hcju/isabelleps huggingface dataset:
equal_mod_timpls_mono: fixes TI :: "('e set \<times> 'e set) list"   and TI' :: "('e set \<times> 'e set) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "set TI \<subseteq> set TI'"   and "equal_mod_timpls TI s t" shows "equal_mod_timpls TI' s t"
dgrad_p_set_exhaust_expl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" shows "F \<subseteq> dgrad_p_set d (Max (d ` pp_of_term ` Keys F))"
TR3_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme"   and m :: "'f"   and n :: "'h"   and a :: "'a" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "a \<in> carrier R" shows "(a \<cdot>\<^sub>s\<^bsub>M\<^esub> m, n) \<plusminus>\<^bsub>P\<^esub> -\<^sub>a\<^bsub>P\<^esub> a \<cdot>\<^sub>s\<^bsub>P\<^esub> (m, n) \<in> TR\<^bsub>R\<^esub> M N P"
fold_eq_nfoldli: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "'b list"   and s :: "'a" shows "RETURN (fold f l s) = nfoldli l (\<lambda>_. True) (\<lambda>x s. RETURN (f x s)) s"
simplex_sat0: fixes cs :: "('i \<times> constraint) list"   and v :: "(nat, rat) mapping" assumes "solve_exec cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)"
Dynkin_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "Dynkin \<Omega> M = \<Inter> {D. Dynkin_system \<Omega> D \<and> M \<subseteq> D}"
bin_ops_same_3: fixes x :: "int" shows "x XOR x = 0"
V_equalityI: fixes a :: "V"   and b :: "V" assumes "\<And>x. (x \<in>\<^sub>\<circ> a) = (x \<in>\<^sub>\<circ> b)" shows "a = b"
integrable_product_swap: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f (y, x))"
sublist_code_1: fixes ys :: "'b list" shows "\<epsilon> \<le>f ys = True"
Rep_inverse': fixes a :: "'b word"   and r :: "bool list" assumes "to_bl a = r" shows "of_bl r = a"
one_step_def: fixes X0 :: "'e set"   and h :: "real"   and m :: "'e set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'e set \<Rightarrow> ('e set \<times> 'd) option nres" shows "one_step X0 h m = CHECK (\<lambda>_. print_msg ''one_step nonneg'') (0 < h) \<bind> (\<lambda>_. iterations_spec \<bind> (\<lambda>its. halve_stepsizes_spec \<bind> (\<lambda>halvs. cert_stepsize m X0 h its halvs \<bind> (\<lambda>(h, res, res_ivl, err). ASSERT (0 < h) \<bind> (\<lambda>_. RETURN (h, err, res_ivl, res))))))"
inj_iff_trivial_ker: shows "inj_on h (carrier R) = (a_kernel R S h = {\<zero>})"
m_comm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x"
Rep_monom: fixes x :: "'c monom" shows "Rep_monom x \<in> Collect monom_inv"
input_not_constrained_gval_swap_inputs: fixes a :: "vname gexp"   and v :: "nat"   and i :: "value list"   and c :: "nat \<Rightarrow>f value option"   and x :: "value" assumes "\<not> gexp_constrains a (V (I v))" shows "gval a (join_ir i c) = gval a (join_ir (i[v := x]) c)"
zero_ideal: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "ideal R {\<zero>\<^bsub>R\<^esub>}"
not_in_sub: fixes var :: "nat"   and p :: "real mpoly"   and q :: "real mpoly" assumes "var \<notin> vars p"   and "var \<notin> vars q" shows "var \<notin> vars (p - q)"
finite_UNIV: assumes "class.finite TYPE('g)" shows "finite UNIV"
HRB_slice_is_SDG_path_HRB_slice: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "n'' \<in> HRB_slice {n}"   and "n' \<in> S" shows "n'' \<in> HRB_slice S"
split: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'e"   and sum :: "'f + 'g" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
Abs_freeword_inverse: fixes y :: "('a \<times> bool) list" assumes "y \<in> {as. proper_signed_list as}" shows "freeword (Abs_freeword y) = y"
assms_2: assumes "\<phi> (map stl \<pi>l)" shows "\<chi>"
simps_7: fixes p :: "pvar"   and \<phi> :: "'a dfmla" shows "wff (Exi p \<phi>) = True"
whn_inv_auto: fixes t :: "trm"   and t' :: "trm" assumes "t \<Down> t'" obtains t :: "trm" where "ta__ \<leadsto> t" and "t \<Down> t'"   | "t' = ta__" and "\<forall>x. \<not> ta__ \<leadsto> x"
lincomb_sum: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> carrier M"   and "a \<in> A \<rightarrow> carrier R"   and "b \<in> A \<rightarrow> carrier R" shows "lincomb (\<lambda>v. a v \<oplus> b v) A = lincomb a A \<oplus>\<^bsub>M\<^esub> lincomb b A"
steps_z_norm_complete: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and u :: "'c \<Rightarrow> real"   and l' :: "'d"   and u' :: "'c \<Rightarrow> real"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"   and "u \<in> [D]\<^bsub>v,n\<^esub>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "\<exists>D'. A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle> \<and> u' \<in> [D']\<^bsub>v,n\<^esub>"
case_prod_eta: fixes f :: "'a \<times> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). f (x, y)) = f"
assms_2: shows "G\<^sub>2 \<simeq> G\<^sub>3"
sym: fixes P :: "conc_interface"   and Q :: "conc_interface" assumes "P \<simeq> Q" shows "Q \<simeq> P"
left_hcomp_closed: fixes \<mu> :: "'a"   and x :: "'a"   and y :: "'a"   and \<nu> :: "'a"   and c :: "'a"   and d :: "'a" assumes "Right.in_hom \<mu> x y"   and "\<guillemotleft>\<nu> : c \<Rightarrow> d\<guillemotright>"   and "\<nu> \<star> \<mu> \<noteq> null" shows "Right.in_hom (\<nu> \<star> \<mu>) (c \<star> x) (d \<star> y)"
filter_trms_inclusion: fixes C :: "'a literal set"   and E :: "'a trm set" shows "filter_trms C E \<subseteq> E"
dim_poly_to_row: fixes ts :: "'d list"   and p :: "'d \<Rightarrow>\<^sub>0 'c" shows "dim_vec (poly_to_row ts p) = length ts"
hyps_2: shows "\<Gamma>\<turnstile>mark_guards f c1 \<down> Normal s"
ide_char': fixes a :: "'a" shows "A.ide a = (A.arr a \<and> (A.dom a = a \<or> A.cod a = a))"
f20: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "f 2 \<noteq> (0::'b)"
iMOD_minus_divisor_mult: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and k :: "nat" assumes "x \<in> [ r, mod m ]"   and "k * m \<le> x - r" shows "x - k * m \<in> [ r, mod m ]"
assms_3: shows "\<forall>\<^sub>F i in I. \<forall>x\<in>X. h x \<le> f i x"
Mapping_RBT_inverse: fixes y :: "('b, 'c) rbt" assumes "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}" shows "mapping_rbt.impl_of (Mapping_RBT y) = y"
add_eq_conv_ex: fixes a :: "'k"   and M :: "'k multiset"   and b :: "'k"   and N :: "'k multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
rel_interior_same_affine_hull: fixes S :: "'a set" assumes "convex S" shows "affine hull rel_interior S = affine hull S"
word_power_less_diff: fixes n :: "nat"   and q :: "'b word"   and m :: "nat" assumes "2 ^ n * q < 2 ^ m"   and "q < 2 ^ (LENGTH('b) - n)" shows "q < 2 ^ (m - n)"
star_denest_var: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> ; (y ; x\<^sup>\<star>)\<^sup>\<star> = (x + y)\<^sup>\<star>"
state_update_after_def: fixes sc :: "'d \<times> 'd \<Rightarrow> bool"   and Ac :: "('c, 'd) action" shows "state_update_after sc Ac = action_of (action.relation_of Ac ;; R (true \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A'))"
assertLiteralEffect_1: fixes state :: "State"   and l :: "Literal"   and d :: "bool" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "getM (assertLiteral l d state) = getM state @ [(l, d)]"
rcosets_subset_PowG: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "rcosets\<^bsub>add_monoid G\<^esub> H \<subseteq> Pow (carrier G)"
simps_1: fixes x1 :: "'m"   and x2 :: "'n"   and y1 :: "'m"   and y2 :: "'n" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
2: fixes g :: "'g"   and p :: "'val"   and q :: "'val"   and r :: "'val" assumes "phiArg g p q"   and "phiArg g p r"   and "distinct [p, q, r]"   and "p \<in> allVars g" shows "\<not> (def_dominates g q p \<and> def_dominates g r p)"
Sum_any_sum_swap: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "finite A"   and "\<And>y. finite {x. f x y \<noteq> (0::'c)}" shows "(\<Sum>x. sum (f x) A) = (\<Sum>y\<in>A. \<Sum>x. f x y)"
hyps_1: shows "0 < nat \<bar>x ^ 3\<bar>"
seqI: fixes f :: "'a2"   and g :: "'a2" assumes "C2.arr f"   and "C2.arr g"   and "C2.dom g = C2.cod f" shows "C2.seq g f"
lcm_eq_lcm_factorial: fixes a :: "'a"   and b :: "'a" shows "lcm a b = lcm_factorial a b"
hom_restrict: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and t :: "'a \<Rightarrow> 'e" assumes "h \<in> hom G H"   and "\<And>g. g \<in> carrier G \<Longrightarrow> h g = t g" shows "t \<in> hom G H"
card_finite_linorders: fixes A :: "'a set" assumes "finite A" shows "card {R. linorder_on A R} = fact (card A)"
subs_def2: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "\<not> SATAxiom (sequent gamma)" shows "subs gamma = (case nforms gamma of [] \<Rightarrow> {} | nA # nAs \<Rightarrow> let (n, A) = nA in case A of FAtom z P vs \<Rightarrow> subsFAtom (atoms gamma) nAs z P vs | FConj z A0 A1 \<Rightarrow> subsFConj (atoms gamma) nAs z A0 A1 | FAll z A \<Rightarrow> subsFAll (atoms gamma) nAs n z A (freeVarsFL (sequent gamma)))"
exponents_le_finite: fixes p :: "nat"   and k :: "nat"   and l :: "nat"   and x :: "real" assumes "1 < p"   and "0 < k" shows "finite {i. real (p ^ (k * i + l)) \<le> x}"
Suc: shows "r\<inverse> ^^ n = (r ^^ n)\<inverse>"
ih: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" assumes "is_final_dstate ((deterministic_RP_step ^^ k) St)" shows "deterministic_RP St \<noteq> None"
vector_smult_hom: fixes x :: "real"   and v :: "(real, 'c) Finite_Cartesian_Product.vec" shows "of_real x *s map_vector of_real v = map_vector of_real (x *s v)"
neg_numeral_less_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n)"
not_dbm_lt_lt_impl: fixes a :: "'c"   and b :: "'c" assumes "\<not> Lt a \<prec> Lt b" shows "b \<le> a"
independent_card_le_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "independent B" shows "card B \<le> dim V"
l_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> ((\<forall>ptr h h'. reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h') \<and> (\<forall>h ptr. type_wf h \<longrightarrow> known_ptr ptr \<longrightarrow> ptr |\<in>| object_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_child_nodes ptr)) \<and> (\<forall>h ptr. h \<turnstile> ok get_child_nodes ptr \<longrightarrow> ptr |\<in>| object_ptr_kinds h) \<and> (\<forall>ptr h. pure (get_child_nodes ptr) h)"
contractible_eq_homotopy_equivalent_singleton_subtopology: fixes X :: "'a topology" shows "contractible_space X = (topspace X = {} \<or> (\<exists>a\<in>topspace X. X homotopy_equivalent_space subtopology X {a}))"
ipresSwapAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresSwapAll h hA MOD \<equiv> ipresSwap h MOD \<and> ipresSwapAbs hA MOD"
finite_Diff: fixes A :: "'b set"   and B :: "'b set" assumes "finite (A::'b set)" shows "finite ((A::'b set) - (B::'b set))"
assms_1: fixes H :: "circline" assumes "circline_diag H" shows "P H"
integrate_insert_result: fixes M :: "('c, 'd) message set"   and m :: "('c, 'd) insert_message"   and s :: "('c, 'd option) insert_message list" assumes "consistent (M \<union> {Insert m})"   and "Insert m \<notin> M"   and "is_associated_string M s" shows "is_certified_associated_string (M \<union> {Insert m}) (integrate_insert m s (P m) (S m))"
k_nin': fixes v' :: "'b" shows "(k, v') \<notin> S"
Tree\<^sub>\<alpha>_equivp: shows "equivp (=\<^sub>\<alpha>)"
diff_Suc: fixes m :: "nat"   and n :: "nat" shows "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
fmdom'I: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "m $$ x = Some y" shows "x \<in> fmdom' m"
simps: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv" shows "interaction_bound consider gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound consider (c input)) else \<Squnion>input. interaction_bound consider (c input))"
assms: fixes i :: "'a" shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
greaterThan_iff: fixes i :: "'e"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'e" shows "(i \<in> ord.greaterThan less k) = less k i"
onlyGoodsAreAllocated: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "the_elem (Outside' {seller} ` (argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) \<in> soldAllocations N (set \<Omega>)"
upt: shows "[0..<m] = 0 # [1..<m]"
left_total_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "left_total R = (\<forall>x. \<exists>y. R x y)"
finfun_upd_apply_code: fixes f :: "('b, 'a) finfun"   and a :: "'b"   and b :: "'a"   and a' :: "'b" shows "finfun_apply (finfun_update_code f a b) a' = (if a = a' then b else finfun_apply f a')"
finalStateCharacterization: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars"   and "isFinalState state decisionVars" shows "\<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> formulaFalse (getF state) (elements (getM state)) \<and> decisions (getM state) = []"
trans: fixes P :: "'b"   and Q :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus"   and R :: "'b" assumes "(P, Q) \<in> cSim_cs Rel Cal"   and "(Q, R) \<in> cSim_cs Rel Cal" shows "(P, R) \<in> cSim_cs Rel Cal"
r_3: fixes w :: "complex" assumes "w \<in> cball z0 r - {z0}" shows "f w = zor_poly f z0 w * (w - z0) powr of_int n"
simps_6: fixes \<sigma> :: "nat \<Rightarrow> val"   and b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "expr_sem_rf \<sigma> (IF b THEN e1 ELSE e2) = (if expr_sem_rf \<sigma> b = TRUE then expr_sem_rf \<sigma> e1 else expr_sem_rf \<sigma> e2)"
InvariantsNoDecisionsWhenConflictNorUnitAfterAssertLiteral: fixes state :: "State"   and decision :: "bool"   and literal :: "Literal" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "decision \<longrightarrow> \<not> getConflictFlag state \<and> getQ state = []" shows "let state' = assertLiteral literal decision state in InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
restrict_out: fixes x :: "'b"   and A :: "'b set"   and m :: "'b \<Rightarrow> 'c option" assumes "x \<notin> A" shows "(m |` A) x = None"
repr_independence: fixes G :: "('d, 'e) monoid_scheme"   and y :: "'d"   and H :: "'d set"   and x :: "'d" assumes "Group.group G"   and "y \<in> H #>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H G" shows "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y"
independent_insertI: fixes a :: "'c"   and S :: "'c set" assumes "a \<notin> vs2.span S"   and "vs2.independent S" shows "vs2.independent (insert a S)"
nth_app_eq: fixes sl :: "'b list"   and x :: "'b"   and l :: "'b list" assumes "sl \<noteq> []"   and "sl ! (length sl - Suc 0) = x" shows "(l @ sl) ! (length l + length sl - Suc 0) = x"
monom: fixes a :: "'a" assumes "a \<in> carrier R" shows "Q (monom P a 0)"
x0_def: shows "x0 = real (nth_prime (nat \<lceil>max 0 C\<rceil>))"
assms_5: shows "(\<lambda>n. f n x) sums l"
prl2: fixes g :: "'f \<times> nat"   and f :: "'f \<times> nat" assumes "prl g" shows "fst (prc g f) = False"
isUCont_isCont: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "isUCont f" shows "isCont f x"
stranspose_nth: fixes k :: "nat"   and ws :: "'a stream list" assumes "k < length ws" shows "smap (\<lambda>xs. xs ! k) (stranspose ws) = ws ! k"
bounded_snd: fixes s :: "('b \<times> 'c) set" assumes "bounded s" shows "bounded (snd ` s)"
exhaust_sel: fixes list :: "'d list" obtains "list = []"   | "list = hd list # tl list"
1: shows "bf_between_arcs a b H (d \<squnion> c) \<longrightarrow> bf_between_arcs a b H d \<or> bf_between_arcs a c H d \<and> bf_between_arcs c b H d"
sel_2: fixes x1 :: "'d"   and x2 :: "'c"   and x3 :: "nat"   and x4 :: "('d, 'c) SkewBinomialTree list" shows "prio (Node x1 x2 x3 x4) = x2"
ord_less_eq_trans: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b < a"   and "b = c" shows "c < a"
psubst_compose: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "good X"   and "goodEnv rho"   and "goodEnv rho'" shows "X #[rho] #[rho'] = X #[rho &[rho']]"
bit_not_zero_iff: fixes a :: "bit" shows "(a \<noteq> 0) = (a = 1)"
assms: shows "\<forall>y. (y, t) \<in> trm_ord \<longrightarrow> y \<noteq> trm_rep y S \<longrightarrow> (trm_rep y S, y) \<in> trm_ord"
lnth_LCons: fixes x :: "'a"   and xs :: "'a llist"   and n :: "nat" shows "lnth (LCons x xs) n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> lnth xs n')"
Units_one_closed: shows "\<one> \<in> Units (Ring_Divisibility.mult_of R)"
swap_self: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "Fun.swap a a f = f"
is_Done_DONE: fixes a :: "'f" shows "Resumption.resumption.is_Done (DONE a)"
bisim1Sync14: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and a' :: "'addr" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v, Addr a'], xs, 7 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
vconn_implies_hpar_2: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D \<mu> = trg\<^sub>D f"
intros_1: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state" shows "evaluate_prog ck env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"
fps_of_ratfps_altdef: fixes x :: "'b ratfps" shows "fps_of_ratfps x = (case quot_of_ratfps x of (a, b) \<Rightarrow> fps_of_poly a / fps_of_poly b)"
Fresh_apply: fixes h :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "\<exists>a. atom a \<sharp> (h, h a)"   and "atom a \<sharp> h" shows "Fresh h = h a"
butterfly_def_one_dim: fixes s :: "'c"   and t :: "'b" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
top_assn_rule: fixes h :: "heap"   and as :: "nat set" shows "(pHeap h as \<Turnstile> true) = in_range (h, as)"
le_utransI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> t P \<tturnstile> u P" shows "le_utrans t u"
iFROM_iprev_diff_const: fixes n :: "nat"   and t :: "nat" assumes "n < t" shows "t - iprev t [n\<dots>] = Suc 0"
Longest_common_prefix_prefix: fixes xs :: "'b list"   and L :: "'b list set" assumes "xs \<in> L" shows "prefix (Longest_common_prefix L) xs"
ps: shows "positive (matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))) R)"
discr_transT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>t s'" shows "s \<approx> s'"
distinct_39: shows "AtOp \<noteq> string_binop.LessEqOp"
idinfo_pwd: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "\<exists>X. \<langle>n, X\<rangle> \<in> spied s \<and> X \<noteq> Pwd n"   and "n \<notin> bad_id_pubkey \<union> bad_id_shakey" shows "\<exists>SK. SesKey SK \<in> spied s \<and> ((Owner n, Crypt (SesK SK) (Pwd n)) \<in> s \<or> (Asset n, Crypt (SesK SK) (Num 0)) \<in> s)"
smult_0_right: fixes a :: "'d" shows "smult a 0 = 0"
create_name_gba_correct: fixes \<phi> :: "'b ltlr" shows "create_name_gba \<phi> \<le> SPEC (\<lambda>\<A>. gba \<A> \<and> finite (g_V \<A>) \<and> (\<forall>\<xi>. gba.accept \<A> \<xi> = \<xi> \<Turnstile>\<^sub>r \<phi>))"
cont_on_left: shows "continuous_on (path_image (\<lambda>y. twoC (y, 0))) F"
subst_lit_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and L :: "'b literal"   and \<sigma> :: "'c" shows "substitution_ops.subst_lit subst_atm L \<sigma> = map_literal (\<lambda>A. subst_atm A \<sigma>) L"
has_unique_inverse: shows "\<exists>!F. inverse_functors (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G"