hcju/isabelleps huggingface dataset:
can_def: fixes u :: "('a, 'a set) Maps.arr arrow_of_spans_data E.term"   and t :: "('a, 'a set) Maps.arr arrow_of_spans_data E.term" shows "Span.can u t \<equiv> Coherence.evaluation_map.eval (\<bullet>) (\<circ>) Span.assoc Span.unit Span.src Span.trg (\<lambda>\<mu>. if Span.arr \<mu> then \<mu> else Span.null) (bicategorical_language.Inv (\<bullet>) (bicategorical_language.red (\<bullet>) Span.src Span.trg u) \<^bold>\<cdot> bicategorical_language.red (\<bullet>) Span.src Span.trg t)"
STCal_divergent_1: fixes S :: "'procS" shows "SourceTerm S \<longmapsto>STCal Source Target\<omega> = S \<longmapsto>Source\<omega>"
h'_definite: fixes h' :: "'b \<Rightarrow> real"   and x' :: "'b"   and H :: "'b set"   and h :: "'b \<Rightarrow> real"   and xi :: "real"   and x :: "'b"   and y :: "'b"   and a :: "real"   and E :: "'b set" assumes "\<And>x. h' x = (let (y, a) = SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi)"   and "x = y + a \<cdot> x'"   and "vectorspace E"   and "Subspace.subspace H E"   and "y \<in> H"   and "x' \<notin> H"   and "x' \<in> E"   and "x' \<noteq> (0::'b)" shows "h' x = h y + a * xi"
intensional_simps_15: fixes P :: "'b \<Rightarrow> bool" shows "P \<longrightarrow> \<not> P \<equiv> \<not> P"
is_TComp_var_instance_closedD: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and x :: "('fun, 'atom) Term.term \<times> nat"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"   and "\<Gamma> (Var x) = Fun f T"   and "is_TComp_var_instance_closed \<Gamma> M" shows "\<exists>g U. Fun g U \<in> set M \<and> \<Gamma> (Fun g U) = \<Gamma> (Var x) \<and> (\<forall>u\<in>set U. is_Var u) \<and> distinct U"
simps: fixes fv1 :: "'a \<Rightarrow> atom set"   and fv2 :: "'b \<Rightarrow> atom set"   and x :: "'a"   and y :: "'b" shows "prod_fv fv1 fv2 (x, y) = fv1 x \<union> fv2 y"
that: fixes d :: "real" assumes "0 < d"   and "\<And>K. \<lbrakk>K \<subseteq> standard_simplex p; \<And>x y i. \<lbrakk>i \<le> p; x \<in> K; y \<in> K\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> d\<rbrakk> \<Longrightarrow> \<exists>U. U \<in> \<C> \<and> K \<subseteq> U" shows "thesis"
Can_implies_coherent_iff_coherent_Inv: fixes t :: "'c local.term" assumes "Can t" shows "coherent t = coherent (Inv t)"
g0_le_f: fixes e :: "'a \<times> 'a" shows "g0 e \<le> f e"
axioms_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "fmpt M T" shows "finite_measure M"
assms_4: shows "l' \<in> labels \<Gamma> q"
snd: shows "\<forall>i. snd (f i) \<in> A2"
simps_7: shows "- \<infinity> * \<infinity> = - \<infinity>"
field_def2: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "field P C F = (D, b, T)"
h2: assumes "\<exists>q\<in>{x. a * x\<^sup>2 + b * x + c = 0}. r < q" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0"
comm_monoid_lift_option: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "comm_monoid f z" shows "comm_monoid (lift_option f) (Some z)"
neqE: fixes x :: "char"   and y :: "char" assumes "x \<noteq> y" obtains "less_char x y"   | "less_char y x"
psubst_Op_simp: fixes inp :: "'b \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and binp :: "'c \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and delta :: "'d" assumes "goodInp inp"   and "goodBinp binp"   and "goodEnv rho" shows "Op delta inp binp #[rho] = Op delta inp %[rho] binp %%[rho]"
simps_79: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c"   and x5 :: "char list" shows "(case Call x5 of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = f5 x5"
exe_trap_st_pc_result: fixes s :: "'b sparc_state" shows "snd (exe_trap_st_pc () s) = False"
NSconvergent_NSBseq: fixes X :: "nat \<Rightarrow> 'a" assumes "NSconvergent X" shows "NSBseq X"
factorial_monoid: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ufd times one plus zero minus uminus" shows "factorial_monoid \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>"
opp_def: fixes p :: "'a point" shows "opp p = (case p of Infinity \<Rightarrow> Infinity | Point x y \<Rightarrow> Point x (\<ominus> y))"
lift_preserves_eq: fixes i :: "nat"   and F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and v :: "(nat \<Rightarrow> 'c) \<times> 'd \<Rightarrow> 'e" shows "(lift i F \<in> preserves v) = (F \<in> preserves (v \<circ> lift_map i))"
Derive_empty: fixes a :: "symbol list" shows "Derive a [] = a"
reduces_ivle_def: fixes X :: "('c \<times> 'd) set" shows "reduces_ivle X = FOREACH\<^bsup>\<lambda>B. (\<subseteq>) X\<^esup> (set Basis_list ::: \<langle>lv_rel\<rangle>list_set_rel) (\<lambda>b X. reduce_ivle X b) X"
factor_sum_cubes: fixes x :: "int"   and y :: "int" shows "x ^ 3 + y ^ 3 = (x + y) * (x\<^sup>2 - x * y + y\<^sup>2)"
nth_nth_transpose_sorted: fixes xs :: "'g list list"   and i :: "nat"   and j :: "nat" assumes "sorted (rev (map length xs))"   and "i < length (transpose xs)"   and "j < length (filter (\<lambda>ys. i < length ys) xs)" shows "transpose xs ! i ! j = xs ! j ! i"
val_Zp0_imp_unit0: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "val_Zp x = 0"   and "x \<in> carrier Zp" shows "x (Suc n) \<in> Units (residue_ring (p ^ Suc n))"
completion_eq: fixes r :: "('a, 'b, 'c) expr"   and r' :: "('a, 'b, 'c) expr"   and \<E> :: "('a, 'b, 'c) cntxt"   and \<E>' :: "('a, 'b, 'c) cntxt" assumes "redex r"   and "redex r'" shows "(\<E> [r] = \<E>' [r']) = (\<E> = \<E>' \<and> r = r')"
M: fixes m :: "nat"   and n :: "nat" assumes "M \<le> m" shows "norm (\<Sum>k = m..n. inverse ((real k)\<^sup>2)) < e'"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_compat_snoc_axioms ext" shows "ext_compat_snoc ext"
measure_countably_zero: fixes c :: "nat \<Rightarrow> real set" assumes "range c \<subseteq> events"   and "\<And>i. prob (c i) = 0" shows "prob (\<Union> (range c)) = 0"
assms: shows "\<forall>i. (\<lambda>n. \<c>\<o>\<l> i (X n)) \<longlonglongrightarrow> L $ i"
subo: shows "Subobjs\<^sub>R P (hd (Cs @ C' # Cs')) (Cs @ C' # Cs')"
arrE: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" obtains "f \<noteq> Null" and "concrete_category.Dom f \<in> Obj" and "Cod f \<in> Obj" and "Map f \<in> Hom (concrete_category.Dom f) (Cod f)"
single_diff: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "monomial (a - b) k = monomial a k - monomial b k"
intro_part: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'b constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>i \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow> \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
1: shows "f = ZFfun (|dom|f) (|cod|f) ((|@|) f)"
rep_eq: fixes x :: "int poly" shows "Rep_real_alg_2_list (roots_of_2_list x) = roots_of_2 x"
derivation_shift_append: fixes A :: "(nat \<times> symbol \<times> symbol list) list"   and B :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "derivation_shift (A @ B) left right = derivation_shift A left right @ derivation_shift B left right"
simp_implies_def: shows "(=simp=>) \<equiv> (\<Longrightarrow>)"
fst_par_onode_comp: fixes i :: "nat"   and s :: "((nat \<Rightarrow> 'a) \<times> 'b, 'd seq_action) automaton"   and I :: "nat"   and t :: "('c, 'd seq_action) automaton"   and R :: "nat set" shows "automaton.trans (\<langle>i : s \<langle>\<langle>\<^bsub>I\<^esub> t : R\<rangle>\<^sub>o) = onode_sos (oparp_sos I (automaton.trans s) (automaton.trans t))"
boffa_aux3: fixes x :: "('b, 'c) boffa_mon" assumes "conway_assm1 x" shows "\<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV} + \<Sum> {x\<^bsub>i\<^esub> \<cdot> x\<^bsub>j\<^esub> |i j. i \<in> UNIV \<and> j \<in> UNIV} = \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV}"
p_leI: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and p' :: "real" assumes "akra_bazzi_params_nonzero k as bs"   and "(\<Sum>i<k. as ! i * bs ! i powr p') \<le> 1" shows "akra_bazzi_exponent as bs \<le> p'"
oInv_less: fixes a :: "ordinal"   and z :: "ordinal" assumes "F 0 \<le> a"   and "a < F z" shows "oInv F a < z"
dom_inv: fixes f :: "'c" assumes "C.iso f" shows "C.dom (C.inv f) = C.cod f"
h: shows "h \<in> borel_measurable (lebesgue_on S)"
order_root_unity: fixes n :: "nat"   and x :: "complex" assumes "n \<noteq> 0" shows "order x (root_unity n) = (if x ^ n = 1 then 1 else 0)"
after_summary_zmset_frontier: fixes A :: "'t zmultiset"   and S :: "'sum antichain" shows "after_summary (zmset_of (mset_set (set_antichain (frontier A)))) S \<subseteq>#\<^sub>z after_summary (zmset_of (mset_pos A)) S"
upE: fixes p :: "'a\<^sub>\<bottom>" obtains "p = \<bottom>"   | x :: "'a" where "p = up\<cdot>x"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>" shows "\<CC>\<lparr>Dom\<rparr> = []\<^sub>\<circ>"
IH_2: fixes inputs :: "real vec list"   and j :: "nat"   and a :: "nat" assumes "map dim_vec inputs = input_sizes m2"   and "valid_net m2"   and "j < output_size m2" shows "polyfun {..<a + count_weights s m2} (\<lambda>f. evaluate_net (insert_weights s m2 (\<lambda>i. f (i + a))) inputs $ j)"
le_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
Sigma_mset_cong: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a \<Rightarrow> 'b multiset"   and D :: "'a \<Rightarrow> 'b multiset" assumes "A = B"   and "\<And>x. x \<in># B \<Longrightarrow> C x = D x" shows "Sigma_mset A C = Sigma_mset B D"
in_second_summand: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CEM"   and "P z (x \<oplus> y) \<and> \<not> O z x" shows "P z y"
tensor_prod_of_id_2_z: shows "Quantum.Id 2 \<Otimes> Z = M5"
False: shows "\<not> 1 << LENGTH('a) - 1 \<le> y"
set_cases_2: fixes e :: "'d"   and a :: "('e, 'd, 'f) generat" assumes "e \<in> generat_outs a" obtains z2 :: "'f" where "a = IO e z2"
le_l: fixes a :: "MIR.num" shows "isrlfm (rsplit le a)"
\<Phi>_def: shows "\<Phi> \<equiv> \<lambda>m M. infinite M \<and> m < \<Sqinter> M"
convex_on_ereal_univ: fixes f :: "'b \<Rightarrow> ereal" shows "Lower_Semicontinuous.convex_on UNIV f = (\<forall>S. Lower_Semicontinuous.convex_on S f)"
sqrt_iteration_bound: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 < real_of_float x" shows "sqrt (real_of_float x) < real_of_float (sqrt_iteration prec n x)"
v11_app_if_vconverse_app: fixes y :: "V"   and x :: "V" assumes "y \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> = x" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = y"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and t :: "('b \<times> real) tree"   and A :: "'b set" shows "rinss (x # xs) t A = insR x t A \<bind> (\<lambda>t. rinss xs t A)"
P2: shows "\<forall>x. x \<in> s \<longrightarrow> P x"
Interval_inverse: fixes y :: "'a \<times> 'a" assumes "y \<in> {(a, b). a \<le> b}" shows "bounds_of_interval (Interval y) = y"
jordan_matrix_dim_2: fixes n_as :: "(nat \<times> 'a) list" shows "dim_col (jordan_matrix n_as) = sum_list (map fst n_as)"
ramsey1: fixes q0 :: "nat"   and q1 :: "nat" shows "\<exists>N. partn_lst {..<N} [q0, q1] 1"
foldli_image: fixes X :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" shows "det_fold_set X (\<lambda>_. True) (\<lambda>x. insert (f x)) {} ((`) f)"
simps_1: fixes uu :: "variable"   and uv :: "typ" shows "is_variable (Fv uu uv) = True"
homotopic_loops_imp_homotopic_paths_null: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and a :: "'a" assumes "homotopic_loops s p (linepath a a)" shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
finlsts_def: shows "finlsts \<equiv> \<lambda>A. {x. finlstsp (\<lambda>x. x \<in> A) x}"
supr_above_Above: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "r.Above B \<noteq> {}" shows "r.Above B = r.above (r.supr B)"
bind_eq_Some_conv: fixes f :: "'c option"   and g :: "'c \<Rightarrow> 'b option"   and x :: "'b" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
semiconfluentp_imp_confluentp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semiconfluentp r" shows "confluentp r"
gval_foldr_equiv_gval_fold: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (foldr gAnd G (Bc True)) s = gval (fold gAnd G (Bc True)) s"
Rats_of_nat: fixes n :: "nat" shows "of_nat n \<in> \<rat>"
prv_imp_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> (imp \<chi> \<psi>)) (imp (imp \<phi> \<chi>) (imp \<phi> \<psi>)))"
mk_linear_orders_decisive_on_refl_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C A"   and "decisive_on A f" shows "refl_on A (linord_of_list r)"
inj_imp_surj_inv: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "surj (inv f)"
le_less_trans: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term"   and z :: "('d, 'e) Term.term" assumes "y \<unrhd> x"   and "z \<rhd> y" shows "z \<rhd> x"
poly_eval_monomial: fixes a :: "'c \<Rightarrow> 'b"   and t :: "'c \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "poly_eval a (monomial c t) = c * (\<Prod>x\<in>keys t. a x ^ lookup t x)"
simps: fixes a1 :: "'b list"   and a2 :: "'b list" shows "prefixList a1 a2 = ((\<exists>x xs. a1 = [] \<and> a2 = x # xs) \<or> (\<exists>xa xb x. a1 = x # xa \<and> a2 = x # xb \<and> prefixList xa xb))"
simps: fixes T :: "ty"   and n :: "nat" shows "alen_of_htype (Array_type T n) = n"
scalar_mult_periodic_arithmetic: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat"   and a :: "complex" assumes "periodic_arithmetic f k" shows "periodic_arithmetic (\<lambda>n. a * f n) k"
wf_syscls_class_Object: fixes P :: "'b prog" assumes "wf_syscls P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
1: shows "\<guillemotleft>\<lparr>Chn = fgh.chine_assoc, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> : (f \<star> g) \<star> h \<Rightarrow> f \<star> g \<star> h\<guillemotright>"
t_ucast: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "CT \<turnstile> D <: C" shows "CT;\<Gamma> \<turnstile> Cast C e0 : C"
wf_less_nmultiset: shows "wf {(X, Y). X < Y}"
par_strict_not_col_4: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "\<not> Col A B D"
Ana_subterm': fixes s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "s \<in> set (snd (Ana t))" shows "s \<sqsubseteq> t"
sinvar_mono_imp_negative_mono: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and N :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and nP :: "'b \<Rightarrow> 'c" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<not> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP" shows "\<not> sinvar \<lparr>nodes = N, edges = E\<rparr> nP"
simps_37: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> x41 ;; x42"
PRED_dist: fixes v :: "nat"   and u :: "nat" assumes "v \<noteq> src"   and "PRED v = Some u" shows "ndist v = Suc (ndist u)"
Fields_i_i_oI: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "((char list \<times> char list) \<times> ty) list" assumes "x \<turnstile> xa has_fields xb" shows "pred.eval (Fields_i_i_o x xa) xb"
ideD_2: fixes a :: "'a arrow_of_spans_data" assumes "V.ide a" shows "V.dom a = a"
fix_least: fixes F :: "'d \<rightarrow> 'd"   and x :: "'d" assumes "F\<cdot>x = x" shows "fix\<cdot>F \<sqsubseteq> x"
lookup_some_upper: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and y :: "'h"   and ys :: "'h list"   and e :: "('f, 'g, 'h) env" assumes "lookup env (xs @ y # ys) = Some e" shows "\<exists>b' es' env'. lookup env xs = Some (Env b' es') \<and> es' y = Some env' \<and> lookup env' ys = Some e"
graph_restrict_simps_3: fixes G :: "('a, 'b) graph_rec_scheme"   and R :: "'a set" shows "g_V0 (graph_restrict G R) = g_V0 G - R"
process_rule_2_traces: fixes xs :: "'a list"   and xs' :: "'a list"   and P :: "'a process" assumes "xs @ xs' \<in> traces P" shows "xs \<in> traces P"
inf_sup_aci_6: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "sup (sup x y) z = sup x (sup y z)"
fls_lr_inverse_X_1: fixes x :: "'a" shows "fls_left_inverse fls_X x = fls_shift 1 (fls_const x)"
inf_apply: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(f \<sqinter> g) x = f x \<sqinter> g x"
IH: assumes "list_constr xa (length qs)" shows "(prod_list (map ((!) (map_sgas sign)) xa) = 1) = (\<forall>p\<in>set (retrieve_polys qs xa). poly p x = 0)"
s_top_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and K :: "'c set" shows "H \<diamondop>\<^bsub>G\<^esub> K = {z. \<exists>x\<in>H. \<exists>y\<in>K. x \<cdot>\<^bsub>G\<^esub> y = z}"
simps: fixes a :: "'d set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
has_vector_derivative_diff: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and net :: "real filter"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') net"   and "(g has_vector_derivative g') net" shows "((\<lambda>x. f x - g x) has_vector_derivative f' - g') net"
simps_11: fixes f1 :: "label \<Rightarrow> 'a"   and f2 :: "label \<Rightarrow> 'a"   and f3 :: "'a" shows "(case Ret of Break x \<Rightarrow> f1 x | Cont x \<Rightarrow> f2 x | Ret \<Rightarrow> f3) = f3"
aval'_correct: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and s :: "char list \<Rightarrow> int"   and S :: "'a st"   and a :: "aexp" assumes "Gamma_semilattice num' plus' \<gamma>"   and "s \<in> \<gamma>_st \<gamma> S" shows "aval a s \<in> \<gamma> (Gamma_semilattice.aval' num' plus' a S)"
swing_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "swing w1 w2" shows "kauff_mat w1 = kauff_mat w2"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at (A.dom a) D) (at (A.cod a') D) (at (a' \<cdot>\<^sub>A a) D)"
weakly_preferred_by_men_refl: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw"   and "StableMarriage.match Pm Pw \<mu>" shows "StableMarriage.weakly_preferred_by_men Pm \<mu> \<mu>"
words_infI_construct_chain: fixes w :: "nat \<Rightarrow> 'a list"   and execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" assumes "chain w"   and "\<And>k. transition_system.path execute enabled (w k) p" shows "transition_system.run execute enabled (Word_Prefixes.limit w) p"
simps_1: fixes x2 :: "bool"   and y2 :: "bool" shows "(Bool x2 = Bool y2) = (x2 = y2)"
zero_less_one: fixes one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
abs_zhmultiset_def: fixes M :: "zhmultiset" shows "\<bar>M\<bar> = (if M < 0 then - M else M)"
qpWWP: shows "is_quantum_predicate (wlp S (wlp (While M S) P))"
\<tau>_mono: fixes i :: "nat"   and j :: "nat"   and s :: "'b Trace.trace" assumes "i \<le> j" shows "\<tau> s i \<le> \<tau> s j"
basic_monos_7: fixes A :: "'y set"   and B :: "'y set"   and x :: "'y" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
nonzero_mult_div_cancel_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide (times a b) a = b"
Sup_eqI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" shows "\<Squnion> A = x"
assms_2: fixes P :: "'d"   and Q :: "'d" assumes "Prop P Q" shows "Prop Q P"
less_imp_not_less: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.preorder less_eq less"   and "less x y" shows "(\<not> less y x) = True"
rep_eq: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b fset" shows "fpairwise x xa = Compiler_Utils.pairwise x (fset xa)"
arg_apps_Cons: fixes s :: "('c, 'd) tm"   and sa :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list" shows "arg (apps (App s sa) ss) = last (sa # ss)"
le_iff_inf: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<subseteq># y) = (x \<inter># y = x)"
prodrg_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" shows "r\<Pi>\<^bsub>I\<^esub> A = \<lparr>carrier = carr_prodag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A, tp = prod_tOp I A, un = prod_one I A\<rparr>"
Fvars_\<phi>R: shows "Fvars \<phi>R = {}"
bb: fixes i' :: "nat"   and j' :: "nat"   and k' :: "nat"   and k :: "nat"   and j :: "nat"   and i :: "nat" assumes "i' < j'"   and "j' \<le> k'"   and "k' < k" shows "b k' (j', i') \<lless> b k (j, i)"
assms_2: shows "matchs xs' ys' = Some env\<^sub>2"
sel_1: fixes x1 :: "real interval"   and x2 :: "real interval" shows "Re_interval (Complex_Interval x1 x2) = x1"