hcju/isabelleps huggingface dataset:
isBackjumpLevel_def: fixes level :: "nat"   and l :: "Literal"   and c :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "isBackjumpLevel level l c M \<equiv> isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M) \<and> 0 \<le> level \<and> level < elementLevel (opposite l) M \<and> (\<forall>l'. l' el c \<and> l' \<noteq> l \<longrightarrow> elementLevel (opposite l') M \<le> level)"
Mu_def: fixes T :: "(bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "Mu T \<equiv> \<lambda>ab. if ab then lfp_trans (\<lambda>t. T (Embed t) ab) else gfp_trans (\<lambda>t. T (Embed t) ab)"
eexp_tendsto': fixes u :: "'a \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'a filter" shows "((\<lambda>n. eexp (u n)) \<longlongrightarrow> eexp l) F = (u \<longlongrightarrow> l) F"
in_strand_fv_subset: fixes x :: "('c, 'd) strand_step"   and S :: "('c, 'd) strand_step list" assumes "x \<in> set S" shows "vars\<^sub>s\<^sub>t\<^sub>p x \<subseteq> vars\<^sub>s\<^sub>t S"
fraction_not_in_ints: fixes n :: "int"   and m :: "int" assumes "\<not> n dvd m"   and "n \<noteq> 0" shows "of_int m / of_int n \<notin> \<int>"
pcompose_conv_poly: fixes p :: "'c poly"   and q :: "'c poly" shows "p \<circ>\<^sub>p q = poly (poly_lift p) q"
mirror_pure: fixes x :: "'a" shows "mirror (pure x) = pure x"
intro: fixes invar :: "'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option" assumes "\<And>m. invar m \<Longrightarrow> finite (dom (\<alpha> m))" shows "finite_map \<alpha> invar"
power_Suc_less_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a"   and "less a one" shows "less (power.power one times a (Suc n)) one"
empty_sets: fixes M :: "'a measure" shows "{} \<in> sets M"
min_dfa_def: shows "min_dfa = (\<lambda>(bd, as). let (os, ns) = mk_eqcl (replicate (length bd) None) [] 0 (fixpt (bd, as) (init_tr (bd, as))) in (map (\<lambda>p. bdd_map ((!) os) (bd ! p)) ns, map ((!) as) ns))"
prob_compl: fixes A :: "'c set"   and M :: "'c pmf" assumes "A \<in> measure_pmf.events M" shows "measure_pmf.prob M (space (measure_pmf M) - A) = 1 - measure_pmf.prob M A"
convex_contains_open_segment: fixes s :: "'b set" shows "convex s = (\<forall>a\<in>s. \<forall>b\<in>s. open_segment a b \<subseteq> s)"
memo_list_trancl_def: fixes r :: "('b \<times> 'b) list" shows "memo_list_trancl r = (let tr = trancl_list_impl r; rm = map (\<lambda>a. (a, tr [a])) ((remdups \<circ> map fst) r) in (\<lambda>a. case map_of rm a of None \<Rightarrow> [] | Some as \<Rightarrow> as))"
add_less_cancel_left: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y \<subset>\<^sub>\<circ> x + z) = (y \<subset>\<^sub>\<circ> z)"
dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S>_at_pre_def: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X.boss@pre) = eval_extract X (deref_oid\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n in_pre_state (deref_assocs\<^sub>2\<B>\<O>\<S>\<S> in_pre_state (select\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S> (deref_oid\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n in_pre_state))))"
streams_Stream: fixes x :: "'d"   and s :: "'d stream"   and A :: "'d set" shows "(x ## s \<in> streams A) = (x \<in> A \<and> s \<in> streams A)"
ksimplex_eq_top: fixes p :: "nat"   and n :: "nat"   and b_s :: "nat \<Rightarrow> nat"   and u_s :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and b_t :: "nat \<Rightarrow> nat"   and u_t :: "nat \<Rightarrow> nat"   and t :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" assumes "kuhn_simplex_pair p n b_s u_s s b_t u_t t"   and "a \<in> s"   and "\<And>a'. a' \<in> s \<Longrightarrow> a' \<le> a"   and "b \<in> t"   and "\<And>b'. b' \<in> t \<Longrightarrow> b' \<le> b"   and "s - {a} = t - {b}" shows "s = t"
less_imp_le_nat: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m \<le> n"
continuous_on_arccos: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f"   and "\<forall>x\<in>s. - 1 \<le> f x \<and> f x \<le> 1" shows "continuous_on s (\<lambda>x. arccos (f x))"
inv_inv: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "category.inv C (category.inv C f) = f"
assms_2: shows "\<exists>k. hamlet (Rep_run sub k c)"
MATCH1_singleton_Var_1: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term option"   and x :: "'f"   and t :: "('d, 'e) Term.term" assumes "\<sigma> x = None" shows "MATCH1 ({#(Var x, t)#}, \<sigma>) ({#}, \<sigma>(x \<mapsto> t))"
nonemptyE: fixes S :: "'b set" assumes "S \<noteq> {}" obtains x :: "'b" where "x \<in> S"
Healthy_carrier_image: fixes A :: "'a upred set"   and \<H> :: "'a upred \<Rightarrow> 'a upred" assumes "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H" shows "\<H> ` A = A"
S1: shows "sf (x + 1) = (x + 1) * (x + 2) div 2"
m_o1: fixes X :: "char list set"   and o1 :: "'a st option"   and o2 :: "'a st option" assumes "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 \<le> o2" shows "m\<^sub>o o2 X \<le> m\<^sub>o o1 X"
runit_simps_3: fixes f :: "'a" assumes "B.ide f" shows "trg\<^sub>B (B.runit f) = trg\<^sub>B f"
Inf_acom_def: fixes c :: "com"   and M :: "'a acom set" shows "Inf_acom c M = annotate (\<lambda>p. INF C\<in>M. anno C p) c"
Image_UN: fixes r :: "('c \<times> 'b) set"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "r `` \<Union> (B ` A) = (\<Union>x\<in>A. r `` B x)"
insert_ran_inj_on: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and j :: "nat" assumes "inj_on f {0..<n}"   and "j < Suc n" shows "inj_on (insert_ran f j) {0..<n}"
that_5: shows "\<forall>i\<le>n. (0::'a) \<le> fwi m n k n n i i"
Gromov_converging_at_boundary_bounded_perturbation: fixes u :: "nat \<Rightarrow> 'b"   and x :: "'b Gromov_completion"   and v :: "nat \<Rightarrow> 'b"   and C :: "real" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"   and "x \<in> Gromov_boundary"   and "\<And>n. dist (u n) (v n) \<le> C" shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> x"
induct: fixes S :: "'a set"   and B :: "'a set set"   and x :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "generated_topology S B x"   and "P S"   and "\<And>U V. \<lbrakk>generated_topology S B U; P U; generated_topology S B V; P V\<rbrakk> \<Longrightarrow> P (U \<inter> V)"   and "\<And>K. \<lbrakk>\<And>U. U \<in> K \<Longrightarrow> generated_topology S B U; \<And>U. U \<in> K \<Longrightarrow> P U\<rbrakk> \<Longrightarrow> P (\<Union> K)"   and "\<And>b. b \<in> B \<and> b \<subseteq> S \<Longrightarrow> P b" shows "P x"
sum: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
eq_for_lvar: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "EqForLVar eq_idx_for_lvar"   and "x \<in> lvars t" shows "EqForLVar.eq_for_lvar eq_idx_for_lvar t x \<in> set t \<and> lhs (EqForLVar.eq_for_lvar eq_idx_for_lvar t x) = x"
distinct_9: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x61 :: "form"   and x62 :: "form" shows "Bound x11 x12 x13 x14 \<noteq> Conj x61 x62"
heap_is_wellformed_impl: shows "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed Shadow_DOM.get_child_nodes get_disconnected_nodes"
dotprod_def: fixes xs :: "nat list"   and ys :: "nat list" shows "xs \<bullet> ys = (\<Sum>i<min (length xs) (length ys). xs ! i * ys ! i)"
le_imp_less_or_eq: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "x < y \<or> x = y"
assms: shows "B \<subseteq>\<^sub>\<circ> A"
keys_of_iff: fixes x :: "'b"   and M :: "('b, 'c) Mapping_Str.map" shows "(x \<in> keys_of M) = (M\<langle>x\<rangle> \<noteq> None)"
is_am_def: fixes am :: "nat \<Rightarrow> 'b list"   and psi :: "'b list Heap.array" shows "is_am am psi \<equiv> \<exists>\<^sub>Al. psi \<mapsto>\<^sub>a l * \<up> (length l = N \<and> (\<forall>i<N. l ! i = am i) \<and> (\<forall>i\<ge>N. am i = []))"
ind': fixes P :: "vbl \<Rightarrow> bool"   and v' :: "vbl" assumes "P zeroX"   and "\<forall>v. P v \<longrightarrow> P (nextX v)" shows "P v'"
le_neq_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
Int_left_absorb: fixes A :: "'h set"   and B :: "'h set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
lincomb_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and a :: "'j \<Rightarrow> 'h"   and A :: "'j set" assumes "module R M" shows "module.lincomb M a A = (\<Oplus>\<^bsub>M\<^esub>v\<in>A. a v \<odot>\<^bsub>M\<^esub> v)"
distinct_sorted_list_of_set: fixes A :: "'b set" shows "distinct (sorted_list_of_set A)"
fstTS_t3: shows "snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) = rev (take i cs) @ [x] @ rev bs @ [x] @ rev as @ h"
ext_inject: fixes wait :: "bool"   and tr :: "'c list"   and ref :: "'c set"   and more :: "'d"   and wait' :: "bool"   and tr' :: "'c list"   and ref' :: "'c set"   and more' :: "'d" shows "(\<lparr>wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr> = \<lparr>wait = wait', tr = tr', ref = ref', \<dots> = more'\<rparr>) = (wait = wait' \<and> tr = tr' \<and> ref = ref' \<and> more = more')"
hyps: fixes it :: "int node ref option"   and s :: "int" shows "<os_is_it l p l' it> os_sum' it s <\<lambda>r. os_list l p * \<up> (r = s + sum_list l')>\<^sub>t"
select_weight_drop_zero: fixes xs :: "(natural \<times> 'b) list" shows "Random.select_weight (filter (\<lambda>(k, uu_). 0 < k) xs) = Random.select_weight xs"
add_param: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s $ Fv x \<tau>') (Abs_fv x \<tau>' t $ Fv x \<tau>')"
lift_Use_Low: fixes V :: "'g"   and L :: "'g set"   and X :: "'h"   and Use :: "'h \<Rightarrow> 'g set"   and E :: "'h"   and H :: "'g set" assumes "V \<in> L" shows "(Node X, V) \<in> lift_Use_set Use E X H L"
term_okD2: fixes \<Theta> :: "((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)) \<times> Core.term set"   and t :: "Core.term" assumes "term_ok \<Theta> t" shows "typ_of t \<noteq> None"
dim_Macaulay_mat_2: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "dim_col (Macaulay_mat ps) = card (Keys (set ps))"
mult_vec_distrib_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and r :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_vec pair_of_term term_of_pair (p + q) r = term_powerprod.mult_vec pair_of_term term_of_pair p r + term_powerprod.mult_vec pair_of_term term_of_pair q r"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R (h x) (h y)"   and "R (h y) (h z)" shows "R (h x) (h z)"
succ_in: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "(i, succ r i) \<in> r"
rdvd_def: fixes x :: "real"   and y :: "real" shows "(x rdvd y) = (\<exists>k. y = x * real_of_int k)"
function_ring_is_ring: fixes R :: "('a, 'b) ring_scheme"   and S :: "'c set" assumes "ring_functions R" shows "ring (function_ring S R)"
mono_ccSup: fixes A :: "'a set" assumes "countable A" shows "Sup (f ` A) \<le> f (Sup A)"
until_mono: fixes \<phi>1 :: "'a stream \<Rightarrow> bool"   and \<psi>1 :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<phi>2 :: "'a stream \<Rightarrow> bool"   and \<psi>2 :: "'a stream \<Rightarrow> bool" assumes "(\<phi>1 until \<psi>1) xs"   and "\<And>xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs"   and "\<And>xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs" shows "(\<phi>2 until \<psi>2) xs"
tr_Delete_constr_iff_aux2: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and Di :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "ground M"   and "\<lbrakk>M; map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) Di @ map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>" shows "(\<forall>d\<in>set Di. (t, s) \<cdot>\<^sub>p \<I> = d \<cdot>\<^sub>p \<I>) \<and> (\<forall>d\<in>set D - set Di. (t, s) \<cdot>\<^sub>p \<I> \<noteq> d \<cdot>\<^sub>p \<I>)"
lm066: fixes a :: "('d \<times> 'c set) set" assumes "\<forall>y\<in>Range a. finite y" shows "\<forall>y\<in>omega ` a. finite y"
2124: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> y = y"
Complete_mk_regular2: fixes k :: "nat" shows "\<exists>P F. Complete k = mk_regular2 P F"
simps_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "subfrmlsn (\<phi> or\<^sub>n \<psi>) = {\<phi> or\<^sub>n \<psi>} \<union> subfrmlsn \<phi> \<union> subfrmlsn \<psi>"
complex_rotationsp_complex_rotations_eq: shows "complex_rotationsp = (\<lambda>x. x \<in> complex_rotations)"
pruned_def: fixes g :: "'g" shows "step.pruned g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> step.phiDefs g n \<longrightarrow> step.liveVal g val)"
distinct_3: fixes x1 :: "nat"   and x31 :: "POPLmark.type"   and x32 :: "trm" shows "Var x1 \<noteq> (\<lambda><:x31. x32)"
igAbsCongSSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igAbsCongSSTR MOD \<equiv> \<forall>xs x x' y X X'. igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
arc_injective: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.arc x" shows "matrix_stone_relation_algebra.injective x"
empty_prob_dom_imp_empty_plan_always_good: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "prob_dom PROB = {}"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "exec_plan s [] = exec_plan s as"
elims: fixes x :: "('a \<times> 'b) \<times> 'a \<times> 'c"   and y :: "'a" assumes "Pre_Port_Graph.edge_begin x = y" obtains v1 :: "'a" and p1 :: "'b" and v2 :: "'a" and p2 :: "'c" where "x = ((v1, p1), v2, p2)" and "y = v1"
iT_Plus_cut_ge2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>\<ge> (t + k) = I \<down>\<ge> t \<oplus> k"
sets_point_measure: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" shows "sets (point_measure A f) = Pow A"
ih_6: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
axiom6_F: fixes f1 :: "'l \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 'q"   and f3 :: "'n \<Rightarrow> 'o" shows "F'set2 \<circ> F'map f2 f3 f1 = (`) f3 \<circ> F'set2"
inorder_split_maxD: fixes t :: "('b \<times> bal) tree"   and t' :: "('b \<times> bal) tree alt"   and a :: "'b" assumes "split_max t = (t', a)"   and "t \<noteq> \<langle>\<rangle>"   and "avl t" shows "Tree2.inorder (tree t') @ [a] = Tree2.inorder t"
bounded_minus_comp: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x - g x) ` S)"
MN_eq_nextl: shows "MyhillNerode Power.language Power.eq_nextl"
rotate_rev: fixes n :: "nat"   and xs :: "'b list" shows "rotate n (rev xs) = rev (rotate (length xs - n mod length xs) xs)"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "wb_lens x"   and "mwb_lens x" shows "vwb_lens x"
rel_cases: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b option"   and b :: "'d option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'b" and y :: "'d" where "a = Some x" and "b = Some y" and "R x y"
removeAll_filter_not: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "\<not> P x" shows "removeAll x (filter P xs) = filter P xs"
switch_hchop1: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and c :: "cars"   and v' :: "view" shows "v=v1\<parallel>v2 \<and> v = c > v' \<longrightarrow> (\<exists>v1' v2'. v1 = c > v1' \<and> v2 = c > v2' \<and> v'=v1'\<parallel>v2')"
take_conc'': fixes xs :: "int list" assumes "xs \<noteq> []" shows "\<exists>ys. xs = Challenge1A.take xs @ ys \<and> (ys \<noteq> [] \<longrightarrow> incr (Challenge1A.take xs) \<and> \<not> last (Challenge1A.take xs) < hd ys \<or> decr (Challenge1A.take xs) \<and> \<not> hd ys \<le> last (Challenge1A.take xs))"
sinvar_no_offending: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" assumes "sinvar G nP" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP = {}"
zn3: fixes z :: "complex" assumes "z \<noteq> 0" shows "\<exists>T. z ^ n \<in> T \<and> open T \<and> 0 \<notin> T \<and> (\<exists>v. \<Union> v = - {0} \<inter> (\<lambda>z. z ^ n) -` T \<and> (\<forall>u\<in>v. open u \<and> 0 \<notin> u) \<and> disjoint v \<and> (\<forall>u\<in>v. Ex (homeomorphism u T (\<lambda>z. z ^ n))))"
cl_inv_lengthD: fixes g :: "'b gState_scheme"   and p :: "pState" assumes "cl_inv (g, p)" shows "length (pState.channels p) \<le> length (gState.channels g)"
simps_4: fixes a :: "'b"   and a' :: "'b"   and f :: "('b, 'c) Kform" shows "subjective a \<^bold>K\<^sub>a' f = (a = a')"
sqnf_kno: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:16} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> sqnf (rt \<xi>) (dip \<xi>) = kno)"
valid_W_invD0_1: fixes r :: "'d"   and s :: "'f process_name \<Rightarrow> ('e, 'f, 'g, 'd) Model.local_state"   and p :: "'f process_name"   and q :: "'f process_name" assumes "r \<in> W (s p)"   and "valid_W_inv s"   and "p \<noteq> q" shows "r \<notin> WL q s"
Quotient_fset: shows "Quotient (\<approx>) abs_fset rep_fset cr_fset"
abstract_correct: fixes \<eta> :: "nat" shows "ped_commit.correct \<eta>"
compE2_jump_ok: fixes e :: "'a expr1"   and n :: "nat"   and n' :: "nat" shows "jump_ok (compE2 e) n (Suc n')"
assms_4: shows "0 < fst (dominant_term_ms_aux F)"
less_not_refl: fixes n :: "nat" shows "\<not> n < n"
ccSUP_least: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and u :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
is_aligned_shiftl_self: fixes p :: "'a word"   and n :: "nat" shows "is_aligned (p << n) n"
in_set_replicate: fixes x :: "'b"   and n :: "nat"   and y :: "'b" shows "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)"
atom_map_inf_pres: fixes x :: "'a"   and y :: "'a" shows "atom_map (x \<sqinter> y) = atom_map x \<inter> atom_map y"
represents_imp_degree: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" shows "degree p \<noteq> 0"
lActive_greater_active: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<and> t\<rangle> < n"   and "\<forall>n''>n'. \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>" shows "\<not> \<parallel>c\<parallel>\<^bsub>t n\<^esub>"
snd_netgmap_netliftl: fixes sr :: "'e \<Rightarrow> 'd \<times> 'c"   and s :: "'e net_state" shows "snd (netgmap sr s) = netliftl sr s"
aC_eq_aa: fixes xs :: "'a list"   and y :: "'a"   and zs :: "'a list" shows "xs @ y # zs = (xs @ [y]) @ zs"
simps_3: fixes v :: "nat"   and va :: "'a"   and vb :: "'a list" shows "T_chop (Suc v) (va # vb) = T_take (Suc v) (va # vb) + T_drop (Suc v) (va # vb) + T_chop (Suc v) (drop (Suc v) (va # vb))"
psimps: fixes i :: "int"   and j :: "int" assumes "opt_bst2_dom (i, j)" shows "opt_bst2 i j = (if j < i then \<langle>\<rangle> else if i = j then \<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle> else let left = root (opt_bst2 i (j - 1)); right = root (opt_bst2 (i + 1) j) in argmin (local.wpl i j) (map (\<lambda>k. \<langle>opt_bst2 i (k - 1), k, opt_bst2 (k + 1) j\<rangle>) [left..right]))"
connectedD_interval: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and "open" :: "'e set \<Rightarrow> bool"   and U :: "'e set"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.linorder_topology less_eq less open"   and "topological_space.connected open U"   and "x \<in> U"   and "y \<in> U"   and "less_eq x z"   and "less_eq z y" shows "z \<in> U"
assms: fixes p :: "pat"   and e :: "exp0"   and s' :: "'a state"   and env :: "v sem_env" assumes "(p, e) \<in> set pes"   and "clock s' \<le> clock s" shows "\<exists>s'' r. evaluate True env s' e (s'', r)"
simps_1: fixes P :: "'b" shows "strip (SKIP {P}) = com.SKIP"
is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"
FAddSA_RootEx: fixes G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and F :: "('d, 'e, 'f) seqauto set"   and SA :: "('d, 'e, 'f) seqauto"   and S :: "'d" assumes "\<Union> (ran G) = F - {Root F G}"   and "dom G = \<Union> (States ` F)"   and "dom G \<inter> States SA = {}"   and "S \<in> dom G"   and "RootEx F G" shows "RootEx (insert SA F) (G [f+] (S, SA))"
eq1: fixes A :: "'a set" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf q) {y. \<exists>x\<in>A. R x y}"
pderivs_lang_Times_aux2: fixes s :: "'b list"   and r2 :: "'b rexp"   and r1 :: "'b rexp" assumes "s \<in> UNIV1" shows "Timess (pderivs s r1) r2 \<subseteq> Timess (pderivs_lang UNIV1 r1) r2"
linear: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
substT_tms: fixes t1 :: "'trm"   and t2 :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (tms t1 t2) t x = tms (substT t1 t x) (substT t2 t x)"
bij_betw_byWitness: fixes A :: "'b set"   and f' :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and A' :: "'c set" assumes "\<forall>a\<in>A. f' (f a) = a"   and "\<forall>a'\<in>A'. f (f' a') = a'"   and "f ` A \<subseteq> A'"   and "f' ` A' \<subseteq> A" shows "bij_betw f A A'"