hcju/isabelleps huggingface dataset:
oseq_step_is_seq_step: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and \<xi> :: "'e" assumes "((\<sigma>, p), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i"   and "\<sigma> i = \<xi>" shows "\<exists>\<xi>'. \<sigma>' i = \<xi>' \<and> ((\<xi>, p), a, \<xi>', p') \<in> seqp_sos \<Gamma>"
is_RB_inI1: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and g :: "'t \<Rightarrow>\<^sub>0 'b"   and d :: "'a \<Rightarrow> nat" assumes "is_canon_rewriter rword G u g"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G (monom_mult (1::'b) (pp_of_term u - lp g) g)" shows "is_RB_in d rword G u"
dom_simp: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.dom f = VxV.dom f"
edom_env: fixes S :: "'a set"   and m :: "'a \<Rightarrow> 'b" shows "edom (m f|` S) = edom m \<inter> S"
B2: fixes f :: "nat \<Rightarrow> real" shows "map dim_vec (drop (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs) = input_sizes m2"
assms: fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "e \<in> A"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) - delta \<le> Gromov_product_at e x z"
callee_invariant': fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (left s x)"   and "I s"   and "x \<in> outs_\<I> \<I>l" shows "I s' \<and> y \<in> responses_\<I> \<I>l x"
assms_5: assumes "P ((\<sigma>, s), a, \<sigma>', s')" shows "Q"
simps: fixes x :: "'b"   and n :: "nat" shows "braun2_of x n = (if n = 0 then (\<langle>\<rangle>, \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>) else let (s, t) = braun2_of x ((n - 1) div 2) in if odd n then (\<langle>s, x, s\<rangle>, \<langle>t, x, s\<rangle>) else (\<langle>t, x, s\<rangle>, \<langle>t, x, t\<rangle>))"
Seq1: fixes P0 :: "'b com"   and s :: "'b"   and t :: "'b"   and P1 :: "'b com" assumes "(Some P0, s) -c\<rightarrow> (None, t)" shows "(Some (Seq P0 P1), s) -c\<rightarrow> (Some P1, t)"
zip_with_index_from_def: fixes n :: "nat"   and xs :: "'a list" shows "zip_with_index_from n xs = zip [n..<n + length xs] xs"
assms_2: shows "kKS m \<notin> specKeysSecrets Q"
epred_llength: fixes xs :: "'a llist" shows "epred (llength xs) = llength (ltl xs)"
is_valid_operator_sas_plus_then_2: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (sas_plus_operator.precondition_of op). \<R>\<^sub>+ \<Psi> v \<noteq> {} \<and> a \<in> \<R>\<^sub>+ \<Psi> v"
aval'_correct: fixes s :: "char list \<Rightarrow> int"   and S :: "'av st"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>s S" shows "aval a s \<in> \<gamma> (aval' a S)"
Abs_preal_cases: fixes x :: "preal" obtains y :: "rat set" where "x = Abs_preal y" and "y \<in> Collect Dedekind_Real.cut"
filtration_space: fixes s :: "nat"   and t :: "nat" assumes "s \<le> t" shows "space (F s) = space (F t)"
q1: shows "q1 = hom_component p (deg_pm s1)"
inf_le1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type)) x"
assms: shows "\<forall>i<n. \<exists>x. P x i"
assms: shows "s \<rightarrow>\<^sub>\<beta>\<^sup>* t"
enum_Agent_def: shows "enum_class.enum = [Child0, Child1, Child2]"
assms_1: shows "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y' g"
get_even_def: fixes n :: "nat" shows "get_even n = (if even n then n else Suc n)"
integral_pmf: fixes X :: "'e set"   and M :: "'e pmf" shows "integral\<^sup>L (count_space X) (pmf M) = measure_pmf.prob M X"
msgrel_refl: fixes X :: "freemsg" shows "X \<sim> X"
edg_simps_4: fixes \<Delta> :: "('a, 'b) network_scheme"   and v :: "'a antiparallel_edges.vertex"   and x :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.edg \<Delta> v (antiparallel_edges.Edge x y) = (v = antiparallel_edges.Vertex x \<and> edge \<Delta> x y)"
dominates_mid: fixes g :: "'g"   and n :: "'node"   and x :: "'node"   and m :: "'node"   and ns :: "'node list" assumes "old.dominates g n x"   and "old.dominates g x m"   and "g \<turnstile> n-ns\<rightarrow>m"   and "invar g" shows "x \<in> set ns"
InvariantConflictFlagCharacterizationAfterApplyBackjump_1: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantUniqC (getC state)"   and "getC state = [opposite (getCl state)]"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state' = applyBackjump state in InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
CFG_SSA_defs_5: fixes g :: "'g" shows "allVars g \<equiv> \<Union>n\<in>set (\<alpha>n g). allDefs g n \<union> allUses g n"
f: shows "f \<in> A \<rightarrow> carrier_vec n"
in_segment_fst_ge: fixes x0 :: "real"   and x1 :: "real"   and x :: "real"   and b :: "real"   and y0 :: "real"   and y1 :: "real" assumes "x0 \<le> x1"   and "(x, b) \<in> closed_segment (x0, y0) (x1, y1)" shows "x0 \<le> x"
localize_recursion: fixes \<pi>' :: "char list \<Rightarrow> com option"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>': (PScope \<pi> c, s) \<Rightarrow> t = \<pi>: (c, s) \<Rightarrow> t"
finite_nat_iff_bounded_le: fixes S :: "nat set" shows "finite S = (\<exists>k. S \<subseteq> {..k})"
in_mono: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B" shows "s2p A x \<longrightarrow> s2p B x"
intro: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and cHeap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "CardinalityPrognosisEdom prognosis"   and "CardinalityPrognosisShape prognosis"   and "CardinalityPrognosisApp prognosis"   and "CardinalityPrognosisLam prognosis"   and "CardinalityPrognosisVar prognosis"   and "CardinalityPrognosisLet prognosis cHeap Aheap"   and "CardinalityPrognosisIfThenElse prognosis"   and "CardinalityHeapSafe cHeap Aheap"   and "ArityAnalysisLetSafe Aexp Aheap" shows "CardinalityPrognosisSafe prognosis cHeap Aheap Aexp"
red: shows "t \<turnstile> s1 -1-ta1\<rightarrow> s1'"
rhs: shows "(x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q = (x * y + (q - (x * ((y + (q - b)) mod q) + (q - r)) mod q)) mod q"
ex_un_null: shows "\<exists>!n. \<forall>f. B.VV.comp n f = n \<and> B.VV.comp f n = n"
cod_def: fixes f :: "'c" shows "C.cod f = (if C.codomains f \<noteq> {} then SOME b. b \<in> C.codomains f else C.null)"
sumle: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n (M 0) (M 1) (denote S) n \<rho>) n \<le>\<^sub>L matrix_sum d (\<lambda>n. denote_while_n (M 0) (M 1) (denote S) n \<rho>) (Suc n)"
lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and vs :: "'c list" shows "lincomb_list c vs = M.sumlist (map (\<lambda>i. c i \<odot>\<^bsub>M\<^esub> vs ! i) [0..<length vs])"
Q: fixes T :: "'a set" assumes "T \<subseteq> S"   and "P T" shows "Q (f ` T)"
Fault_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Fault f" shows "t = Fault f"
assms: shows "A \<subseteq> {s. conv_abscissa f < ereal (s \<bullet> (1::'a))}"
alphaAbs_preserves_qGoodAbs1: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "A $= A'"   and "qGoodAbs A" shows "qGoodAbs A'"
epsilonG_def: fixes uu :: "'a itself" shows "epsilonG uu = ln 2 / (2 + 2 * deltaG TYPE('a))"
Some_com: fixes y :: "'c"   and x :: "'c option" shows "(Some y = x) = (x = Some y)"
returns_result_select_result: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
lt_setI: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "f \<in> F"   and "f \<noteq> 0" shows "lt f \<in> lt_set F"
inner_dirsum_append: fixes As :: "'a set list"   and Bs :: "'a set list" assumes "add_independentS (As @ Bs)"   and "(0::'a) \<in> (\<Sum>X\<leftarrow>Bs. X)" shows "(\<Oplus>X\<leftarrow>As @ Bs. X) = (\<Oplus>X\<leftarrow>As. X) \<oplus> (\<Oplus>X\<leftarrow>Bs. X)"
n_def: shows "n = (LEAST n. 0 < n \<and> fds_nth g n \<noteq> (0::'a))"
F: fixes i :: "'i"   and j :: "'j" assumes "(i, j) \<in> A" shows "F i j \<in> sets borel"
unop_onto_def: fixes A :: "V"   and f :: "V" shows "unop_onto A f \<equiv> nop_onto A (1\<^sub>\<nat>) f"
tendsto_uminus_nhds: fixes a :: "'a" shows "(uminus \<longlongrightarrow> - a) (nhds a)"
Not_ltlp_def: fixes \<phi> :: "'b pltl" shows "not\<^sub>p \<phi> \<equiv> \<phi> implies\<^sub>p false\<^sub>p"
eq: shows "set_mset (\<Sum>bl\<in>#remove1_mset blv \<B>. mset_set (bl \<inter> blv)) = blv"
set_cases: fixes e :: "'a"   and a :: "'a pvar" assumes "e \<in> set_pvar a" obtains "a = PWt e"   | z2 :: "nat" where "a = PCoef e z2"
f0: fixes s :: "real"   and x :: "(real, 'i) vec" assumes "s \<in> T" shows "f s x $ i = 0"
A0: fixes n0 :: "nat"   and n1 :: "nat" assumes "N < n0 \<and> N < n1" shows "s n0 n = s n1 n"
ring_distribs_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) * c = a * c + b * c"
eq: shows "rev y \<^sup>@ b \<cdot> rev x \<^sup>@ a = rev z \<^sup>@ c"
eq_Max_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Max less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
complex_i_not_neg_numeral: fixes w :: "num" shows "\<i> \<noteq> - numeral w"
SD_pref_profile: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents" shows "SD (R i) q p = (p \<in> lotteries \<and> q \<in> lotteries \<and> (\<forall>x\<in>alts. lottery_prob q (preferred_alts (R i) x) \<le> lottery_prob p (preferred_alts (R i) x)))"
nofail_inf_serialize: fixes a :: "'e nres"   and b :: "'e nres" assumes "nofail a"   and "nofail b" shows "inf a b = a \<bind> (\<lambda>x. ASSUME (inres b x) \<bind> (\<lambda>_. RETURN x))"
in_out_arcs_conv: fixes e :: "'a"   and G :: "('b, 'a) pre_digraph"   and v :: "'b" shows "(e \<in> out_arcs G v) = (e \<in> arcs G \<and> tail G e = v)"
subprob_not_empty: shows "space (M \<bind> f) \<noteq> {}"
not_eq_complement: fixes a :: "'a" shows "NOT a = - a - (1::'a)"
ImpL: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>"
deriv_cong_ev: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<forall>\<^sub>F x in nhds x. f x = g x"   and "x = y" shows "deriv f x = deriv g y"
mergeable_envs_def: fixes n :: "nat"   and S :: "'c list set" shows "mergeable_envs n S = (\<forall>v1\<in>S. \<forall>v2\<in>S. \<forall>A B f. (\<forall>x\<in>A. x < n \<and> v1 ! x = f x) \<and> (\<forall>x\<in>B. x < n \<and> v2 ! x = f x) \<longrightarrow> (\<exists>v\<in>S. \<forall>x\<in>A \<union> B. v ! x = f x))"
finite_cartesian_product: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "finite (A \<times> B)"
prefix_of_node_history_def: fixes history :: "nat \<Rightarrow> 'c list"   and xs :: "'c list"   and i :: "nat" assumes "node_histories history" shows "node_histories.prefix_of_node_history history xs i \<equiv> \<exists>ys. xs @ ys = history i"
fmsubset_alt_def: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" shows "(m \<subseteq>\<^sub>f n) = fmpred (\<lambda>k v. fmlookup n k = Some v) m"
exhaust: fixes y :: "'b trm" obtains x1 :: "'b" where "y = Var x1"   | x2 :: "'b" where "y = Const x2"   | x31 :: "'b trm" and x32 :: "'b trm" where "y = x31 \<cdot> x32"
coeff_Poly_eq: fixes xs :: "'b list" shows "coeff (Poly xs) = nth_default (0::'b) xs"
simps_2: fixes A :: "hfm"   and B :: "hfm"   and i :: "name"   and x :: "fm" shows "(A HOR B)(i:::=x) = A(i:::=x) HOR B(i:::=x)"
rvars_pivot: shows "rvars (\<T> (pivot' x\<^sub>i x\<^sub>j s)) = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}"
lincomb_dim: fixes X :: "'b vec set"   and n :: "nat"   and a :: "'b vec \<Rightarrow> 'b" assumes "finite X"   and "X \<subseteq> carrier_vec n" shows "dim_vec (module.lincomb (module_vec TYPE('b) n) a X) = n"
comp: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "P M1"   and "P M2" shows "P (M1 + M2)"
h0: shows "\<forall>i<length (take_indices signs S). col (alt_matrix_A_R (take_indices signs S) subsets) i = col (take_cols_from_matrix (alt_matrix_A_R signs subsets) S) i"
inversion_unit_disc_compl: shows "inversion ` unit_disc_compl = unit_disc"
Suc_pred': fixes n :: "nat" assumes "0 < n" shows "n = Suc (n - 1)"
sumE: fixes s :: "'c + 'd" obtains x :: "'c" where "s = Inl x"   | y :: "'d" where "s = Inr y"
forward_sim_execs: fixes f :: "'b \<Rightarrow> 'c set"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa"   and e_B :: "'b \<times> ('d \<times> 'b) list" assumes "is_forward_sim f B A"   and "is_exec_of B e_B" shows "\<exists>e_A. is_exec_of A e_A \<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
Rep_hf_inverse: fixes x :: "hf" shows "Abs_hf (Rep_hf x) = x"
distinct_concat_map_I: fixes xs :: "'x list"   and f :: "'x \<Rightarrow> 'y list" assumes "distinct xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> distinct (f x)"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; x \<noteq> y\<rbrakk> \<Longrightarrow> set (f x) \<inter> set (f y) = {}" shows "distinct (concat (map f xs))"
simps_2: fixes m :: "'v"   and ms :: "'v list" shows "RSpan (m # ms) = {r \<cdot> m |r. r \<in> UNIV} + RSpan ms"
is_segment_def: fixes X :: "'p set" shows "is_segment X \<equiv> \<exists>x y. X = {x, y}"
simps_2: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int"   and l :: "int tree"   and k :: "int"   and r :: "int tree" shows "wpl.wpl w i j \<langle>l, k, r\<rangle> = wpl.wpl w i (k - 1) l + wpl.wpl w (k + 1) j r + w i j"
body: shows "c_add + c_swap + 5 \<le> body_cost"
fwi_out_of_bounds2: fixes n :: "nat"   and j' :: "nat"   and j :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "n < j'"   and "j \<le> n" shows "fwi M n k i j i' j' = M i' j'"
split_minD: fixes t :: "'c tree"   and x :: "'c"   and t' :: "'c tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # inorder t' = inorder t"
rep_eq: fixes x :: "'a mpoly"   and xa :: "nat"   and xb :: "nat" shows "mapping_of (isolate_variable_sparse x xa xb) = (\<Sum>m | m \<in> monomials x \<and> lookup m xa = xb. monomial (MPoly_Type.coeff x m) (Poly_Mapping.update xa 0 m))"
at_ds5: fixes a :: "'a"   and b :: "'a" assumes "at TYPE('a)" shows " [(a, b)] \<triangleq> [(b, a)] "
in_reg_mod_def: fixes data_w32 :: "word32"   and win :: "'b word"   and ur :: "word5"   and state :: "'b sparc_state" shows "in_reg_mod data_w32 win ur state \<equiv> let state' = state\<lparr>user_reg := (user_reg state)(win := (user_reg state win)(ur := data_w32))\<rparr>; win' = next_window win; ur' = ur - 16 in state'\<lparr>user_reg := (user_reg state')(win' := (user_reg state' win')(ur' := data_w32))\<rparr>"
imageE: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "b \<in> f ` A" obtains x :: "'f" where "b = f x" and "x \<in> A"
Abs_eqD: fixes x :: "'b"   and y :: "'b" assumes "Abs x = Abs y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
add_not_less_TC_self: fixes x :: "V"   and y :: "V" shows "\<not> x + y \<sqsubset> x"
env_unconc: fixes a :: "'a el_step list"   and b :: "'a el_step list" shows "env (a @ b) = env a @ env b"
3: fixes x :: "'a formula"   and y :: "'a formula" assumes "P x"   and "P y" shows "P (x \<sqinter> y)"
is_lifting_lift: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "basis_wf (b # basis)" shows "is_lifting lift_ms basis (b # basis)"
simps_1: fixes L :: "'c \<Rightarrow> ('a \<times> 'a) set"   and R :: "'c \<Rightarrow> ('a \<times> 'a) set"   and E :: "'c \<Rightarrow> ('a \<times> 'a) set" shows "lconv L R E [] = Id"
induct: fixes P :: "'b rule list \<Rightarrow> bool"   and a0 :: "'b rule list" assumes "P []"   and "\<And>uu_ rs. P rs \<Longrightarrow> P (Rule uu_ Empty # rs)"   and "\<And>uv_ rs. P rs \<Longrightarrow> P (Rule uv_ Log # rs)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Accept # rs)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Drop # rs)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Reject # rs)"   and "\<And>v vb rs. P rs \<Longrightarrow> P (Rule v (Call vb) # rs)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Return # rs)"   and "\<And>v vb rs. P rs \<Longrightarrow> P (Rule v (Goto vb) # rs)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Unknown # rs)" shows "P a0"
hyps: shows "Q \<parallel> R \<longmapsto> \<alpha> \<prec> QR'"
leg1_simps_5: shows "local.dom r\<^sub>1 = r\<^sub>1"
val: fixes v' :: "v" assumes "do_log op v e = Some (Val v')" shows "thesis"
div_closed: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "field R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "y \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<oslash>\<^bsub>R\<^esub> y \<in> carrier R"
hbasep_hbase_eq: fixes b :: "nat" shows "hbasep b = (\<lambda>x. x \<in> hbase b)"
convol_o: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'd" shows "BNF_Def.convol f g \<circ> h = BNF_Def.convol (f \<circ> h) (g \<circ> h)"
eWlsAbs_nchotomy: fixes eA :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and us :: "'varSort"   and s :: "'sort" shows "(\<exists>A. eA = OK A \<and> igWlsAbs MOD (us, s) A) \<or> \<not> eWlsAbs MOD (us, s) eA"
RestrictedP_NotInDom: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and j :: "tm" shows "{RestrictedP s k s', Neg (j IN k)} \<turnstile> NotInDom j s'"
transI: fixes r :: "('c \<times> 'c) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
concat_map_map_index: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'c list"   and ys :: "'b list" shows "concat (map (\<lambda>n. map (f n) xs) ys) = map (\<lambda>i. f (ys ! (i div length xs)) (xs ! (i mod length xs))) [0..<length xs * length ys]"
assms: shows "\<Union> (S ` I) \<subseteq> carrier G"
g_to_rev_list_impl: shows "map_to_rev_list \<alpha> invar g_to_rev_list"
test_bit_sbintrunc: fixes i :: "nat"   and n :: "nat" assumes "i < LENGTH('a)" shows "word_of_int (signed_take_bit n (uint w)) !! i = (if n < i then w !! n else w !! i)"
stern_brocot_coprime: fixes path :: "dir list" shows "case root (traverse_tree path stern_brocot_recurse) of (m, n) \<Rightarrow> coprime m n"
intro: fixes R :: "('b list \<times> 'b list) set"   and L :: "'b list set"   and A :: "'b list set"   and h :: "'b list set \<Rightarrow> hf"   and n :: "nat" assumes "equiv UNIV R"   and "right_invariant R"   and "L = R `` A"   and "bij_betw h (UNIV // R) (hfset (ord_of n))" shows "MyhillNerode_dfa L R A n h"
mgu_eliminates_dom: fixes x :: "('b, 'a) Term.term"   and y :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a" assumes "mgu x y = Some \<theta>"   and "v \<in> subst_domain \<theta>" shows "subst_elim \<theta> v"
order_iff_strict: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less" shows "less_eq a b = (less a b \<or> a = b)"
exhaust: fixes y :: "'b wait_set_status" obtains x1 :: "'b" where "y = InWS x1"   | x2 :: "wake_up_status" where "y = PostWS x2"
X1: shows "X1 = {x. le_vec 0 x \<and> norm x = 1}"
properfactor_mult_l: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "monoid_cancel G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (c \<otimes>\<^bsub>G\<^esub> a) (c \<otimes>\<^bsub>G\<^esub> b) = properfactor G a b"
mset_ran_combine1: fixes r\<^sub>1 :: "int set"   and r\<^sub>2 :: "int set"   and a :: "int \<Rightarrow> 'b" assumes "finite r\<^sub>1"   and "finite r\<^sub>2"   and "r\<^sub>1 \<inter> r\<^sub>2 = {}" shows "mset_ran a r\<^sub>1 + mset_ran a r\<^sub>2 = mset_ran a (r\<^sub>1 \<union> r\<^sub>2)"
subsetD: fixes x :: "'a" assumes "x \<in> U" shows "x \<in> V"
singular_relcycle_minus: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X S (- c) = singular_relcycle p X S c"
subbicategory_axioms: shows "subbicategory (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B Arr"
hoare_partial_complete: fixes P :: "complex mat"   and S :: "com"   and Q :: "complex mat" assumes "\<Turnstile>\<^sub>p {P} S {Q}"   and "well_com S"   and "is_quantum_predicate P"   and "is_quantum_predicate Q" shows "\<turnstile>\<^sub>p {P} S {Q}"
lappend_code_1: fixes ys :: "'b llist" shows "lappend LNil ys = ys"
span_breakdown_eq: fixes x :: "'c"   and a :: "'c"   and S :: "'c set" shows "(x \<in> ideal (insert a S)) = (\<exists>k. x - k * a \<in> ideal S)"
e: assumes "x \<sqsubseteq> y"   and "x .\<noteq> y" shows "P"