hcju/isabelleps huggingface dataset:
load_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> (fst instr = load_store_type LDSB \<or> fst instr = load_store_type LDUB \<or> fst instr = load_store_type LDUBA \<or> fst instr = load_store_type LDUH \<or> fst instr = load_store_type LD \<or> fst instr = load_store_type LDA \<or> fst instr = load_store_type LDD) \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (load_instr instr s1)) \<and> t2 = snd (fst (load_instr instr s2))" shows "low_equal t1 t2"
field_inverse: fixes a :: "'d" assumes "a \<noteq> (0::'d)" shows "inverse a * a = (1::'d)"
minus_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "f - g = (\<lambda>(x, y). x - y) \<circ>$ ($f, g$)"
rtrancl_eq_or_trancl: fixes x :: "'d"   and y :: "'d"   and R :: "('d \<times> 'd) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
r_ball: shows "proots (p - [:b:]) \<subseteq> ball z0 r"
E_wf_1: fixes G :: "'a graph" assumes "wf_graph G" shows "fst ` edges G \<subseteq> nodes G"
sum_image_mset_mono: fixes A :: "'b multiset"   and B :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "A \<subseteq># B" shows "\<Sum>\<^sub># (image_mset f A) \<le> \<Sum>\<^sub># (image_mset f B)"
set_drop_subset_set_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list" assumes "n \<le> m" shows "set (drop m xs) \<subseteq> set (drop n xs)"
mult_distr_tensor: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat"   and C :: "complex Matrix.mat"   and D :: "complex Matrix.mat" assumes "dim_col A = dim_row B"   and "dim_col C = dim_row D"   and "0 < dim_col A"   and "0 < dim_col B"   and "0 < dim_col C"   and "0 < dim_col D" shows "A * B \<Otimes> C * D = (A \<Otimes> C) * (B \<Otimes> D)"
starfun_exp_HInfinite_Infinitesimal_disj: fixes x :: "real star" assumes "x \<in> HInfinite" shows "(*f* exp) x \<in> HInfinite \<or> (*f* exp) x \<in> Infinitesimal"
actions_def: fixes asig :: "'a set \<times> 'a set \<times> 'a set" shows "actions asig \<equiv> inputs asig \<union> outputs asig \<union> internals asig"
is_zero_iarray_eq_iff: fixes A :: "('d, 'e) vec" shows "(A = 0) = is_zero_iarray (vec_to_iarray A)"
flow_of_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and e :: "nat \<times> nat" shows "flow_of_cf cf e \<equiv> if e \<in> E then c e - cf e else (0::'capacity)"
HPow_mono_iff: fixes x :: "hf"   and y :: "hf" shows "(HPow x \<le> HPow y) = (x \<le> y)"
add_ac_3: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "b \<oplus> (a \<oplus> c) = a \<oplus> (b \<oplus> c)"
evContainsEx0_allInstances: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula"   and i :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (0, FAll Neg A))" shows "EV (contains f (0, instanceF (X i) A))"
union_ts_def: fixes A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" shows "A \<Rightarrow>\<union> B = (\<lambda>k. A k \<union> B k)"
ba_minus: fixes x :: "'a"   and y :: "'a" shows "- x - - y = - (- - x \<squnion> - y)"
sf_aux4: fixes s :: "nat"   and t :: "nat" assumes "s < t" shows "sf s + s < sf t"
bad_def: fixes x :: "'a" shows "local.bad x = (\<exists>f. local.inf_chain f \<and> f 0 = x)"
K_cfg_trans: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and l :: "'s"   and R :: "'c \<Rightarrow> real"   and cfg' :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and l' :: "'s"   and R' :: "'c \<Rightarrow> real" assumes "cfg \<in> MDP.cfg_on (l, R)"   and "cfg' \<in> set_pmf (K_cfg cfg)"   and "state cfg' = (l', R')" shows "local.trans R R'"
ind_hyp: assumes "sorted_by cmp l'"   and "\<And>y. \<lbrakk>y \<in> set l'; \<not> cmp e y\<rbrakk> \<Longrightarrow> cmp y e" shows "sorted_by cmp (insert_sort_insert_by cmp e l')"
red_supsetD: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "A \<sqsupset>p B"   and "is_red B p" shows "is_red A p"
mset_update: fixes i :: "nat"   and ls :: "'c list"   and v :: "'c" assumes "i < length ls" shows "mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
RB: fixes s :: "'b"   and s' :: "'a" assumes "(s, s') \<in> R'"   and "I' s'"   and "I s"   and "b' s'" shows "b s"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "(inv hom x = inv hom y) = (x = y)"
Ewx: shows "e\<^sup>2 + 3 * f\<^sup>2 = x * w"
simpfm: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (simpfm p) = Ifm bs p"
foldl_un_empty_eq: fixes i :: "'c set"   and ww :: "'c set list" shows "foldl (\<union>) i ww = i \<union> foldl (\<union>) {} ww"
sas_plus_equivalent_to_strips_i_a_II: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and \<phi>\<^sub>s :: "('c, 'd) sas_plus_problem \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool option"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable (\<phi>\<^sub>s \<Psi> s) ops' \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops'" shows "SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')"
d_lb1: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y \<le> d x"
fixp_induct: fixes P :: "(('d, 'a, 'b) gpv \<Rightarrow> 's1 \<Rightarrow> 's2 \<Rightarrow> (('d \<times> 's1) \<times> 'd \<times> 's2) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>exec_until_bad. P (curry (curry exec_until_bad)))"   and "P (\<lambda>exec_until_bad gpv s1. return_pmf None)"   and "\<And>s2. P s2 \<Longrightarrow> P (\<lambda>a aa aaa. if bad1 aa \<or> bad2 aaa then pair_spmf (exec_gpv oracle1 a aa) (exec_gpv oracle2 a aaa) else the_gpv a \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf ((x, aa), x, aaa) | IO out f \<Rightarrow> joint_oracle (aa, aaa) out \<bind> (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (x, s1') \<Rightarrow> \<lambda>a. case a of (y, s2') \<Rightarrow> if bad1 s1' \<or> bad2 s2' then pair_spmf (exec_gpv oracle1 (f x) s1') (exec_gpv oracle2 (f y) s2') else s2 (f x) s1' s2') b)))" shows "P local.exec_until_bad"
going_to_mono: fixes F :: "'a filter"   and G :: "'a filter"   and A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "F \<le> G"   and "A \<subseteq> B" shows "f going_to F within A \<le> f going_to G within B"
separable_i_def: fixes ops :: "'a arith_ops_record"   and xs :: "'a list" shows "separable_i ops xs \<equiv> gcd_poly_i ops xs (pderiv_i ops xs) = one_poly_i ops"
isOK_def: fixes m :: "'a + 'b" shows "isOK m = (case m of Inl e \<Rightarrow> False | Inr x \<Rightarrow> True)"
up_filter_injective: fixes x :: "'c"   and y :: "'c" assumes "up_filter x = up_filter y" shows "x = y"
lmeasurable_iff_integrable: fixes S :: "'c set" shows "(S \<in> lmeasurable) = integrable lebesgue (indicat_real S)"
dests_inv: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:11, PRreq-:20, PRrep-:7, PRerr-:1} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>))) \<and> (l \<in> {PAodv-:16..PAodv-:17} \<union> {PPkt-:8..PPkt-:9} \<union> {PRreq-:12..PRreq-:13} \<union> {PRreq-:21..PRreq-:22} \<union> {PRrep-:8..PRrep-:9} \<union> {PRerr-:2..PRerr-:4} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> iD (rt \<xi>) \<and> the (dests \<xi> ip) = sqn (rt \<xi>) ip)) \<and> (l = PPkt-:12 \<longrightarrow> dip \<xi> \<in> iD (rt \<xi>)))"
axioms_1: fixes Ax :: "(ZF, ZF) Axioms" assumes "ZFAxioms Ax" shows "Axioms Ax"
check_tableau_index_valid: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and as :: "('a \<times> 'b atom) set" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "index_valid as (check s) = index_valid as s"
subfield_iff_1: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "ring R"   and "field (R\<lparr>carrier := K\<rparr>)"   and "K \<subseteq> carrier R" shows "subfield K R"
bij_betw_function_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (function_of A enum) {xs. set xs \<subseteq> B \<and> length xs = card A \<and> distinct xs} {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A}"
givenBy_eq_all: fixes v :: "'d \<Rightarrow> 'e" shows "givenBy v = {A. \<forall>x\<in>A. \<forall>y. v x = v y \<longrightarrow> y \<in> A}"
nofail_def: fixes S :: "'c nres" shows "nofail S \<equiv> S \<noteq> FAIL"
InvariantUniqC_def: fixes clause :: "Literal list" shows "InvariantUniqC clause = uniq clause"
bounded_linear_Blinfun_apply: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "blinfun_apply (Blinfun f) = f"
alpha'_2: fixes a :: "vrs"   and x :: "'a"   and b :: "vrs"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
map_filter_on_UNIV: shows "map_filter_on UNIV = filtermap"
compact_space_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "compact_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> (\<forall>i\<in>I. compact_space (X i)))"
block_size_t: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and index :: "int"   and u_block_size :: "int" assumes "k_\<Lambda>_PBD point_set block_collection index u_block_size" shows "2 \<le> u_block_size"
Im_cos: fixes z :: "complex" shows "Im (cos z) = sin (Re z) * (exp (- Im z) - exp (Im z)) / 2"
assms: fixes x :: "'a" assumes "x \<in> A" shows "0 \<le> f x"
h1: shows "sortedTree h t1 \<longrightarrow> sortedTree h (remove h e t1)"
edges_of_walk_middle: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list"   and xs' :: "'a list" shows "(v, w) \<in> edges_of_walk (xs @ v # w # xs')"
sa_iso_pair: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d" assumes "triple phi" shows "(- - x, phi (- x) \<squnion> up_filter y) \<in> triple.pairs phi"
a_2: shows "\<forall>a'\<ge>a. a' \<in> \<nat> \<longrightarrow> f \<in> O(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a' (real x)))"
set_cases: fixes e :: "'b"   and a :: "'b list" assumes "e \<in>. a" obtains z2 :: "'b list" where "a = e # z2__"   | z1 :: "'b" and z2 :: "'b list" where "a = z1 # z2" and "e \<in>. z2"
dsjAssoc1: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A OR B OR C" shows "H \<turnstile> (A OR B) OR C"
atLeast_eq_UNIV_iff: fixes x :: "'a" shows "({x..} = UNIV) = (x = bot)"
step_z_valid_dbm: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "valid_dbm D'"
measure_count_space: fixes A :: "'c set"   and X :: "'c set" shows "Sigma_Algebra.measure (count_space A) X = (if X \<subseteq> A then real (card X) else 0)"
tensor_mat_dim_col: fixes dims :: "nat list"   and vars :: "nat set"   and m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_col (tensor_mat dims vars m1 m2) = state_sig.d dims"
\<delta>ss_def: shows "\<delta>ss \<equiv> Collect \<delta>ssp"
finite_imageD: fixes f :: "'l \<Rightarrow> 'k"   and A :: "'l set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
ImpR_in_SNa: fixes M :: "trm"   and x :: "name"   and a :: "coname"   and b :: "coname" assumes "SNa M" shows "SNa (ImpR x.<a>.M b)"
lit_ord_def: shows "lit_ord = {(x, y). (mset_lit x, mset_lit y) \<in> mult trm_ord}"
map_upd_nonempty: fixes t :: "'c \<Rightarrow> 'd option"   and k :: "'c"   and x :: "'d" shows "t(k \<mapsto> x) \<noteq> Map.empty"
substitute_substitute'': fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "(xs \<in> paths (substitute f T t)) = (\<exists>xs'\<in>paths t. substitute'' f T xs' xs)"
scale_right_imp_eq: fixes x :: "'d"   and a :: "complex"   and b :: "complex" assumes "x \<noteq> (0::'d)"   and "a *\<^sub>C x = b *\<^sub>C x" shows "a = b"
ctr_1: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lnull xs"   and "lnull ys" shows "lappend xs ys = LNil"
SubstArityAnalysis_axioms: shows "SubstArityAnalysis Aexp"
1: shows "[a] \<preceq>\<^sub>F v @ w"
simps: fixes a1 :: "'a trm"   and a2 :: "'a trm" shows "a1 \<rightarrow>\<beta> a2 = ((\<exists>x T A M. a1 = App (Fn x T A) M \<and> a2 = A[x ::= M]) \<or> (\<exists>A A' B. a1 = App A B \<and> a2 = App A' B \<and> A \<rightarrow>\<beta> A') \<or> (\<exists>B B' A. a1 = App A B \<and> a2 = App A B' \<and> B \<rightarrow>\<beta> B') \<or> (\<exists>A A' x T. a1 = Fn x T A \<and> a2 = Fn x T A' \<and> A \<rightarrow>\<beta> A') \<or> (\<exists>A A' B. a1 = SimplyTyped.Pair A B \<and> a2 = SimplyTyped.Pair A' B \<and> A \<rightarrow>\<beta> A') \<or> (\<exists>B B' A. a1 = SimplyTyped.Pair A B \<and> a2 = SimplyTyped.Pair A B' \<and> B \<rightarrow>\<beta> B') \<or> (\<exists>P P'. a1 = Fst P \<and> a2 = Fst P' \<and> P \<rightarrow>\<beta> P') \<or> (\<exists>A B. a1 = Fst (SimplyTyped.Pair A B) \<and> a2 = A) \<or> (\<exists>P P'. a1 = Snd P \<and> a2 = Snd P' \<and> P \<rightarrow>\<beta> P') \<or> (\<exists>A B. a1 = Snd (SimplyTyped.Pair A B) \<and> a2 = B))"
abs_eq: fixes xa :: "(char list \<times> 'a) list"   and x :: "(char list \<times> 'a) list" shows "St xa \<triangle> St x = St (map2_st_rep (\<triangle>) xa x)"
prems_2: shows "(l # ts) ! i = (p, is, xs, Write\<^sub>s\<^sub>b True a sop v A L R W # sb, \<D>, \<O>, \<R>)"
that: fixes k :: "nat" assumes "\<^bold>|q\<^bold>| = k * \<^bold>|\<rho> u\<^bold>|" shows "thesis"
local_time_unbounded2: fixes A :: "'a set"   and i :: "nat"   and B :: "'a set"   and k :: "nat" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>"   and "A \<subseteq> (T ^^ i) --` B" shows "(\<lambda>n. emeasure M {x \<in> (T ^^ n) --` A. local_time B n x < k}) \<longlonglongrightarrow> 0"
is_path'_snoc: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> b"   and x :: "'a"   and ys :: "'a list"   and y :: "'a"   and z :: "'a" shows "is_path' r x (ys @ [y]) z = (is_path' r x ys y \<and> r y z = T)"
bury_correct2: fixes c :: "com"   and X :: "char list set"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(bury c X, s) \<Rightarrow> s'"   and "s = t on L c X" shows "\<exists>t'. (c, t) \<Rightarrow> t' \<and> s' = t' on X"
iso_runit: fixes f :: "'a" assumes "ide f" shows "local.iso \<r>[f]"
bound: fixes x :: "'a"   and y :: "real" assumes "x \<in> A"   and "a \<le> y" shows "norm (f x y) \<le> g y"
igFreshIGAbs2_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gFreshGAbs2 MOD" shows "igFreshIGAbs2 (fromMOD MOD)"
lookupDef_cases: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)" obtains "v \<in> defs g n" and "lookupDef g n v = (v, n, SimpleDef)"   | "v \<notin> defs g n" and "length (predecessors g n) \<noteq> 1" and "lookupDef g n v = (v, n, PhiDef)"   | m :: "'node" where "v \<notin> defs g n" and "predecessors g n = [m]" and "m \<in> set (\<alpha>n g)" and "lookupDef g n v = lookupDef g m v"
closed_segment_of_real: fixes x :: "real"   and y :: "real" shows "closed_segment (of_real x) (of_real y) = of_real ` closed_segment x y"
rbt_lookup_keys: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "dom (ord.rbt_lookup less t) = set (RBT_Impl.keys t)"
fundamental_theorem_of_calculus': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "(f has_vderiv_on f') {a..b}" shows "(f' has_integral f b - f a) {a..b}"
single_subset_iff: fixes a :: "'a"   and M :: "'a multiset" shows "({#a#} \<subseteq># M) = (a \<in># M)"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> M"   and "y \<in> M" shows "\<eta> x \<cdot>' \<eta> y = \<eta> (x \<cdot> y)"
simps_9: fixes va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and d :: "('a, 'b) rbt" shows "balance (Branch B va vb vc vd) w x (Branch R b s t (Branch R c y z d)) = Branch R (Branch B (Branch B va vb vc vd) w x b) s t (Branch B c y z d)"
assm': fixes v :: "nat \<times> char list"   and b :: "contour" shows "case ve (v, b) of None \<Rightarrow> {} | Some x \<Rightarrow> |x| \<lessapprox> ve_a (v, |b|)"
proj_iff: fixes P :: "'a \<Rightarrow> bool"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "equivP P \<phi>"   and "P x"   and "P y" shows "(proj \<phi> x = proj \<phi> y) = \<phi> x y"
L: fixes t :: "'a" assumes "t \<in> cball a u \<inter> T" shows "L-lipschitz_on (cball b u \<inter> X) (f t)"
rat_to_int_factor_normalized_int_poly: fixes p :: "rat poly"   and g :: "rat poly"   and h :: "rat poly"   and i :: "rat"   and ip :: "int poly" assumes "p = g * h"   and "rat_to_normalized_int_poly p = (i, ip)" shows "\<exists>g' h'. ip = g' * h' \<and> degree g' = degree g"
same_fw_spec: fixes ip1 :: "'b word"   and ip2 :: "'b word"   and rs :: "'b simple_rule list"   and c :: "'c parts_connection_scheme" assumes "same_fw_behaviour ip1 ip2 rs" shows "same_fw_behaviour_one ip1 ip2 c rs"
runit_naturality: 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 \<mu> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "V \<mu> (bicategory.runit V H \<a> \<i> src (partial_magma.dom V \<mu>)) = V (bicategory.runit V H \<a> \<i> src (partial_magma.cod V \<mu>)) (H \<mu> (src \<mu>))"
rtn: shows "the (rt dip) = (dsn\<^sub>n, dsk\<^sub>n, f\<^sub>n, hops\<^sub>n, nhip\<^sub>n, pre\<^sub>n)"
uop_ueval: fixes f :: "'g \<Rightarrow> 'e"   and x :: "('g, 'f) uexpr"   and b :: "'f" shows "\<lbrakk>uop f x\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b)"
prime_gauss_int_of_nat: fixes n :: "nat" assumes "prime n"   and "[n = 3] (mod 4)" shows "prime (of_nat n)"
simps: fixes f :: "'c \<Rightarrow> 'd list"   and key :: "'d \<Rightarrow> 'c"   and u :: "nat"   and ns :: "nat list"   and xs :: "'d list" shows "stab_inv f key (u, ns, xs) = (\<forall>k. filter (\<lambda>x. key x = k) xs = f k)"
assms_3: fixes c :: "real set" assumes "c \<in> C" shows "mono_on f c"
prime_n_pd_principal: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and j :: "nat" assumes "distinct_pds K n P"   and "j \<le> n" shows "P\<^bsub>K P n\<^esub> j = (O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p (Kb\<^bsub>K n P\<^esub>) j"
sat_the_restrict: fixes \<phi> :: "'a MFOTL.formula"   and A :: "nat set"   and \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a option list"   and i :: "nat" assumes "fv \<phi> \<subseteq> A" shows "MFOTL.sat \<sigma> (map the (restrict A v)) i \<phi> = MFOTL.sat \<sigma> (map the v) i \<phi>"
simps_3: fixes v :: "prim" shows "contours_in_d (DP v) = {}"
collect_locks'I: fixes las :: "'c lock_actions"   and l :: "'c" assumes "must_acquire_lock (las $ l)" shows "l \<in> collect_locks' las"
iter_tensor_suc: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "1 \<le> n" shows "A \<otimes>\<^bsup>Suc n\<^esup> = A \<Otimes> A \<otimes>\<^bsup>n\<^esup>"
mat_to_polys_to_mat: fixes ts :: "'d list"   and A :: "'e mat" assumes "distinct ts"   and "length ts = dim_col A" shows "polys_to_mat ts (mat_to_polys ts A) = A"
lexordp_eq_conv_lexord: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp_eq xs ys = (xs = ys \<or> ord_class.lexordp xs ys)"
joinI: fixes x :: "'c"   and sup :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> sup"   and "y \<sqsubseteq> sup"   and "\<And>z. \<lbrakk>x \<sqsubseteq> z; y \<sqsubseteq> z\<rbrakk> \<Longrightarrow> sup \<sqsubseteq> z" shows "x \<squnion> y = sup"
Suc_nat_eq_nat_zadd1: fixes z :: "int" assumes "0 \<le> z" shows "Suc (nat z) = nat (1 + z)"
lesspoll_Pow_self: fixes A :: "'a set" shows "A \<prec> Pow A"
primefact_def: fixes ps :: "'a list"   and n :: "'a" shows "primefact ps n = (foldr (*) ps (1::'a) = n \<and> (\<forall>p\<in>set ps. prime p))"
simps_15: fixes x3 :: "'b" shows "Stuck \<noteq> Fault x3"
map_def: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set" shows "Set_Theory.map \<alpha> S T \<equiv> \<alpha> \<in> S \<rightarrow>\<^sub>E T"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P 1"   and "\<And>n. \<lbrakk>2 \<le> n; P (nat \<lfloor>real n / 2\<rfloor>); P (nat \<lceil>real n / 2\<rceil>)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
segments_of_aform_line_segment: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and e :: "nat \<Rightarrow> real" assumes "segments_of_aform X = [x, y]"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "aform_val e X \<in> closed_segment (fst x) (snd x)"
prios_pqueue: fixes xs :: "('d, 'c) bintree option list" assumes "distinct (vals xs)" shows "set (prios xs) = set \<parallel>pqueue xs\<parallel>"
SAT_deduction: fixes x :: "formula list" assumes "SATAxiom x" shows "x \<in> deductions CutFreePC"
nonrestrictD: fixes a :: "hf"   and s :: "hf"   and X :: "hf" assumes "a \<^bold>\<in> nonrestrict s X" shows "a \<^bold>\<in> s"
dagger_trans_eq: fixes x :: "'b \<times> 'c" shows "times_prod' (prod.qstar x) (prod.qstar x) = prod.qstar x"
is_swap_values_1: assumes "is_swap" shows "f 0 = 1"
Conseq': fixes P :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "('d, 'f, 'g) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s. s \<in> P \<longrightarrow> (\<exists>P' Q' A'. (\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)) \<and> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A))" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
Red_I_of_Red_F_subset: fixes N' :: "'g set"   and N :: "'g set" assumes "N' \<subseteq> Red_F_G N" shows "Red_I_G N \<subseteq> Red_I_G (N - N')"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "P TBase"   and "P TUnit"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"