hcju/isabelleps huggingface dataset:
SCallNone: fixes P :: "(char list list \<times> char list exp) prog"   and ps :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>"   and "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchMethodError,s\<^sub>2\<rangle>"
continuous_on_LINT_pmf: fixes M :: "'d pmf"   and A :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and B :: "real" assumes "\<And>i. i \<in> set_pmf M \<Longrightarrow> continuous_on A (f i)"   and "\<And>a i. \<lbrakk>a \<in> A; i \<in> set_pmf M\<rbrakk> \<Longrightarrow> norm (f i a) \<le> B" shows "continuous_on A (\<lambda>a. measure_pmf.expectation M (\<lambda>i. f i a))"
staysuntouched: fixes S :: "nat list \<times> nat list"   and x :: "nat"   and y :: "nat"   and qs :: "nat list" assumes "distinct (fst S)"   and "x \<in> set (fst S)"   and "y \<in> set (fst S)"   and "set qs \<subseteq> set (fst S)"   and "x \<notin> set qs"   and "y \<notin> set qs" shows "x < y in fst (Partial_Cost_Model.config' (rTS []) S qs) = x < y in fst S"
is_coext_def: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" shows "\<epsilon>x.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>D G x\<guillemotright> \<and> f = \<epsilon> x \<cdot>\<^sub>C F g"
simps: fixes p :: "'a Point"   and q :: "'a Point" shows "line joining p to q = \<lparr>basepoint = p, direction = from p to q\<rparr>"
set_concat: fixes xs :: "'b list list" shows "set (concat xs) = \<Union> (set ` set xs)"
l1_inv1_def: shows "l1_inv1 \<equiv> {s. \<forall>Ra A B gny. 0 < signalsInit s (Commit A B (Exp gny (NonceF (Ra $ nx)))) \<longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))}"
assms_1: shows "u\<^sub>1 =\<^sub>F u\<^sub>2"
finite_number_of_roots: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "p \<in> carrier (poly_ring M)" shows "finite {x. M.is_root p x}"
bit_not_one_imp: fixes a :: "bit" shows "(a \<noteq> 1) = (a = 0)"
fst_clock_sim: fixes t :: "'s Trace" assumes "t \<in> Clock.jkbpC" shows "fst (clock_sim t) = clock_commonAbs t"
2: shows "pderiv f \<bullet> \<one> = [n] \<cdot> \<one>"
scc_non_empty: fixes E :: "('a \<times> 'a) set" shows "\<not> is_scc E {}"
assms_1: shows "optim 0 [Ubx.instr.IPush d\<^sub>1, Ubx.instr.IPush d\<^sub>2, Ubx.instr.IStore y] [] = Ok (xs, ys)"
"case": fixes f :: "meth_sig \<Rightarrow> meth_body \<Rightarrow> 'a"   and x1 :: "meth_sig"   and x2 :: "meth_body" shows "(case meth_def_def x1 x2 of meth_def_def x xa \<Rightarrow> f x xa) = f x1 x2"
no_abs_induct: fixes t :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "no_abs t"   and "\<And>name. P (free name)"   and "\<And>name. P (const name)"   and "\<And>t\<^sub>1 t\<^sub>2. \<lbrakk>P t\<^sub>1; no_abs t\<^sub>1; P t\<^sub>2; no_abs t\<^sub>2\<rbrakk> \<Longrightarrow> P (app t\<^sub>1 t\<^sub>2)" shows "P t"
interval_upperbound_def: fixes s :: "'a set" shows "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x\<in>s. x \<bullet> i) *\<^sub>R i)"
assoc_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "trg\<^sub>B \<a>\<^sub>B[f, g, h] = trg\<^sub>B f"
mdqdb: shows "bound0 (MIR.disj (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (unit p)))))) [1..snd (snd (unit p))]) (evaldjf (\<lambda>t. simpfm (subst0 t (fst (unit p)))) (remdups (map (\<lambda>(b, j). simpnum (Add b (C j))) (concat (map (\<lambda>b. map (Pair b) [1..snd (snd (unit p))]) (fst (snd (unit p)))))))))"
leg1_commutes: shows "\<nu>.cod.leg1 \<cdot> \<nu>.chine = \<nu>.leg1"
rank_less_i: shows "rank A \<le> mod_type_class.to_nat i"
cts_step_def: fixes a :: "real"   and b :: "real"   and x :: "real" shows "cts_step a b x \<equiv> if x \<le> a then 1 else if b \<le> x then 0 else (b - x) / (b - a)"
scf_lowering_axioms: shows "scf_lowering agents alts scf agents' alts'"
push_down_state_rank_token_run: fixes q :: "'b"   and n :: "nat"   and i :: "nat" assumes "state_rank q n = Some i" shows "\<exists>x. token_run x n = q \<and> x \<le> n"
vars0_def: shows "st2.vars0 = {0..<Suc k} \<union> {Suc k}"
gfp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "gfp f = f (gfp f)"
chain_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" assumes "A \<subseteq> B"   and "chain_on P C A" shows "chain_on P C B"
cases: fixes a1 :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c memref list \<times> 'd \<times> 'e \<times> 'f) list"   and a2 :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> 'g \<times> 'h) list" assumes "a1 \<sim> a2 " obtains ts :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c memref list \<times> 'd \<times> 'e \<times> 'f) list" and ts\<^sub>d :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> 'g \<times> 'h) list" where "a1 = ts" and "a2 = ts\<^sub>d" and "length ts = length ts\<^sub>d" and "\<forall>i<length ts. \<exists>\<O>' \<D>' \<R>'. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>d ! i in ts ! i = (p, is, \<theta>, [], \<D>', \<O>', \<R>')"
i_s: shows "i < dim_row (two_block_non_interfering A A\<^sup>T)"
waiting_code_2: fixes w :: "wake_up_status" shows "waiting \<lfloor>PostWS w\<rfloor> = False"
birkhoff_aux2: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "AE x in M. limsup (\<lambda>n. ereal (birkhoff_sum f n x / real n)) \<le> ereal (real_cond_exp M Invariants f x)"
undual_inject: fixes x :: "'d dual"   and y :: "'d dual" shows "(undual x = undual y) = (x = y)"
bigthetaI_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "f \<in> \<Theta>[F](g)"
reachable_trace: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a"   and r :: "(nat \<times> 'a) stream" assumes "q \<in> greachable A w V p"   and "grun A w V r q" shows "sset (gtrace r q) \<subseteq> greachable A w V p"
fds_right_inverse_unique: fixes f :: "'a fds"   and g :: "'a fds" assumes "f * g = 1" shows "g = inverse f"
vec_of_poly_n_add: fixes a :: "'a poly"   and b :: "'a poly"   and n :: "nat" shows "vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n"
intro: fixes t0 :: "real"   and t1 :: "real"   and x :: "real \<Rightarrow> 'b"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "nat"   and B :: "real"   and r :: "real"   and L :: "real"   and t :: "nat \<Rightarrow> real" assumes "consistent_one_step t0 t1 x incr p B r L"   and "max_step t t1 p L B r"   and "convergent_one_step_axioms t0 t" shows "convergent_one_step t0 t1 x incr p B r L t"
eqpoll_sym: fixes A :: "'c set"   and B :: "'d set" assumes "A \<approx> B" shows "B \<approx> A"
de_morgan4: fixes x :: "'a"   and y :: "'a" shows "n (t x \<oplus> t y) = n x \<cdot> n y"
hom_closed: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h x \<in> carrier H"
assms_3: shows "e \<le> v * - v\<^sup>T"
elims_2: fixes x :: "term list" assumes "linears x" obtains "x = []"   | t :: "term" and ts :: "term list" where "x = t # ts" and "linear t \<and> fdisjnt (frees t) (freess ts) \<and> linears ts"
set_sel_1: fixes a :: "'a stream" shows "shd a \<in> i.R a"
termMOD_igSubstIGOp: shows "igSubstIGOp termMOD"
inf_not_lt: fixes x :: "'c DBMEntry" shows "(\<infinity> \<prec> x) = False"
induced_automorphism_halfmorphism_fopp_to_fimage: shows "ChamberComplexMorphism folding_f.Y (f \<turnstile> X) \
"
sqrt_greater_zero_iff: fixes n :: "nat" shows "(0 < local.sqrt n) = (0 < n)"
symp_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "symp r" shows "symp r\<^sup>*\<^sup>*"
comp_inv_arr: fixes f :: "'s"   and g :: "'s" assumes "inverse_arrows f g" shows "g \<cdot> f = local.dom f"
l2wi_wi2l: fixes r :: "'b wordinterval" shows "wordinterval_to_set (l2wi (wi2l r)) = wordinterval_to_set r"
subadditive_converges_ereal': fixes u :: "nat \<Rightarrow> real"   and N0 :: "nat" assumes "eventually_subadditive u N0" shows "(\<lambda>m. ereal (u m / real m)) \<longlonglongrightarrow> Inf {ereal (u n / real n) |n. N0 < n}"
rel_frontier_affine_hull: fixes S :: "'a set" shows "rel_frontier S \<subseteq> affine hull S"
vec_of_dim_0: fixes v :: "'a Matrix.vec" shows "(dim_vec v = 0) = (v = 0\<^sub>v 0)"
Entry_no_call_source: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "sourcenode a = Entry" shows "False"
intro: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real" assumes "finite_total_preorder_on carrier le"   and "vnm_utility_axioms carrier le u" shows "vnm_utility carrier le u"
simps: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and t :: "'c" shows "free_thread_id ts t = (ts t = None)"
fps_inverse_one_minus_fps_X: shows "inverse (1 - fps_X) = Abs_fps (\<lambda>n. 1::'a)"
Sup_le_iff: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'g set"   and b :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
ext_cf_def: fixes S :: "('e, 'f) Ring_scheme"   and n :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "ext_cf S n c = (n + fst c, \<lambda>i. if n \<le> i then snd c (sliden n i) else \<zero>\<^bsub>S\<^esub>)"
simply_connected_eq_homotopic_circlemaps3: fixes S :: "'a set" assumes "path_connected S"   and "\<And>f. \<lbrakk>continuous_on (sphere 0 1) f; f ` sphere 0 1 \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>a. homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f (\<lambda>x. a)" shows "simply_connected S"
induct: fixes P :: "'c tree \<Rightarrow> bool"   and a0 :: "'c tree" assumes "P \<langle>\<rangle>"   and "\<And>A a. P \<langle>A, a, \<langle>\<rangle>\<rangle>"   and "\<And>A a B b CD. (CD \<noteq> \<langle>\<rangle> \<Longrightarrow> P CD) \<Longrightarrow> P \<langle>A, a, \<langle>B, b, CD\<rangle>\<rangle>" shows "P a0"
merge_guards_simps_11: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Catch c\<^sub>1 c\<^sub>2) = Catch (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
iprev_nth_image: fixes I :: "nat set"   and f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "SetInterval2.strict_mono_on f I" shows "(f ` I) \<leftarrow> n = f (I \<leftarrow> n)"
simps: fixes I :: "('sf, 'sc, 'sz) interp"   and R :: "(real, 'sz) vec" shows "extendf I R = \<lparr>Functions = case_sum (Functions I) (\<lambda>f' _. R $ f'), Predicates = Predicates I, Contexts = Contexts I, Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
obs_none_no_kth_obs: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" assumes "obs \<sigma> k = None" shows "\<nexists>i. is_kth_obs (path \<sigma>) k i"
safeassms_1: shows "(a, b) = aform_val e X"
weakBisimParSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> R \<parallel> P \<approx> R \<parallel> Q"
np_specE: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "np_spec np"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "p \<noteq> q" obtains q_in_bs :: "bool" where "(q_in_bs, p, q) \<in> set (np gs bs hs data)"   | p_in_bs :: "bool" where "(p_in_bs, q, p) \<in> set (np gs bs hs data)"
execution_def: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list" shows "execution trans sends start exec trace \<equiv> (1 \<le> length exec \<and> length exec - 1 = length trace) \<and> asynchronousSystem.initial start (hd exec) \<and> (\<forall>i cfg1 cfg2. i < length exec - 1 \<longrightarrow> cfg1 = exec ! i \<longrightarrow> cfg2 = exec ! (i + 1) \<longrightarrow> asynchronousSystem.steps trans sends cfg1 (trace ! i) cfg2)"
getFrN: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "set (getFrN xs ts \<phi>s n) \<subseteq> var"
exhaust_sel: fixes item :: "item" obtains "item = Item (item_rule item) (item_dot item) (item_origin item) (item_end item)"
finite_ord_stays_finite: fixes a :: "'a" assumes "a \<in> carrier G"   and "G.ord a \<noteq> 0" shows "H.ord (h a) \<noteq> 0"
equal_refl: fixes equal :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "class.equal equal" shows "equal x x = True"
assms: shows "((\<sigma>, ns), a, \<sigma>', ns') \<in> automaton.trans (opnet np p)"
intro: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equal = (=)" shows "equal equal"
assms_14: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
assms_3: shows "inv_implications_nonneg (prop_config c1 p)"
set_replicate: fixes n :: "nat"   and x :: "'a" assumes "n \<noteq> 0" shows "set (replicate n x) = {x}"
acc_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "acc r = wf {(y, x). x \<sqsubset>\<^bsub>r\<^esub> y}"
strict_subset_subsumption_grounding_redundant_clss: fixes D :: "'a literal multiset"   and \<sigma> :: "'s"   and C :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "D \<cdot> \<sigma> \<subset># C"   and "D \<in> CC" shows "grounding_of_cls C \<subseteq> sr.Rf (grounding_of_clss CC)"
char_one_nz: shows "\<chi> \<one> \<noteq> 0"
finite_FV_FVoption: fixes t :: "sterm"   and s :: "sterm option" shows "finite (FV t) \<and> finite (FVoption s)"
C_def: shows "C = {b \<in> B. space M - f -` b \<in> null_sets M}"
ConjI: fixes A :: "fm"   and B :: "fm" assumes "se_fm A"   and "se_fm B" shows "se_fm (A AND B)"
hyp2: shows "matrix_to_iarray (snd (echelon_form_of_upt_k_det bezout (1::'a, A) k)) = snd (echelon_form_of_upt_k_det_iarrays (1::'a, matrix_to_iarray A) k bezout)"
disc_2: fixes x2 :: "'a" shows "is_Sym (Sym x2)"
le_degree: fixes p :: "'a poly"   and n :: "nat" assumes "poly.coeff p n \<noteq> (0::'a)" shows "n \<le> Polynomial.degree p"
inj_Inner: fixes \<Gamma> :: "('k, 'l) web_scheme"   and A :: "'m set" assumes "countable_bipartite_web \<Gamma>" shows "inj_on bi.Inner A"
order_prop: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod x y = (\<exists>z. sup_prod x z = y)"
arg_less_sf_at_Suc_of_c_sum: fixes u :: "nat" shows "u < sf (c_sum u + 1)"
awalkI_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.apath u p v" shows "G.awalk u p v"
later_2: fixes x :: "'a"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x \<in> A"   and "\<forall>xa y. xa \<in> A \<and> y \<in> A \<and> (P xa y \<or> P x xa) \<longrightarrow> Q xa y" shows "af A Q"
vsubstEnv_psubstEnv_idEnv: fixes ys :: "'varSort"   and z :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho &[z // y]_ys = rho &[idEnv [y \<leftarrow> Var ys z]_ys]"
complete_dual: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "UNIV-complete A less_eq" shows "UNIV-complete A (\<lambda>x y. less_eq y x)"
le_Suc_numeral: fixes n :: "nat"   and k :: "num" shows "(Suc n \<le> numeral k) = (n \<le> pred_numeral k)"
Infinitesimal_add: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "x + y \<in> Infinitesimal"
select_convs_1: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>\<V> = variables_of"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a OR b) OR c = a OR b OR c"
deny_pfun_def: fixes pf :: "'d \<Rightarrow> 'e option" shows "D\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>deny y\<rfloor>"
triangle_paths_unique: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<exists>!ab. path ab a b"
continuous_map_componentwise: fixes X :: "'d topology"   and Y :: "'e \<Rightarrow> 'f topology"   and I :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "continuous_map X (product_topology Y I) f = (f ` topspace X \<subseteq> extensional I \<and> (\<forall>k\<in>I. continuous_map X (Y k) (\<lambda>x. f x k)))"
p1: shows "insert (OrdP (Var k)) (quote_all p (Vs - {j})) \<turnstile> (SyntaxN.All j' (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)))(j::=Var j)"
hyps_2: fixes w :: "letter list" assumes "is_prefix (w # ws) f" shows "\<exists>vs. is_prefix vs f \<and> good vs"
Abs_dlist_inject: fixes x :: "'d list"   and y :: "'d list" assumes "x \<in> {xs. distinct xs}"   and "y \<in> {xs. distinct xs}" shows "(Abs_dlist x = Abs_dlist y) = (x = y)"
inj_on_rowB: shows "inj_on (row B) {0..<m}"
tiny_dg_length: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "tiny_digraph \<alpha> \<CC>" shows "vcard \<CC> = 4\<^sub>\<nat>"
fs: shows "class P C = \<lfloor>(D, fs, ms)\<rfloor>"
assms: shows "C.internal_equivalence f g \<eta> \<epsilon>"
intros_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
lcm_eq_0_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(lcm a b = zero) = (a = zero \<or> b = zero)"
SourcesA8_L0: shows "Sources level0 sA8 = {sA6, sA7, sA8, sA9}"
mat_eq_is_eq_dj: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" shows "[\<^bold>\<forall>x. \<^bold>\<box>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>G,x\<^sup>P\<rparr>) in v] = ((\<lambda>x. eval\<Pi>\<^sub>1 F x dj) = (\<lambda>x. eval\<Pi>\<^sub>1 G x dj))"
slp_indexl_length: fixes xs :: "'a list"   and i :: "nat" shows "slp_indexl (length xs) i = slp_index xs i"
HARootStates_notmem_Range_ChiRel: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<in> States (HARoot A)" shows "S \<notin> Range (ChiRel A)"
minus_minus: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<ominus> (\<ominus> x) = x"
point_injective: fixes x :: "'a" assumes "(x \<sqinter> top \<sqinter> id (x \<sqinter> top) \<le> top \<and> top \<sqinter> (x \<sqinter> top) = top) \<and> id x \<sqinter> top \<sqinter> id (id x \<sqinter> top) \<le> top \<and> top \<sqinter> (id x \<sqinter> top) = top" shows "id x \<sqinter> top \<sqinter> x \<le> top"
rat_to_normalized_int_poly_exists: fixes p :: "rat poly" assumes "p \<noteq> 0" obtains q :: "int poly" and lc :: "rat" where "p = Polynomial.smult lc (Ring_Hom_Poly.of_int_poly q)" and "0 < lc" and "Polynomial.content q = 1"
rtrancl_beta_AppR: fixes t :: "dB"   and t' :: "dB"   and s :: "dB" assumes "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s \<degree> t'"
rel_nondetT_expand: fixes M :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('f, 'd) nondetT"   and m' :: "('g, 'e) nondetT" assumes "M (run_nondet m) (run_nondet m')" shows "rel_nondetT M m m'"
ddh_1_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.ddh_1 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> z))))"
bit_of_nat_iff: fixes m :: "nat"   and n :: "nat" shows "bit (of_nat m) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> bit m n)"
ctrm_degree: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "deg R (monom P (p 0) 0) = 0"
PS\<^sub>R_def: shows "PS\<^sub>R = {p \<in> ps\<^sub>R. dist (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))}"
D_comm: fixes x :: "'a"   and y :: "'a" shows "D x y = D y x"
adopt_node_preserves_known_ptrs: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "known_ptrs h'"
HaddP_Zero_D2: fixes x :: "tm"   and y :: "tm"   and H :: "fm set" shows "insert (HaddP x Zero y) H \<turnstile> x EQ y"
join_return: fixes xs :: "'d\<cdot>'e" shows "join\<cdot>(return\<cdot>xs) = xs"
vars0_def: shows "c.vars0 = vars2 \<union> vars3"