hcju/isabelleps huggingface dataset:
coprime_power_right_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"   and n :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a (power.power one times b n) = (algebraic_semidom.coprime times one a b \<or> n = 0)"
parPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim> Q" shows "P \<parallel> R \<sim> Q \<parallel> R"
eq_order_le: fixes x :: "'c"   and y :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = y"   and "order r" shows "x \<sqsubseteq>\<^bsub>r\<^esub> y"
simps_2: fixes I :: "nat set"   and n :: "nat" shows "I \<leftarrow> Suc n = iprev (I \<leftarrow> n) I"
degree_pCons_eq_if: fixes a :: "'b"   and p :: "'b poly" shows "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
MPAIR: fixes X :: "freemsg"   and X' :: "freemsg"   and Y :: "freemsg"   and Y' :: "freemsg" assumes "X \<sim> X'"   and "Y \<sim> Y'" shows "MPAIR X Y \<sim> MPAIR X' Y'"
rat_number_expand_5: fixes k :: "num" shows "- numeral k = Fract (- numeral k) 1"
fac_pref: fixes u :: "'b list"   and v :: "'b list" shows "u \<le>f v \<equiv> \<exists>p. p \<cdot> u \<le>p v"
2: shows "\<eta>2 $$ X maps\<^bsub>NTCatCod \<eta>1\<^esub> NTDom \<eta>2 @@ X to NTCod \<eta>2 @@ X"
that: fixes k :: "nat"   and z :: "'a list" assumes "k < l"   and "z \<le>np w"   and "w \<^sup>@ k \<cdot> z = u" shows "thesis"
exec_gpv_fixp_induct_strong: fixes P :: "(('b, 'd, 'e) gpv \<Rightarrow> 'f \<Rightarrow> ('b \<times> 'f) spmf) \<Rightarrow> bool"   and callee :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv'. \<lbrakk>\<And>c s. ord_spmf (=) (exec_gpv' c s) (exec_gpv callee c s); P exec_gpv'\<rbrakk> \<Longrightarrow> P (\<lambda>c s. the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv' (c x) y)))" shows "P (exec_gpv callee)"
has_integral_set_lebesgue: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S" shows "(f has_integral set_lebesgue_integral lebesgue S f) S"
appliedBackjump_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedBackjump stateA stateB \<equiv> \<exists>bc bl level. isUnitClause bc bl (elements (prefixToLevel level (getM stateA))) \<and> formulaEntailsClause (getF stateA) bc \<and> var bl \<in> vars (getF stateA) \<union> vars (elements (getM stateA)) \<and> 0 \<le> level \<and> level < currentLevel (getM stateA) \<and> getF stateB = getF stateA \<and> getM stateB = prefixToLevel level (getM stateA) @ [(bl, False)]"
Entry_reaches: fixes n :: "'d"   and g :: "('d, 'e, 'f) gen_cfg_wf" assumes "n \<in> set (gen_wf_\<alpha>n g)"   and "True" shows "\<exists>ns. gen_wf.path2 g (gen_wf_Entry g) ns n"
subst_all: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT t" shows "subst (all x \<phi>) t y = all x (subst \<phi> t y)"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>A.map f : A.map a \<rightarrow> A.map b\<guillemotright>"
infinite_finite_induct: fixes P :: "'c set \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
inorder_insert: fixes t :: "'b tree"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (insert x t) = ins_list x (inorder t)"
1: shows "eintT (resOf f) (eintF (Oldf f) (map2 ntsem (arOf f) al))"
tendsF: fixes x :: "'a" assumes "x \<notin> NF" shows "(\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else (0::'b))"
simps_17: fixes f1 :: "i32 \<Rightarrow> 'a"   and f2 :: "i64 \<Rightarrow> 'a"   and f3 :: "f32 \<Rightarrow> 'a"   and f4 :: "f64 \<Rightarrow> 'a"   and x1 :: "i32" shows "(case ConstInt32 x1 of ConstInt32 x \<Rightarrow> f1 x | ConstInt64 x \<Rightarrow> f2 x | ConstFloat32 x \<Rightarrow> f3 x | ConstFloat64 x \<Rightarrow> f4 x) = f1 x1"
h: shows "\<forall>f\<in>F. Inf_pres f"
set_insort_insert: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (FWNormalisationCore.insort x xs l) \<subseteq> insert x (set xs)"
eq_fold: fixes M :: "'a multiset" shows "\<Sum>\<^sub># M = fold_mset (+) (0::'a) M"
FSwSbImorph_def: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme" shows "FSwSbImorph h hA MOD MOD' \<equiv> FSwImorph h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
less_not_sym: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "\<not> y < x"
lSup_strict_prefixes: fixes xs :: "'c llist" assumes "\<not> lfinite xs" shows "lSup {ys. ys \<sqsubseteq> xs \<and> ys \<noteq> xs} = xs"
last_of_prefix_in_set: fixes n :: "nat"   and E :: "'a list"   and D :: "'a list"   and F :: "'a list" assumes "n < length E"   and "D = E @ F" shows "last E \<in> set (drop n D)"
plus_vwb_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "vwb_lens x"   and "vwb_lens y"   and "x \<bowtie> y" shows "vwb_lens (x +\<^sub>L y)"
declassified_prefix_subset: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and I :: "'var \<Rightarrow> ('fun, 'a) Term.term" assumes "prefix A B" shows "declassified\<^sub>l\<^sub>s\<^sub>t A I \<subseteq> declassified\<^sub>l\<^sub>s\<^sub>t B I"
subst_cls_comp_subst: fixes C :: "'a literal multiset"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "C \<cdot> (\<tau> \<odot> \<sigma>) = C \<cdot> \<tau> \<cdot> \<sigma>"
simps: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and F :: "atom fm" shows "substInfinitesimalLinear_fm var b c F = liftmap (\<lambda>x. substInfinitesimalLinear (var + x) (liftPoly 0 x b) (liftPoly 0 x c)) F 0"
preserves_isomorphic: fixes f :: "'b"   and g :: "'b" assumes "C.isomorphic f g" shows "D.isomorphic (G f) (G g)"
comp_simp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "seq (fst g) (fst f)"   and "seq (snd g) (snd f)" shows "VxV.comp g f = (fst g \<cdot> fst f, snd g \<cdot> snd f)"
simps: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and a :: "('a, 'b, 'c) trans list" shows "IO_Automaton.valid step a = ((\<exists>trn. a = [trn] \<and> IO_Automaton.validTrans step trn) \<or> (\<exists>trn tr. a = trn # tr \<and> IO_Automaton.validTrans step trn \<and> tgtOf trn = srcOf (hd tr) \<and> IO_Automaton.valid step tr))"
I_intT_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "I_intT \<sigma> a \<equiv> infTp \<sigma> \<longrightarrow> intT \<sigma> a"
preserves_cod: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.cod (\<nu>\<tau>.map f) = H (\<tau>.A.cod f)"
6: shows "t \<le> v * v\<^sup>T \<sqinter> - - g"
lexext_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "lexext gt ys xs" shows "lexext gt (map f ys) (map f xs)"
arr_MkArr: fixes A :: "'obj"   and B :: "'obj"   and f :: "'obj" assumes "A \<in> Obj"   and "B \<in> Obj"   and "f \<in> (if A = B then {A} else {})" shows "arr (MkArr A B f)"
order_r: fixes one :: "'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.impl_lr_ord_algebra one imp_l imp_r less_eq less" shows "less_eq a b = (imp_r a b = one)"
arcs_blocks_iff: fixes x :: "'a"   and y :: "'a" shows "({x, y} \<in># arcs_blocks) = (x \<rightarrow> y \<and> y \<rightarrow> x)"
update_apply: fixes destr :: "'c \<Rightarrow> 'd"   and constr :: "'e \<Rightarrow> 'c"   and n :: "'f"   and f :: "'d \<Rightarrow> 'e"   and s :: "'f \<Rightarrow> 'c"   and x :: "'f" shows "update destr constr n f s x = (if x = n then constr (f (destr (s n))) else s x)"
set_intros_3: fixes ya :: "'d"   and x32 :: "'d regex"   and x31 :: "'d regex" assumes "ya \<in> atms x32" shows "ya \<in> atms (Plus x31 x32)"
cap_def: fixes e :: "'v option \<times> 'v option" shows "cap e = (case e of (None, None) \<Rightarrow> 0 | (None, \<langle>y\<rangle>) \<Rightarrow> if y \<in> \<^bold>V then u y else 0 | (\<langle>x\<rangle>, None) \<Rightarrow> 0 | (\<langle>x\<rangle>, \<langle>y\<rangle>) \<Rightarrow> if edge \<Gamma> x y \<and> x \<noteq> a then f (x, y) else if edge \<Gamma> y x then max (weight \<Gamma> x) (weight \<Gamma> y) + 1 else 0)"
F1set2_bd: fixes x :: "('e, 'd, 'f) F1" shows "|F1set2 x| \<le>o IFbd"
aug_ipp_closedTr2: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "x \<in> add_set f (aug_pm_set z i A) n" shows "\<^sub>i- x \<in> add_set f (aug_pm_set z i A) n"
apartment_simplex_in_max: fixes X :: "'a set set"   and y :: "'a set" assumes "X \<in> \<A>"   and "y \<in> X" shows "\<exists>x. SimplicialComplex.maxsimp X x \<and> order.greater_eq x y"
case_5: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x51 :: "nat"   and x52 :: "'e"   and x53 :: "'e" shows "(case RecvMarker x51 x52 x53 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
rbt_getmin_ismin: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "rbt t"   and "t \<noteq> \<langle>\<rangle>" shows "is_min2 (pst_getmin t) (Tree2.set_tree t)"
measure_orthogonal_image: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set" assumes "orthogonal_transformation f"   and "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue (f ` S) = Sigma_Algebra.measure lebesgue S"
supremum_def: fixes A :: "'a set" shows "supremum A = {a \<in> upper_bound A. \<forall>x\<in>upper_bound A. a \<le> x}"
funpow_shift1_isnpoly: fixes p :: "poly"   and n :: "nat" assumes "isnpoly p"   and "p \<noteq> 0\<^sub>p" shows "isnpoly ((shift1 ^^ n) p)"
Qeq: fixes h :: "'c \<Rightarrow> 'b"   and k :: "'c \<Rightarrow> 'b" assumes "\<And>x. x \<in> U \<Longrightarrow> h x = k x" shows "Q h = Q k"
aff_dim_lt_full: fixes S :: "'a set" shows "(aff_dim S < int DIM('a)) = (affine hull S \<noteq> UNIV)"
simps: fixes x :: "int"   and n :: "int" shows "root_newton_int_main x n = (if x < 0 \<or> n < 0 then (0, False) else if x ^ p \<le> n then (x, x ^ p = n) else root_newton_int_main ((n div x ^ pm + x * int pm) div int p) n)"
assoc'_naturality: fixes f0 :: "'c local.term set"   and f1 :: "'c local.term set"   and f2 :: "'c local.term set" assumes "arr f0"   and "arr f1"   and "arr f2" shows "tensor (tensor f0 f1) f2 \<cdot> assoc' (local.dom f0) (local.dom f1) (local.dom f2) = assoc' (cod f0) (cod f1) (cod f2) \<cdot> tensor f0 (tensor f1 f2)"
simps_1: fixes p :: "fm"   and q :: "fm" shows "iszlfm (And p q) = (\<lambda>bs. iszlfm p bs \<and> iszlfm q bs)"
independent_not_zero_rows_rref: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "vec.independent {row i A |i. row i A \<noteq> 0}"
smult_0_right: fixes a :: "'a" shows "smult a 0 = 0"
bn_tau_fresh: fixes P :: "'b" shows "bn \<tau> \<sharp>* P"
div_2_gt_zero: fixes n :: "nat" assumes "Suc 0 < n" shows "0 < n div 2"
SCF_R2: fixes xs :: "'a option list"   and P :: "'a option process"   and X :: "'a option set"   and Y :: "'a option set"   and Q :: "'a option process" assumes "xs \<in> sentences P"   and "(xs, X) \<in> failures P"   and "([], Y) \<in> failures Q" shows "(xs, insert None X \<inter> Y) \<in> seq_comp_failures P Q"
set_plus_elim: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<in> A + B" obtains a :: "'b" and b :: "'b" where "x = a + b" and "a \<in> A" and "b \<in> B"
init_valuation_def: fixes vd :: "String.literal list"   and x :: "String.literal" shows "init_valuation vd x \<equiv> if x \<in> set vd then Some 0 else None"
smap_less_refl: fixes m :: "'a \<Rightarrow> 'b set" shows "m \<subseteq>. m"
Rep_tensor: fixes x :: "'b tensor" shows "Rep_tensor x \<in> {t. length (snd t) = prod_list (fst t)}"
extended_Gromov_distance_PInf_boundary_2: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" assumes "x \<in> Gromov_boundary" shows "extended_Gromov_distance y x = \<infinity>"
Rep_process_cases: fixes y :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "y \<in> {p. is_process p}" obtains x :: "'a process" where "y = Rep_process x"
diff_eq_diff_less_eq: 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 a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less_eq a b = less_eq c d"
emeasure_sigma: fixes \<Omega> :: "'c set"   and A :: "'c set set" shows "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"
star_rtc_least: fixes x :: "'a list set"   and y :: "'a list set" assumes "p_one \<union> x \<union> p_prod y y \<subseteq> y" shows "p_star x \<subseteq> y"
assms_2: shows "ptr \<noteq> new_shadow_root_ptr"
cmp_simps_3: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "trg\<^sub>D (\<Phi>\<^sub>F (f, g)) = trg\<^sub>D (F f)"
mat1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_match ck env s2 v1 pes (Conv (Some (''Bind'', TypeExn (Short ''Bind''))) []) bv" shows "evaluate ck env s1 (Mat e pes) bv"
finite_enumerate_step: fixes S :: "'b set"   and n :: "nat" assumes "finite S"   and "Suc n < card S" shows "wellorder_class.enumerate S n < wellorder_class.enumerate S (Suc n)"
lead_coeff_not_zero: fixes K :: "'c set"   and a :: "'c"   and p :: "'c list" assumes "polynomial\<^bsub>A\<^esub> K (a # p)" shows "a \<in> K - {\<zero>\<^bsub>A\<^esub>}"
IH: shows "higher_differentiable_on UNIV f k \<and> (\<forall>t\<le>0. nth_derivative k f t 1 = 0) \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
prim_exp_one: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "primitive u"   and "r \<^sup>@ k = u" shows "k = 1"
term_ok'_code: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and p :: "Core.term" assumes "exe_osig_conds (exesorts (ExeSignature cto tao sa))" shows "(term_ok' (translate_signature (ExeSignature cto tao sa)) p \<and> typ_of p = Some propT) = (exeterm_ok' (ExeSignature cto tao sa) p \<and> typ_of p = Some propT)"
iteration_1l: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<infinity> \<le> x\<^sup>\<infinity>"
assms_1: shows "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"
Rep_covector_inject: fixes x :: "'b covector"   and y :: "'b covector" shows "(Rep_covector x = Rep_covector y) = (x = y)"
compute_flow_val_impl_hnr: shows "(uncurry (compute_flow_val_impl s N), uncurry (PR_CONST compute_flow_val)) \<in> local.am_assn\<^sup>k *\<^sub>a local.cf_assn\<^sup>k \<rightarrow>\<^sub>a cap_assn"
simps_5: fixes b :: "bexp"   and c :: "com" shows "ccomp (WHILE b DO c) = (let cc = ccomp c; cb = bcomp b False (size cc + 1) in cb @ cc @ [JMP (- (size cb + size cc + 1))])"
natural_isomorphism_\<rr>: shows "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<rr>"
mlex_fst_decrI: fixes a :: "nat"   and a' :: "nat"   and b :: "nat"   and N :: "nat"   and b' :: "nat" assumes "a < a'"   and "b < N"   and "b' < N" shows "a * N + b < a' * N + b'"
sat_scale_rat_real: fixes v :: "nat \<Rightarrow> real"   and c :: "constraint"   and r :: "rat" assumes "v \<Turnstile>\<^sub>c c" shows "v \<Turnstile>\<^sub>c (r *R c)"
some_betw4b: fixes P :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "P \<in> \<P>"   and "a \<in> P"   and "b \<in> P"   and "c \<in> P"   and "d \<in> P"   and "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d"   and "\<not> ([[b a c d]] \<or> [[b a d c]] \<or> [[b c a d]] \<or> [[b d a c]] \<or> [[c a b d]] \<or> [[c b a d]])" shows "[[a b c d]] \<or> [[a b d c]] \<or> [[a c b d]] \<or> [[a c d b]] \<or> [[a d b c]] \<or> [[a d c b]]"
S4: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
isDERIV_approx': fixes xs :: "real list"   and vs :: "float interval option list"   and x :: "nat"   and ivl :: "float interval"   and X :: "real"   and prec :: "nat"   and f :: "floatarith" assumes "bounded_by xs vs"   and "vs ! x = Some ivl"   and "X \<in>\<^sub>r ivl"   and "isDERIV_approx prec x f vs" shows "isDERIV x f (xs[x := X])"
a: fixes h' :: "'a"   and h'' :: "'a" assumes "h \<oplus> h' = Some h''"   and "P h'" shows "Q h''"
assms_2: shows "(\<lambda>x. \<bar>f x\<bar>) integrable_on A"
nth_dec_above: fixes xs :: "'b list"   and i :: "nat"   and ys :: "'b list"   and j :: "nat"   and k :: "nat"   and zs :: "'b list" assumes "length xs = i"   and "length ys = j"   and "k \<notin> {i..<i + j}" shows "nth_default (0::'b) (xs @ zs) (dec_above i j k) = nth_default (0::'b) (xs @ ys @ zs) k"
edge_rev_simps: fixes u :: "'b"   and v :: "'b" assumes "(u, v) \<in> parcs (list_digraph G_list)" shows "edge_rev (to_map' (snd G_list) xss) (u, v) = (v, u)"
alphaAbs_sym: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and A' :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "A $= A'" shows "A' $= A"
weak_group_morphism_axioms_def: fixes f :: "'c \<Rightarrow> 'd"   and H :: "'c set"   and G :: "('c, 'e) monoid_scheme" shows "weak_group_morphism_axioms f H G \<equiv> \<forall>a b. a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> (f a = f b) = (a \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> b \<in> H)"
is_basis_def: fixes S :: "('a, 'c) vec set" shows "is_basis S \<equiv> vec.independent S \<and> vec.span S = UNIV"
rreq_rrep_fresh_1: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "rreq_rrep_fresh crt (Rreq hops dip dsn dsk oip osn sip) = (sip \<noteq> oip \<longrightarrow> oip \<in> kD crt \<and> (osn < sqn crt oip \<or> sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val))"
Union_down: fixes L :: "'b \<Rightarrow> ('c \<times> 'c) set" shows "(\<Union>i. \<Union> (L ` under q i)) = \<Union> (range L)"
frees_const: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "frees (const name) = {||}"
wilsons_construction_proper: fixes I :: "'c set"   and w :: "nat"   and \<K>' :: "int set"   and f :: "'a set \<Rightarrow> ('a \<times> 'c) set multiset" assumes "card I = w"   and "0 < w"   and "\<And>n. n \<in> \<K>' \<Longrightarrow> 2 \<le> n"   and "\<And>B. B \<in># \<B> \<Longrightarrow> K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}" shows "proper_design (\<V> \<times> I) (\<Sum>\<^sub># (image_mset f \<B>))"
initial_error: shows "dist (x t 0) x0 \<le> B * (exp 1 - 1) * stepsize 0 ^ p / L"
tllist_all2_tlengthD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys" shows "tlength xs = tlength ys"
Weierstrass_m_test': fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> real" assumes "\<And>n x. x \<in> A \<Longrightarrow> norm (f n x) \<le> M n"   and "summable M" shows "uniformly_convergent_on A (\<lambda>n x. \<Sum>i<n. f i x)"
assms_1: shows "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
naturality: fixes f :: "'a" assumes "A.arr f" shows "B (\<sigma>' (A.cod f)) (G f) = B (F f) (\<sigma>' (A.dom f))"
d\<^sub>3_def: shows "Semantics.d\<^sub>3 \<equiv> map_fun eval\<Pi>\<^sub>3 id Some"
isom_gch_cmp: fixes n :: "nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and h :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and f1 :: "nat \<Rightarrow> nat"   and f2 :: "nat \<Rightarrow> nat" assumes "Gchain n g"   and "Gchain n h"   and "f1 \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"   and "f2 \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"   and "isom_Gchains n (cmp f2 f1) g h" shows "isom_Gchains n f1 g (cmp h f2)"
continuous_at_right_real_increasing: fixes f :: "real \<Rightarrow> real"   and a :: "real" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "continuous (at_right a) f = (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
VA': shows "Vagree \<nu> \<omega> (\<Union>x\<in>{x. Inl (Inr x) \<in> SIGP \<alpha>}. FVT (\<sigma> x))"
precedesAntisymmetry: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "a \<in> set l"   and "b \<in> set l"   and "precedes a b l"   and "precedes b a l" shows "a = b"
Quotient_\<Pi>\<^sub>1: shows "Quotient (=) make\<Pi>\<^sub>1 eval\<Pi>\<^sub>1 cr_\<Pi>\<^sub>1"
chine_hcomp_props_4: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "C.commutative_square \<p>\<^sub>0[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)] (Chn \<mu>) (chine_hcomp \<nu> \<mu>) \<p>\<^sub>0[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]"