hcju/isabelleps huggingface dataset:
assms_3: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "rho \<in> set rhoF" shows "wlsEnv rho"
face_of_polytope_polytope: fixes S :: "'a set"   and F :: "'a set" assumes "polytope S"   and "F face_of S" shows "polytope F"
take_Cons': fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
2: shows "inj_on ((+o) w) (genby ` Pow S)"
that: fixes t :: "'a" assumes "v = t \<oplus> u" shows "thesis"
hd_coeffs1_def: fixes as :: "atom list" shows "hd_coeffs1 as = (let m = zlcms (map hd_coeff as) in Dvd m 0 [1] # map (hd_coeff1 m) as)"
A: shows "Deliver (i, Append i e2) \<in> set pre \<or> (\<exists>mo2. Deliver (i, Store e2 mo2 i) \<in> set pre)"
sufD: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v" shows "\<exists>z. z \<cdot> u = v"
iwlsFSwSTR_termFSwImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSwSTR MOD" shows "termFSwImorph h hA MOD = ipresCons h hA MOD"
mono_times_nat: fixes n :: "nat" assumes "0 < n" shows "incseq ((*) n)"
union_code: fixes xs :: "'b list"   and ys :: "'b list" shows "mset xs + mset ys = mset (xs @ ys)"
add_base: fixes k :: "nat"   and m :: "nat"   and ns :: "nat list" shows "foldl (+) (k + m) ns = foldl (+) m ns + k"
ts_tuple_rel_intro: fixes as :: "'a option list"   and X :: "'a option list set"   and t :: "nat"   and ys :: "(nat \<times> 'a option list set) set" assumes "as \<in> X"   and "(t, X) \<in> ys" shows "(t, as) \<in> ts_tuple_rel ys"
rtranclp_mono: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r \<le> s" shows "r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
sources_char: fixes a :: "'a" assumes "weak_unit a" shows "sources a = {x. x \<cong> a}"
rtrancl_idemp: fixes r :: "('x \<times> 'x) set" shows "(r\<^sup>*)\<^sup>* = r\<^sup>*"
upper: fixes N :: "nat" assumes "n \<le> N" shows "u N < x + ereal r"
open_INT: fixes "open" :: "'a set \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'a set" assumes "class.topological_space open"   and "finite A"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Inter> (B ` A))"
linear_independent_extend: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B" shows "\<exists>g. linear (*a) (*b) g \<and> (\<forall>x\<in>B. g x = f x)"
is_processT5_S1: fixes s :: "'b event list"   and X :: "'b event set"   and P :: "'b process"   and Y :: "'b event set" assumes "(s, X) \<in> F P"   and "(s, X \<union> Y) \<notin> F P" shows "\<exists>c. c \<in> Y \<and> (s @ [c], {}) \<in> F P"
Inf_lower: fixes x :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" assumes "s2p A x" shows "less_eq_prod (Inf_prod A) x"
m_inv_consistent: fixes H :: "'a set"   and x :: "'a" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "x \<in> H" shows "inv\<^bsub>G\<lparr>carrier := S, carrier := H\<rparr>\<^esub> x = inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x"
implementsE: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol"   and t :: "'e Trace" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "IncrEnvironment.implements jkbp envInit envAction envTrans envVal jview envObs jp"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview" shows "set \<circ> (\<lambda>a. pAct (jp a) (IncrEnvironment.runJP envObs jp t a)) = set \<circ> JKBP.jAction jkbp (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview)) t"
amor_delete: fixes t :: "'b tree"   and a :: "'b" assumes "bst t" shows "real (T_delete a t) + \<Phi> (delete a t) - \<Phi> t \<le> 6 * \<phi> t + 3"
rtranclp_subset: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "R \<le> S"   and "S \<le> R\<^sup>*\<^sup>*" shows "S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
simps: fixes \<sigma> :: "'a \<times> (bool \<times> 'b \<times> 'a) list" shows "conv_mirror \<sigma> = (let (s, ss) = \<sigma> in case ss of [] \<Rightarrow> (s, ss) | x # xs \<Rightarrow> let (d, \<alpha>, t) = x in (fst (conv_mirror (t, xs)), snd (conv_mirror (t, xs)) @ [(\<not> d, \<alpha>, s)]))"
WHILEIT_rule: fixes R :: "('c \<times> 'c) set"   and I :: "'c \<Rightarrow> bool"   and s :: "'c"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s \<le> SPEC \<Phi>"
h1c1: shows "\<forall>i. Var x ^ i \<noteq> 0"
Infinitesimal_hypreal_sqrt_imp_Infinitesimal: fixes x :: "real star" assumes "0 \<le> x"   and "(*f* sqrt) x \<in> Infinitesimal" shows "x \<in> Infinitesimal"
assms_5: shows "\<forall>v\<in>C. \<phi> (f v) = \<phi> v"
consistentDP_iter_and_compute: fixes f :: "'b \<Rightarrow> ('a, 'c) state"   and x :: "'b"   and y :: "'b" assumes "consistentDP f" shows "crel_vs (=) (dp x) (iter_state f y \<bind> (\<lambda>_. f x))"
eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(x = y) = (suffix x y \<and> suffix y x)"
fmdom_map: fixes f :: "'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmdom (fmmap f m) = fmdom m"
intf_of_assn_def: fixes a :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and b :: "'c itself" shows "intf_of_assn a b = True"
collapse_1: fixes dining_event :: "dining_event" assumes "is_picks dining_event" shows "picks (phil dining_event) (fork dining_event) = dining_event"
smult_add_left: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "smult (a + b) p = smult a p + smult b p"
cmp_assoc: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and g :: "'i \<Rightarrow> 'j"   and C :: "'j set"   and h :: "'j \<Rightarrow> 'k"   and D :: "'k set"   and x :: "'h" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "h \<in> C \<rightarrow> D"   and "x \<in> A" shows "cmp h (cmp g f) x = cmp (cmp h g) f x"
igSwapIPresIGWlsSTR_imp_igSwapIPresIGWls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'i, 'j, 'k, 'l, 'm, 'n, 'o) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igSwapIPresIGWlsSTR MOD" shows "igSwapIPresIGWls MOD"
connected_component_mono: fixes S :: "'a set"   and T :: "'a set"   and x :: "'a" assumes "S \<subseteq> T" shows "connected_component_set S x \<subseteq> connected_component_set T x"
src_inv: fixes \<mu> :: "'b" assumes "D.iso \<mu>" shows "src\<^sub>D (D.inv \<mu>) = src\<^sub>D \<mu>"
iter_tensor_of_H_on_zero_tensor: fixes n :: "nat" assumes "1 \<le> n" shows "H \<otimes>\<^bsup>n\<^esup> * |Deutsch.zero\<rangle> \<otimes>\<^bsup>n\<^esup> = Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n))"
two_le_card: assumes "class.card2 TYPE('a)" shows "2 \<le> CARD('a)"
eq_iff: fixes a :: "'t"   and b :: "'t" shows "(a = b) = (b \<preceq>\<^sub>t a \<and> a \<preceq>\<^sub>t b)"
rep_eq: fixes x :: "integer" shows "Rep_uint32 (Uint32 x) = word_of_int (int_of_integer x)"
hom_det: fixes A :: "'a poly mat" shows "det (map_poly_hom.mat_hom A) = map_poly hom (det A)"
diag_mat_vec_nth_simps_2: fixes i :: "'a"   and j :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "i \<noteq> j" shows "diag_mat f $ i $ j = (0::'b)"
wf: shows "wf_prog (\<lambda>P C (M, b, Ts, T\<^sub>r, mxs, mxl\<^sub>0, is, xt). wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C M)) P"
exhaust: fixes y :: "indices" obtains "y = indices.Left"   | "y = indices.Right"
nth_default_Cons_Suc: fixes dflt :: "'a"   and x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "nth_default dflt (x # xs) (Suc n) = nth_default dflt xs n"
lsided_def3: fixes x :: "'b" shows "(\<nu> x = x) = (\<nu>\<^sup>\<natural> x = x)"
eval_throw_nonLAss: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "lass_val_of e = None"
microstep_choiceI2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" shows "(p1 \<oplus> p2) \<leadsto>\<^bsub>\<Gamma>\<^esub> p2"
distinct_3: fixes x31 :: "'b pltl"   and x32 :: "'b pltl" shows "false\<^sub>p \<noteq> x31 implies\<^sub>p x32"
simps: fixes x :: "'a extC"   and y :: "'a extC" shows "x * y = x \<sqinter> y"
variety_of_radical_ideal: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "\<V> (\<surd>ideal F) = \<V> F"
simps_1: fixes ys :: "'a list" shows "vec_vec_Tensor [] ys = []"
f'g': fixes x :: "'n"   and y :: "'n" assumes "x = y" shows "g' (f' a x) = y"
rev_concat: fixes xs :: "'d list list" shows "rev (concat xs) = concat (map rev (rev xs))"
last_append: fixes xs :: "'b list"   and ys :: "'b list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
netlimit_at: fixes a :: "'b" shows "netlimit (at a) = a"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
simps_9: fixes a :: "MIR.num" shows "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if 0 \<le> v then T else F | _ \<Rightarrow> Ge (reducecoeff a'))"
set1_FGf_def: fixes x :: "('a, 'b, 'e, 'f, 'g, 'h, 'i, 'j, 'k, ('l, 'e, 'm, 'h, 'n, 'k, 'o) G) F" shows "set1_FGf x = set1_F x"
prf_domain_nonempty: shows "prf_domain \<noteq> {}"
lift_option_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a' :: "'b option"   and b' :: "'c option" shows "lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))"
assertionAux_axioms: shows "assertionAux (\<otimes>) SImp (\<bottom>) SChanEq"
wf_ts_def: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and j :: "nat"   and \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula"   and ts :: "nat list" shows "wf_ts \<sigma> j \<phi> \<psi> ts = list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j] ts"
locale_this: shows "impl_lasso_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G"
all_trms_irreducible_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and E :: "'c trm set"   and f :: "'c trm \<Rightarrow> 'c trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "all_trms_irreducible E f = (\<forall>x y. x \<in> E \<longrightarrow> occurs_in y x \<longrightarrow> f y = y)"
finite_bind: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "finite S"   and "\<forall>x\<in>S. finite (f x)" shows "finite (S \<bind> f)"
sup_continuous_lfp: fixes F :: "'a \<Rightarrow> 'a" assumes "sup_continuous F" shows "lfp F = (\<Squnion>i. (F ^^ i) \<bottom>)"
del_unEdge_valid': fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and u :: "'a" assumes "valid_unSimpGraph G" shows "valid_unSimpGraph (del_unEdge v w u G)"
Rep_Abs_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "Rep (Abs 1) = 1"
expandTrans_1: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and P' :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> P' = R \<longmapsto> \<tau> \<prec> P'"
assms_1: shows "create_optimal_solutions A b c = Inr X"
pax2: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "|~ \<circle>(F \<longrightarrow> G) \<longrightarrow> \<circle>F \<longrightarrow> \<circle>G"
insert_bind_set: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "insert a A \<bind> f = f a \<union> (A \<bind> f)"
positive: shows "\<forall>n w. 0 < prices Mkt risk_free_asset n w"
gL_rel_def: fixes x :: "'a"   and y :: "'a" shows "(x \<preceq>\<^sub>L y) = (\<exists>z. D z x \<and> z \<oplus> x = y)"
l_comb_mem_submoduleTr: fixes A :: "'b set"   and N :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "submodule R M N" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> carrier M \<and> (\<forall>j\<le>n. s j \<cdot>\<^sub>s f j \<in> N) \<longrightarrow> l_comb R M n s f \<in> N"
unit_not_var: fixes x :: "'b" shows "Unit \<noteq> Var x"
wprepare_goto_start_pos_nonempty_snd: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list" assumes "lm \<noteq> []" shows "wprepare_goto_start_pos m lm (b, []) = False"
semIntAll_prSubstAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSubstAll (semInt SEM) (semIntAbs SEM) SEM"
VWO_TC_le: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "(\<beta>, \<alpha>) \<in> VWO" shows "\<beta> \<sqsubseteq> \<alpha>"
associatedE: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a \<sim>\<^bsub>G\<^esub> b" obtains "a divides\<^bsub>G\<^esub> b" and "b divides\<^bsub>G\<^esub> a"
has_integral_implies_lebesgue_measurable_real: fixes f :: "'b \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'b set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. f x * indicat_real \<Omega> x) \<in> borel_measurable lebesgue"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "(lookup f k \<noteq> (0::'a)) = (k \<in> keys f)"
list_rest_coinc: fixes s2 :: "'a list"   and s1 :: "'a list"   and r1 :: "'a list"   and r2 :: "'a list" assumes "length s2 \<le> length s1"   and "s1 @ r1 = s2 @ r2" shows "\<exists>r1p. r2 = r1p @ r1"
f_2_def: fixes x :: "int" shows "f_2 x = int p * int q - x"
normalize_quot_quot_of_fract: fixes x :: "'b fract" shows "normalize_quot (quot_of_fract x) = quot_of_fract x"
inf_le1: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y \<subseteq># x"
divisionTr2: fixes g :: "'a"   and n :: "nat" assumes "Corps S"   and "g \<in> carrier R"   and "g \<noteq> \<zero>"   and "0 < deg_n R S X g" shows "\<forall>f. div_condn R S X n g f"
subtopology_eq_discrete_topology_sing: fixes X :: "'d topology"   and a :: "'d" shows "(X = discrete_topology {a}) = (topspace X = {a})"
float_defs_34: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a \<doteq> b = feq a b"
distr_bst_of_list_random_bst: shows "distr (random_perm xs) (count_space (trees A)) bst_of_list = restrict_space (measure_pmf (random_bst (set xs))) (trees A)"
chine_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.arr C (Chn \<mu>)"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and setter :: "('m \<Rightarrow> 'm) \<Rightarrow> ('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> ('c, 'a, 'b, 'k option) RShadowRoot_scheme"   and v :: "'m" assumes "h \<turnstile> ok put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr setter v" shows "ptr |\<in>| shadow_root_ptr_kinds h"
crsp_steps: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (as, lm) (s, l, r) ires" shows "\<exists>stp. crsp ly (abc_steps_l (as, lm) ap n) (steps0 (s, l, r) tp stp) ires"
correct_14: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "funct_OT_14 msgs C = protocol_14_OT msgs C"
cSup_atLeastAtMost: 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 y :: "'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "less_eq y x" shows "Sup (ord.atLeastAtMost less_eq y x) = x"
setadd_subset_G: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier R"   and "K \<subseteq> carrier R" shows "H <+> K \<subseteq> carrier R"
DBM_reset_sound': fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and c :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd"   and u'' :: "'c \<Rightarrow> 'd" assumes "\<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 "v c \<le> n"   and "DBM_reset M n (v c) d M'"   and "u \<turnstile>\<^bsub>v,n\<^esub> M'"   and "u'' \<turnstile>\<^bsub>v,n\<^esub> M" obtains d' :: "'d" where "u(c := d') \<turnstile>\<^bsub>v,n\<^esub> M"
check_extension_l_side_cond_err_impl_def: fixes v :: "'a"   and p :: "'b"   and r :: "'c"   and s :: "'d" shows "check_extension_l_side_cond_err_impl v p r s = ''Error while checking side conditions of extensions polynow, var is '' @ show v @ '' polynomial is '' @ show p @ ''side condition p*p - p = '' @ show s @ '' and should be 0''"
Nset_img0: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and B :: "'e set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> B"   and "f (Suc n) \<in> f ` {j. j \<le> n}" shows "f ` {j. j \<le> Suc n} = f ` {j. j \<le> n}"
test_bit_rsplit_alt: fixes i :: "nat"   and w :: "'d word"   and m :: "nat" assumes "i < length (word_rsplit w)"   and "m < size (hd (word_rsplit w))"   and "0 < length (word_rsplit w)" shows "word_rsplit w ! i !! m = w !! ((length (word_rsplit w) - Suc i) * size (hd (word_rsplit w)) + m)"
insertion_fun_mult: fixes f :: "nat \<Rightarrow> 'a"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "insertion_fun f (poly_mapping.lookup (p * q)) = insertion_fun f (poly_mapping.lookup p) * insertion_fun f (poly_mapping.lookup q)"
Rep_hf_hinsert: fixes a :: "hf"   and b :: "hf" assumes "a \<^bold>\<notin> b" shows "Rep_hf (b \<triangleleft> a) = 2 ^ Rep_hf a + Rep_hf b"
leD: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "\<not> x < y"
alw_sconst: fixes P :: "'a stream \<Rightarrow> bool"   and x :: "'a" shows "alw P (sconst x) = P (sconst x)"
simps: fixes x :: "'d"   and xs :: "'d list" shows "last (x # xs) = (if xs = [] then x else last xs)"
plusU_error_strict: shows "plusU\<cdot>\<bottom> = \<bottom>"
SV_inv2_inductive_2: shows "{Vinv2 \<inter> Vinv1} TS.trans sv_TS {> Vinv2}"
min_alg2_def: fixes s1 :: "('e, 'f, 'd) F1 \<Rightarrow> 'f"   and s2 :: "('e, 'f, 'd) F2 \<Rightarrow> 'd" shows "min_alg2 s1 s2 = (\<Union>i\<in>Field SucFbd. snd (min_algs s1 s2 i))"
chainD: fixes ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f set"   and x :: "'f"   and y :: "'f" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" shows "ord x y \<or> ord y x"
D: shows "f = (\<lambda>t2. {n. \<exists>t1. x = t1 @ t2 \<and> t1 @ [tick] \<in> T S \<and> t2 \<in> D (Y n)})"
eq: fixes x :: "'a"   and y :: "'a" shows "(cmp x y = Eq) = (x = y)"
finsert_case: fixes x :: "'a"   and S' :: "'a fset" assumes "S = finsert x S'" shows "P"
distinct_1: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Skip \<noteq> Assume x2"
code_const1: fixes c :: "nat" shows "code_const1 c = encode (r_const c)"
Rep_formula_eqvt: fixes x :: "perm"   and xa :: "('e, 'f, 'g) formula" shows "x \<bullet> Rep_formula xa = Rep_formula (x \<bullet> xa)"
list_splits_append: fixes xs :: "'a list"   and ys :: "'a list" shows "list_splits (xs @ ys) = map (\<lambda>(xs', x', ys'). (xs', x', ys' @ ys)) (list_splits xs) @ map (\<lambda>(xs', x', ys'). (xs @ xs', x', ys')) (list_splits ys)"
bind_assoc_pmf: fixes A :: "'c pmf"   and B :: "'c \<Rightarrow> 'b pmf"   and C :: "'b \<Rightarrow> 'a pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
NonUni: shows "(\<exists>q. val[q,c] \<noteq> {True}) \<and> (\<exists>q. val[q,c] \<noteq> {False})"
meld_queue_invar: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_invar (meld q q')"
csols_eq: fixes s :: "'a" shows "csols 0 s = {(x, t). t \<in> T \<and> x \<in> Sols (\<lambda>t. f) (\<lambda>s. {0--t}) S 0 s}"
lead_coeff_power: fixes p :: "'a poly"   and n :: "nat" shows "lead_coeff (p ^ n) = lead_coeff p ^ n"
fun_merge: fixes Xs :: "'a set list"   and \<tau> :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<forall>i<length Xs. \<forall>j<length Xs. i \<noteq> j \<longrightarrow> Xs ! i \<inter> Xs ! j = {}" shows "\<exists>\<sigma>. \<forall>i<length Xs. \<forall>x\<in>Xs ! i. \<sigma> x = \<tau> i x"
bind_pure_returns_result_I2: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog"   and y :: "'aa" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
FAcc_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and F :: "String.literal"   and D :: "String.literal" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e\<bullet>F{D}, xs) (e'\<bullet>F{D}, xs')"
sizeF_diseq: fixes p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula" assumes "sizeF p \<noteq> sizeF q" shows "p \<noteq> q"
VSLuckiestRepeat_def: shows "VSLuckiestRepeat = repeatAmountOfQuantifiers VSLuckiest"
prVar_def: fixes g :: "('b, 'c, 'varSort, 'var, 'd) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'e) \<Rightarrow> 'e"   and SEM :: "('f, 'g, 'h, 'sort, 'i, 'e, 'j) semDom_scheme" shows "prVar g SEM \<equiv> \<forall>xs x val. sWlsVal SEM val \<longrightarrow> g (Var xs x) val = val xs x"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<rho> (src w) (g \<star> w)"
angle_commute: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "angle a b c = angle c b a"