hcju/isabelleps huggingface dataset:
fresh_fresh_subst: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "good X"   and "fresh ys y Y" shows "fresh ys y (X #[Y / y]_ys)"
aext_var: fixes x :: "'b \<Longrightarrow> 'd"   and a :: "'d \<Longrightarrow> 'c" shows "x \<oplus>\<^sub>p a = x ;\<^sub>L a"
that: fixes T :: "'a set" assumes "affine T"   and "T \<subseteq> S"   and "aff_dim T = d" shows "thesis"
simps_5: fixes v :: "ternaryvalue" shows "NNF_ternary (TernaryNot (TernaryValue v)) = TernaryValue (eval_ternary_Not v)"
assms: shows "x \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>"
rounding_error: fixes j :: "nat" assumes "t j \<le> t0 + e1'" shows "sqrt (real DIM('a)) * 2 powr real_of_int (- e) \<le> euler_C TYPE('a) / 2 * stepsize j"
prems_2: shows "oid \<in> set (interp_ins (xs @ ys @ [y]))"
sWlsValNE_def: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and x :: "'var" shows "sWlsValNE SEM x \<equiv> \<exists>val. sWlsVal SEM val"
PiE_ext: fixes x :: "'b \<Rightarrow> 'c"   and k :: "'b set"   and s :: "'b \<Rightarrow> 'c set"   and y :: "'b \<Rightarrow> 'c" assumes "x \<in> Pi\<^sub>E k s"   and "y \<in> Pi\<^sub>E k s"   and "\<And>i. i \<in> k \<Longrightarrow> x i = y i" shows "x = y"
trg_def: fixes \<mu> :: "'c" shows "C'.trg \<mu> = (if C'.arr \<mu> then trg\<^sub>C \<mu> else C'.null)"
abs_summable_on_reindex_iff: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f :: "'c \<Rightarrow> 'd" assumes "inj_on g A" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on g ` A)"
rel_state_update: shows "((=) ===> (=) ===> local.rel_state (=)) update_st update"
nAct_eq_proj: fixes n :: "nat"   and c :: "'id"   and t :: "cnf llist" assumes "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>)" shows "\<langle>c #\<^bsub>enat n\<^esub>t\<rangle> = llength (\<pi>\<^bsub>c\<^esub>t)"
col_formula: fixes M1 :: "'a list list"   and M2 :: "'a list list" shows "\<forall>j. j < length M1 * length M2 \<and> mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> col (M1 \<otimes> M2) j = vec_vec_Tensor (col M1 (j div length M2)) (col M2 (j mod length M2))"
inf_add_left: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "\<infinity>\<^sub>h +\<^sub>h z = \<infinity>\<^sub>h"
paths_good_def: shows "paths_good \<equiv> \<lambda>(paths', P_new'). ProofStepInduct G v0 v1 paths' P_new' sep_size"
nth_member: fixes n :: "nat"   and L :: "'a list" assumes "n < length L" shows "List.member L (L ! n)"
update_convs_1: fixes getM' :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getM_update getM' \<lparr>getM = getM, \<dots> = more\<rparr> = \<lparr>getM = getM' getM, \<dots> = more\<rparr>"
cases: fixes x :: "tv" obtains "x = Det True"   | "x = Det False"   | n :: "nat" where "x = Indet n"
fds_ln_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b fds"   and l' :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> fds_nth (f x) 1 \<noteq> (0::'b)"   and "sum l' A = l" shows "fds_ln l (prod f A) = (\<Sum>x\<in>A. fds_ln (l' x) (f x))"
MP': fixes ps :: "('c, 'd) fm list"   and p :: "('c, 'd) fm"   and q :: "('c, 'd) fm" assumes "ps \<turnstile> p \<^bold>\<longrightarrow> q"   and "ps \<turnstile> p" shows "ps \<turnstile> q"
Diff_subset: fixes A :: "'g set"   and B :: "'g set" shows "A - B \<subseteq> A"
add2_self: fixes p :: "real \<times> real"   and q :: "real \<times> real" shows "ccw' 0 (p + q) p = ccw' 0 q p"
substNext_code_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst.substNext_code p g next \<equiv> \<lambda>v. if v = snd next then uninst.substitution_code p g next else v"
u_gt_s_if_ui: fixes ui :: "('s, 'v) tm" assumes "ui \<in> set (args u)"   and "ui \<ge>\<^sub>t t" shows "u >\<^sub>t s"
Conj\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" shows "p \<bullet> Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = Conj\<^sub>\<alpha> (p \<bullet> tset\<^sub>\<alpha>)"
redT_updTs_preserves_lock_thread_ok: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and nts :: "('e, 'f, 'g) new_thread_action list" assumes "lock_thread_ok ls ts" shows "lock_thread_ok ls (redT_updTs ts nts)"
has_chain_integral_chain_integral4: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and e :: "complex" assumes "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i"
system_automaton_simps_3: fixes cstep :: "'a \<Rightarrow> 'e + 'f \<Rightarrow> 'a \<Rightarrow> bool"   and en :: "'b \<times> 'c \<Rightarrow> 'e \<Rightarrow> bool option"   and ex :: "'b \<times> 'c \<Rightarrow> 'e \<Rightarrow> ('b \<times> 'c) option"   and ginit :: "('a, 'b, 'c) Gen_Scheduler.global_config set"   and glabel :: "('a, 'b, 'c) Gen_Scheduler.global_config \<Rightarrow> 'd" shows "g_V0 (Gen_Scheduler_linit.system_automaton cstep en ex ginit glabel) = Gen_Scheduler_linit.init ginit"
hcomp_in_vhom: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a"   and \<nu> :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow>\<^sub>C k\<guillemotright>"   and "src\<^sub>C h = trg\<^sub>C f" shows "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : h \<star>\<^sub>C f \<Rightarrow>\<^sub>C k \<star>\<^sub>C g\<guillemotright>"
ts''_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [$Load t tp a off] : ts'' _> ts'"
aff_dim_parallel_subspace_aux: fixes B :: "'a set"   and a :: "'a" assumes "\<not> affine_dependent B"   and "a \<in> B" shows "finite B \<and> card B - 1 = dim (span ((+) (- a) ` (B - {a})))"
pwd_spy_sesk_def: fixes n :: "nat" shows "pwd_spy_sesk n \<equiv> let (s, S, A, B, C, D) = pwd_spy_sep_agr n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, SesKey SK) s, S, A, B, C, D)"
aff\<M>: fixes C :: "'a set" assumes "C \<in> \<M>" shows "aff_dim C \<le> int n"
connected_space_clopen_in: fixes X :: "'c topology" shows "connected_space X = (\<forall>T. openin X T \<and> closedin X T \<longrightarrow> T = {} \<or> T = topspace X)"
rtrancl_def: shows "rtrancl \<equiv> \<lambda>r. {(xa, x). (\<lambda>x xa. (x, xa) \<in> r)\<^sup>*\<^sup>* xa x}"
mark_guards_simps_8: fixes f :: "'f"   and c :: "'d \<Rightarrow> ('d, 'e, 'g) com" shows "mark_guards f (DynCom c) = DynCom (\<lambda>s. mark_guards f (c s))"
map_of_zip_map: fixes f :: "'a \<Rightarrow> 'b"   and rs :: "'a list"   and k :: "'b"   and v :: "'a" assumes "map_of (zip (map f rs) rs) k = Some v" shows "k = f v"
times_in_PolysD: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "p * q \<in> P[X]"   and "p \<in> P[X]"   and "p \<noteq> 0" shows "q \<in> P[X]"
ereal_plus_eq_PInfty: fixes a :: "ereal"   and b :: "ereal" shows "(a + b = \<infinity>) = (a = \<infinity> \<or> b = \<infinity>)"
map_of_reinsert_SomeD: fixes D :: "'c"   and fs :: "('b \<times> 'a) list"   and F :: "'b"   and T :: "'a" assumes "map_of (map (\<lambda>(F, y). ((F, D), y)) fs) (F, D) = \<lfloor>T\<rfloor>" shows "map_of fs F = \<lfloor>T\<rfloor>"
lnth_Suc_LCons: fixes x :: "'b"   and xs :: "'b llist"   and n :: "nat" shows "lnth (LCons x xs) (Suc n) = lnth xs n"
oth_class_taut_3_e: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<or> \<psi> \<^bold>\<equiv> \<psi> \<^bold>\<or> \<phi> in v]"
suminf_NSsuminf_iff: fixes f :: "nat \<Rightarrow> real" shows "suminf f = NSsuminf f"
aligned_bl_add_size: fixes n :: "nat"   and x :: "'b word"   and y :: "'b word" assumes "n \<le> size x"   and "drop (size x - n) (to_bl x) = replicate n False"   and "take (size x - n) (to_bl y) = replicate (size x - n) False" shows "to_bl (x + y) = take (size x - n) (to_bl x) @ drop (size x - n) (to_bl y)"
rchop_def: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<equiv> left r = left s \<and> right s = left t \<and> right r = right t"
osum_Linear_order: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Linear_order r"   and "Linear_order r'" shows "Linear_order (r +o r')"
seqE: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "B.VxVxV.seq g f" obtains "B.seq (fst g) (fst f)" and "B.VxV.seq (snd g) (snd f)"
splitAt_dist_ram11: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a, []) = splitAt ram2 (fst (splitAt ram1 vs))"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q M xvec N Q'"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M xvec N (P \<parallel> Q')"
quasi_inverse_simps_3: fixes f :: "'a" assumes "equivalence_map f" shows "src f\<^sup>~ = trg f"
assms_3: shows "num_params (tm_poly t1) \<le> length I"
assms_1: shows "\<Phi>.lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = \<Phi>.lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<psi>"
row_def: fixes m :: "'c list list"   and i :: "nat" shows "row m i \<equiv> map (\<lambda>w. w ! i) m"
not_hidden_index_nth: fixes i :: "nat"   and Vs :: "'c list" assumes "i < length Vs"   and "\<not> hidden Vs i" shows "index Vs (Vs ! i) = i"
y_def': shows "y = f \<lparr>[b, a]\<^sub>\<circ>\<lparr>1\<^sub>\<nat>\<rparr>, [b, a]\<^sub>\<circ>\<lparr>[]\<^sub>\<circ>\<rparr>\<rparr>\<^sub>\<bullet>"
element_put_get_preserved: fixes getter :: "('l, 'm, 'n, 'o, 'p option) RElement_scheme \<Rightarrow> 'q"   and setter :: "('r \<Rightarrow> 'r) \<Rightarrow> ('l, 'm, 'n, 'o, 'p option) RElement_scheme \<Rightarrow> ('l, 'm, 'n, 'o, 'p option) RElement_scheme"   and v :: "'r"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and element_ptr' :: "(_) element_ptr" assumes "\<And>x. getter (setter (\<lambda>_. v) x) = getter x"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'"
hypnat_add_self_le: fixes x :: "nat star"   and n :: "nat star" shows "x \<le> n + x"
\<phi>_\<psi>: fixes t' :: "'s" assumes "t' \<in> S.Univ" shows "\<psi> (INV.\<psi> t') = t'"
n1_n_n2: shows "n1 \<noteq> n2"
row_rank_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_rank A = vec.dim (row_space A)"
length_sign_vec: fixes ps :: "rat poly list"   and x :: "real" shows "length (sign_vec ps x) = length ps"
deriv_R: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c R,B"
dvd_0_right: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and a :: "'m" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
unambigous_prefix_routing_weak_mono: fixes rr :: "('c, 'd) routing_rule_scheme"   and rtbl :: "('c, 'd) routing_rule_scheme list"   and rr' :: "('c, 'd) routing_rule_scheme" assumes "is_longest_prefix_routing (rr # rtbl)"   and "rr' \<in> set rtbl" shows "routing_rule_sort_key rr \<le> routing_rule_sort_key rr'"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = min a b" shows "a \<le> b"
mpdouble_def: fixes m :: "int"   and a :: "int"   and p :: "int \<times> int \<times> int" shows "mpdouble m a p = (let (x, y, z) = p in if z = 0 then p else let l = 2 mod m **\<^bsub>m\<^esub> y **\<^bsub>m\<^esub> z; n = 3 mod m **\<^bsub>m\<^esub> x ^^^\<^bsub>m\<^esub> 2 ++\<^bsub>m\<^esub> a **\<^bsub>m\<^esub> z ^^^\<^bsub>m\<^esub> 2 in (l **\<^bsub>m\<^esub> (n ^^^\<^bsub>m\<^esub> 2 --\<^bsub>m\<^esub> 4 mod m **\<^bsub>m\<^esub> x **\<^bsub>m\<^esub> y **\<^bsub>m\<^esub> l), n **\<^bsub>m\<^esub> (6 mod m **\<^bsub>m\<^esub> x **\<^bsub>m\<^esub> y **\<^bsub>m\<^esub> l --\<^bsub>m\<^esub> n ^^^\<^bsub>m\<^esub> 2) --\<^bsub>m\<^esub> 2 mod m **\<^bsub>m\<^esub> y ^^^\<^bsub>m\<^esub> 2 **\<^bsub>m\<^esub> l ^^^\<^bsub>m\<^esub> 2, l ^^^\<^bsub>m\<^esub> 3))"
l_inv_ex: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
resolute_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "resolute_on A f = (\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> (\<exists>a. f B = {a}))"
eq_\<I>_converterD_WT: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and conv2 :: "('i, 'j, 'k, 'l) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd> = \<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
\<sigma>2_1: fixes v :: "'a" assumes "v \<in> V\<^bsub>G'\<^esub>" shows "G'.strategy p \<sigma>2"
LLL_inv_wD_6: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs" shows "length fs = m"
prob0_bell_snd: fixes v :: "complex mat" assumes "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>" shows "prob0 2 v 1 = 1 / 2"
new_pseudodeterministic: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and v :: "('r, 'l, 'v) val"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])" shows "revision_step r s s' = (\<exists>l. l \<notin> LID\<^sub>G s \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)])))"
da_elim_cases_23: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
intros: fixes p :: "'b poly"   and x :: "'b" assumes "lead_coeff p = (1::'b)"   and "\<forall>i. coeff p i \<in> \<int>"   and "poly p x = (0::'b)" shows "algebraic_int x"
sumPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto><Rel> Q"   and "Id \<subseteq> Rel"   and "Rel \<subseteq> Rel'" shows "P \<oplus> R \<leadsto><Rel'> Q \<oplus> R"
real_polynomial_function_power: fixes f :: "'c \<Rightarrow> real"   and n :: "nat" assumes "real_polynomial_function f" shows "real_polynomial_function (\<lambda>x. f x ^ n)"
triangle_in_hom_4: shows "\<guillemotleft>\<r>[g] \<bullet> (g \<star> \<epsilon>) \<bullet> \<a>[g, f, g] \<bullet> (\<eta> \<star> g) \<bullet> \<l>\<^sup>-\<^sup>1[g] : g \<Rightarrow> g\<guillemotright>"
f': shows "\<forall>i\<in>Basis. \<forall>x\<in>s. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>z. z *\<^sub>R f' x \<bullet> i)) (at x) \<and> continuous_on s f'"
ccSUP_upper2: 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 A :: "'d set"   and i :: "'d"   and u :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
set_sel_2: fixes a :: "'a inference" shows "concl_of a \<in> set_inference a"
unit_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "src (\<i> a) = a"
refl: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l, Z\<rangle>"
list_emb_ConsD: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "list_emb P (x # xs) ys" shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs"
some_unit_def: fixes f :: "'a" shows "some_unit f \<equiv> SOME \<eta>. \<exists>\<epsilon>. adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* \<eta> \<epsilon>"
locateI_def: shows "locateI \<equiv> locate2 brn"
d: shows "(d, t) \<rightarrow>*c (d'', t'')"
simps: fixes a :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" shows "Formula.hereditarily_fs a = ((\<exists>tset\<^sub>\<alpha>. a = Formula.Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<and> finite (supp tset\<^sub>\<alpha>) \<and> (\<forall>x. x \<in> set_bset tset\<^sub>\<alpha> \<longrightarrow> Formula.hereditarily_fs x)) \<or> (\<exists>t\<^sub>\<alpha>. a = Formula.Not\<^sub>\<alpha> t\<^sub>\<alpha> \<and> Formula.hereditarily_fs t\<^sub>\<alpha>) \<or> (\<exists>\<phi>. a = Formula.Pred\<^sub>\<alpha> \<phi>) \<or> (\<exists>t\<^sub>\<alpha> \<alpha>. a = Formula.Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> \<and> Formula.hereditarily_fs t\<^sub>\<alpha>))"
integrable_norm: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
f_map: fixes r :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "m \<in> V" shows "T (r \<cdot> m) = r \<star> T m"
emeasure_pair_measure_Times: fixes M :: "'b measure"   and A :: "'c set"   and N :: "'c measure"   and B :: "'b set" assumes "sigma_finite_measure M"   and "A \<in> sets N"   and "B \<in> sets M" shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
rtrancl_Image_step: fixes a :: "'b"   and r :: "('b \<times> 'b) set"   and A :: "'b set"   and b :: "'b" assumes "a \<in> r\<^sup>* `` A"   and "(a, b) \<in> r\<^sup>*" shows "b \<in> r\<^sup>* `` A"
l_inv_ex: fixes x :: "nat \<Rightarrow> nat" assumes "x \<in> carrier (alt_group n)" shows "\<exists>y\<in>carrier (alt_group n). y \<otimes>\<^bsub>alt_group n\<^esub> x = \<one>\<^bsub>alt_group n\<^esub>"
card_nyinitcls_bound: fixes G :: "prog"   and s :: "state" shows "card (nyinitcls G s) \<le> card {C. is_class G C}"
map_upd_def: fixes k :: "'a"   and v :: "'b"   and m :: "'a \<Rightarrow> 'b option" shows "map_upd k v m = m(k \<mapsto> v)"
distrib_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
mkarr_Lunit: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.mkarr \<^bold>\<l>\<^bold>[t\<^bold>] = \<F>C.\<ll> (\<F>C.mkarr t)"
Methd: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and sig :: "sig"   and n :: "nat"   and v :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n\<rightarrow> s1"
RSubmodule_set_plus: fixes L :: "'m set"   and N :: "'m set" assumes "RSubmodule L"   and "RSubmodule N" shows "RSubmodule (L + N)"
netglobalE: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and P :: "(nat \<Rightarrow> 'e) \<Rightarrow> bool"   and s :: "'c net_state"   and Q :: "(nat \<Rightarrow> 'e) \<Rightarrow> bool" assumes "openproc np onp sr"   and "openproc.netglobal np sr P s"   and "\<And>\<sigma>. \<lbrakk>P \<sigma>; fst (openproc.initmissing np sr (netgmap sr s)) = \<sigma>\<rbrakk> \<Longrightarrow> Q \<sigma>" shows "openproc.netglobal np sr Q s"
pbdom_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1" shows "elementary_category_with_pullbacks.pbdom C prj0 f g \<equiv> partial_magma.dom C (prj0 f g)"
ladder_every_stage_1: fixes index :: "nat" assumes "index < length L" shows "ladder_i L' index = ladder_i L index"
poly_weak_anti_mono_def: fixes p :: "('b monom \<times> 'c) list"   and v :: "'b" shows "poly_weak_anti_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<beta> v \<le> \<alpha> v \<longrightarrow> eval_poly \<alpha> p \<le> eval_poly \<beta> p"
terminal_arr_unique: fixes f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "CC.par f f'"   and "CC.terminal_arr f"   and "CC.terminal_arr f'" shows "f = f'"
outsideUnion: fixes R :: "('a \<times> 'b) set"   and x :: "'a" shows "R = Outside' {x} R \<union> {x} \<times> R `` {x}"
hyps_2: shows "liftAll (\<lambda>a. a $[[x \<and> y]]_zs $[[x \<and> y]]_zs = a) binp"
rbt_lookup_rbt_insert': fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and k :: "'b"   and v :: "'c" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "ord.rbt_lookup less (ord.rbt_insert less k v t) = ord.rbt_lookup less t(k \<mapsto> v)"
inf_pp_semi_commute: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - - y \<le> - - (x \<sqinter> y)"
min_maxsimpchainD_maxsimpchain: fixes xs :: "'a set list" assumes "domain.min_gallery xs" shows "domain.gallery xs"
n2a_def: shows "n2a \<equiv> \<Lambda> m n. up\<cdot>(m *\<^sub># n)"
un: shows "adjoint x * x \<le>\<^sub>L 1\<^sub>m d"
large: shows "2 ^ (5 * (degree F div 2) * (degree F div 2)) * \<parallel>F\<parallel>\<^sup>2 ^ (2 * (degree F div 2)) < pl\<^sup>2"
content_cball_gt_0_iff: fixes z :: "'c"   and r :: "real" shows "(0 < content (cball z r)) = (0 < r)"
ltake_LCons: fixes n :: "enat"   and x :: "'c"   and xs :: "'c llist" shows "ltake n (LCons x xs) = (case n of 0 \<Rightarrow> LNil | eSuc n' \<Rightarrow> LCons x (ltake n' xs))"