightarrow> Q' s" shows "\<turnstile>\<^sub>3 {P} x ::= a { Q'}"
MWLsSteps_det_cases_5: fixes \<iota> :: "nat"   and b :: "'exp"   and c :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>while\<^bsub>\<iota>\<^esub> b do c od,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "\<alpha> = []" and "p = Some (c;while\<^bsub>\<iota>\<^esub> b do c od)" and "m' = m" and "BMap (E b m)"   | "\<alpha> = []" and "p = None" and "m' = m" and "\<not> BMap (E b m)"
extreme_bound_mono: fixes X :: "'b set"   and Y :: "'b set"   and A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sX :: "'b"   and sY :: "'b" assumes "X \<subseteq> Y"   and "extreme_bound A less_eq X sX"   and "extreme_bound A less_eq Y sY" shows "less_eq sX sY"
cqt_further_10: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<alpha> :: "'a" shows "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<equiv> \<beta> \<^bold>= \<alpha>) in v]"
ivl_disj_un_one_1: fixes l :: "'b"   and u :: "'b" assumes "l < u" shows "{..l} \<union> {l<..<u} = {..<u}"
change_respecting_doesnt_modify: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and g :: "('Var \<Rightarrow> 'Val option) \<Rightarrow> 'Var \<Rightarrow> 'Val option"   and f :: "'Var \<Rightarrow> 'Val option"   and x :: "'Var" assumes "change_respecting (cms, mem) (cms', mem') X g"   and "(cms, mem) \<leadsto> (cms', mem')"   and "dom f = X"   and "x \<in> dom (g f)"   and "doesnt_read (fst cms) x" shows "mem x = mem' x"
winding_number_big_meets: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and w :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "1 \<le> \<bar>Re (winding_number \<gamma> z)\<bar>"   and "w \<noteq> z" shows "\<exists>a>0. z + complex_of_real a * (w - z) \<in> path_image \<gamma>"
comp_same: fixes x :: "'a" shows "comparator_of x x = Eq"
starfun_def: fixes f :: "'a \<Rightarrow> 'b" shows "*f* f \<equiv> Ifun (star_of f)"
Inf_top_conv_2: fixes A :: "'al set" assumes "A \<subseteq> U" shows "(\<top>\<^sub>o\<^sub>w = \<Sqinter>\<^sub>o\<^sub>w A) = (\<forall>x\<in>A. x = \<top>\<^sub>o\<^sub>w)"
right_options: fixes l :: "game zet"   and r :: "game zet" shows "right_options (Game l r) = r"
in_hhom_def: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright> \<equiv> D.arr \<mu> \<and> src\<^sub>D \<mu> = a \<and> trg\<^sub>D \<mu> = b"
Inf_admS_def: fixes A :: "'a admS set" shows "Inf A \<equiv> mklr (\<Inter> (unlr ` A))"
simps_1: fixes x2 :: "'c"   and y2 :: "'c" shows "(Some x2 = Some y2) = (x2 = y2)"
create_get: fixes x :: "'a \<Longrightarrow> 'b"   and v :: "'a" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (create\<^bsub>x\<^esub> v) = v"
prod_prime_factors'': fixes n :: "nat" assumes "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f_prime p ^ multiplicity p n)"
f4: shows "virt_to_phys addr (mmu (add_data_cache s1 addr data bm)) (mem (add_data_cache s1 addr data bm)) = None \<or> virt_to_phys addr (mmu (add_data_cache s1 addr data bm)) (mem (add_data_cache s1 addr data bm)) = Some (pp (virt_to_phys addr (mmu (add_data_cache s1 addr data bm)) (mem (add_data_cache s1 addr data bm))))"
strict_iff_order: fixes ko :: "'b key_order"   and b :: "'b"   and a :: "'b" shows "ko.lt ko b a = (ko.le ko b a \<and> a \<noteq> b)"
cod_src_eq_dom_src: shows "\<eta>.cod.src = \<eta>.dsrc"
chop_subst_Sym: fixes s :: "('a, 'b) tm"   and \<rho> :: "'b \<Rightarrow> ('a, 'b) tm" assumes "is_App s"   and "is_Sym (head s)" shows "chop (subst \<rho> s) = subst \<rho> (chop s)"