hcju/isabelleps huggingface dataset:
mem_cartesian: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and m :: "'e"   and n :: "'g" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N" shows "(m, n) \<in> M \<times>\<^sub>c N"
tranclp_into_tranclp2: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
intro: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" assumes "StdSet ops1"   and "StdSet ops2" shows "g_set_xy_defs_loc ops1 ops2"
reachable_awalkI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
setleD: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "S *<= x"   and "y \<in> S" shows "y \<le> x"
nonneg_mat_def: fixes A :: "'b mat" shows "nonneg_mat A \<equiv> \<forall>a\<in>elements_mat A. (0::'b) \<le> a"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_subst': fixes t :: "('d, 'e) Term.term"   and S :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" assumes "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>s\<in>trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t S. \<exists>X. set X \<subseteq> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<and> t = s \<cdot> rm_vars (set X) \<delta>"
sup_bot_left: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
arrI: fixes f1 :: "'d"   and f2 :: "'c" assumes "Dop.arr f1"   and "C.arr f2" shows "DopxC.arr (f1, f2)"
Tarski_proof_theoretic: fixes T :: "'fmla" assumes "T \<in> fmla"   and "Fvars T = {xx}"   and "\<And>\<phi>. \<lbrakk>\<phi> \<in> fmla; Fvars \<phi> = {}\<rbrakk> \<Longrightarrow> prv (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)" shows "\<not> consistent"
contextually_equivalent_def: fixes e1 :: "db"   and e2 :: "db" shows "e1 \<approx> e2 \<equiv> e1 \<unlhd> e2 \<and> e2 \<unlhd> e1"
mismatchPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<simeq> Q" shows "[a\<noteq>b]P \<simeq> [a\<noteq>b]Q"
freshness_lemma: fixes h :: "'a \<Rightarrow> 'b" assumes "\<exists>a. atom a \<sharp> (h, h a)" shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
simplex_imp_polytope: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "polytope S"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "dim_col (Matrix.mat nr nc f) = nc"
finite_degenerate_lotteries: shows "finite local.degenerate_lotteries"
cil_\<alpha>nl_cons_helper: fixes fg :: "('d, 'b, 'e, 'a, 'f) flowgraph_rec_scheme"   and wb :: "('b, 'c) label list el_step list" shows "mon_pl (map (\<alpha>nl fg) wb) = mon_ww fg (map le_rem_s wb)"
"case": fixes f :: "'b extended \<Rightarrow> 'b \<Rightarrow> 'b extended \<Rightarrow> 'c \<Rightarrow> 'a"   and x1 :: "'b extended"   and x2 :: "'b"   and x3 :: "'b extended"   and x4 :: "'c" shows "(case InsertMessage x1 x2 x3 x4 of InsertMessage x xa xb xc \<Rightarrow> f x xa xb xc) = f x1 x2 x3 x4"
new_tv_def: fixes n :: "nat"   and ts :: "'b" shows "new_tv n ts = (\<forall>m. m \<in> free_tv ts \<longrightarrow> m < n)"
iT_Mult_Int: fixes k :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "0 < k" shows "A \<inter> B \<otimes> k = (A \<otimes> k) \<inter> (B \<otimes> k)"
rep_eq: fixes x :: "'c"   and xa :: "('a, 'b, 'c) Formula.formula" shows "Formula.formula.Rep_formula (Formula.Act x xa) = Formula.Act\<^sub>\<alpha> x (Formula.formula.Rep_formula xa)"
simps_5: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({I/x} WHILE b DO C) Q = (\<lambda>l s. \<exists>n. I (l(x := n)) s)"
is_root_pass2: fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "is_root' (pass\<^sub>2' hs)"
dirichlet_prod_prime_power: fixes p :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "prime p" shows "(f \<star> g) (p ^ k) = (\<Sum>i\<le>k. f (p ^ i) * g (p ^ (k - i)))"
\<gamma>_def: shows "\<gamma> = (\<lambda>m. rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \<i>) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \<i>))"
rc2: shows "root_cond (p2, l2, r2) y"
take_append: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "take n (xs \<cdot> ys) = take n xs \<cdot> take (n - \<^bold>|xs\<^bold>|) ys"
Response2: fixes p :: "'b"   and ou :: "'d"   and i :: "nat" assumes "Resp p ou t t' \<and> s = s'"   and "id1 \<le> i \<and> i < id2"   and "a = Response i p ou" shows "thesis"
7: shows "i < length y \<longrightarrow> i mod length y = i"
c4_1: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "g_approx i x \<le> c4 * g x"
fake_parts_extr_singleton: fixes X :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and IK' :: "msg set" assumes "X \<in> fake bad IK chan" shows "parts (extr bad IK' {X}) \<subseteq> parts IK \<union> parts (extr bad IK' chan)"
already_in_span: fixes A :: "'c set"   and v :: "'c" assumes "A \<subseteq> carrier V"   and "v \<in> span A" shows "span A = span (A \<union> {v})"
Abs_pupdate_inverse: fixes y :: "'b data \<Rightarrow> 'b pdata" assumes "y \<in> pupdate" shows "Rep_pupdate (Abs_pupdate y) = y"
bigo_mult3: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and j :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "g \<in> O(j)" shows "f * g \<in> O(h * j)"
compat_S_NS_point: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "compat_pair S NS"   and "(x, y) \<in> S"   and "(y, z) \<in> NS" shows "(x, z) \<in> S"
prod_pOp_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" shows "prod_pOp I A = (\<lambda>f\<in>carr_prodag I A. \<lambda>g\<in>carr_prodag I A. \<lambda>x\<in>I. f x \<plusminus>\<^bsub>A x\<^esub> g x)"
bernoulli_even_nonzero: fixes n :: "nat" assumes "even n" shows "bernoulli n \<noteq> 0"
containsPSeq0D: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and A :: "formula" assumes "branch subs (pseq fs) f"   and "contains f (i, A) 0" shows "i = 0"
is_unitE: 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 c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains b :: "'b" where "a \<noteq> zero" and "b \<noteq> zero" and "dvd.dvd times b one" and "divide one a = b" and "divide one b = a" and "times a b = one" and "divide c a = times c b"
compos_bijec: fixes E :: "('c, 'd) aGroup_scheme"   and F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g" assumes "aGroup E"   and "aGroup F"   and "aGroup G"   and "bijec\<^bsub>E,F\<^esub> f"   and "bijec\<^bsub>F,G\<^esub> g" shows "bijec\<^bsub>E,G\<^esub> compos E g f"
sequentially_offset: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat" assumes "eventually P sequentially" shows "\<forall>\<^sub>F i in sequentially. P (i + k)"
derivation_sound: fixes Cs :: "fterm literal set set"   and Cs' :: "fterm literal set set"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "resolution_deriv Cs Cs'"   and "eval\<^sub>c\<^sub>s F G Cs" shows "eval\<^sub>c\<^sub>s F G Cs'"
multi_drop_mem_not_eq: fixes c :: "'b"   and B :: "'b multiset" assumes "c \<in># B" shows "B - (c, {#}) \<noteq> B"
str_max: shows "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)"
is_valid_problem_sas_plus_dom_sas_plus_problem_range_of: fixes \<Psi> :: "('d, 'e) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). v \<in> dom (range_of \<Psi>)"
ty_proc_decl_def: fixes \<Gamma>g :: "String.literal \<Rightarrow> unit option"   and pd :: "proc_decl" shows "ty_proc_decl \<Gamma>g pd \<equiv> let \<Gamma>l = \<lambda>x. if x \<in> set (local_vars pd) then Some () else None in ty_cmd ((\<Gamma>l, \<Gamma>g), False) (body pd)"
DBM_reset_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and c :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   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 "v c \<le> n"   and "DBM_reset (FW M n) n (v c) d M'" shows "([FW M n]\<^bsub>v,n\<^esub> = {}) = ([M']\<^bsub>v,n\<^esub> = {})"
subst_imp_simp: fixes p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (p \<rightarrow> q) \<sigma> = (Fsubst p \<sigma> \<rightarrow> Fsubst q \<sigma>)"
\<pi>_def: shows "\<pi> = prime_sum_upto (\<lambda>p. 1)"
bisim1BinOp1: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and bop :: "bop"   and e2 :: "'addr expr1" assumes "P,e1,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1 \<guillemotleft>bop\<guillemotright> e2,h \<turnstile> (e' \<guillemotleft>bop\<guillemotright> e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
differentiable_within_eqI: fixes g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "g differentiable at x within X"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x"   and "x \<in> X"   and "open X" shows "f differentiable at x within X"
nonneg_power_less: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(a ^ n < b ^ n) = (0 < n \<and> a < b)"
lemma_2_5_9_b: fixes a :: "'b"   and b :: "'b" shows "a \<le> b l\<rightarrow> a"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case EV_CONSUME_ALL of EV_CONSUME_ALL \<Rightarrow> f1 | EV_CONSUME_ONE \<Rightarrow> f2) = f1"
interpret_floatarith_times: fixes f :: "floatarith"   and g :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (f * g) xs = interpret_floatarith f xs * interpret_floatarith g xs"
simps: fixes f :: "'b \<rightarrow> 'c \<rightarrow> 'a"   and x :: "'c"   and y :: "'b" shows "flip\<cdot>f\<cdot>x\<cdot>y = f\<cdot>y\<cdot>x"
mk_doubleton_image_def: fixes X :: "V"   and Y :: "V" shows "mk_doubleton_image X Y = set (mk_doubleton Y ` elts X)"
comm_primroots: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "v \<noteq> \<epsilon>" shows "(u \<cdot> v = v \<cdot> u) = (\<rho> u = \<rho> v)"
leg0_composite: shows "\<nu>\<pi>.leg0 = \<pi>.leg0 \<cdot> \<nu>\<pi>.prj\<^sub>0"
simps_2: shows "X5d X5.Xd1' = D1"
obs_singleton: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and n :: "'g"   and S :: "'g set" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (CFG_wf.wod_backward_slice sourcenode targetnode valid_edge Def Use S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (CFG_wf.wod_backward_slice sourcenode targetnode valid_edge Def Use S) = {}"
Ord_vsubset_in_Vset_succI: fixes \<alpha> :: "V"   and A :: "V" assumes "Ord \<alpha>"   and "A \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "A \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
stream_dec_Inl: fixes i :: "nat"   and FO :: "nat set"   and n :: "nat"   and s :: "('a \<times> bool list) stream" assumes "i \<in> FO"   and "i < n" shows "\<exists>p. stream_dec n FO s ! i = Inl p"
lex_prodp_well_founded: fixes r1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "well_founded r1"   and "well_founded r2" shows "well_founded (lex_prodp r1 r2)"
rel_set_pos_distr_cond_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_set_pos_distr_cond A A' = (rel_set A OO rel_set A' \<le> rel_set (A OO A'))"
hyps: assumes "fold (\<or>?) l true = true" shows "fold (\<or>?) (rev l) false \<noteq> invalid"
SCConcEnd: fixes xs :: "('a, 'b) Combinators list"   and xa :: "('a, 'b) Combinators" assumes "singleCombinators (xs @ [xa])" shows "singleCombinators xs"
sorted_map_remove1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and x :: "'d" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)" shows "linorder.sorted less_eq (map f (remove1 x xs))"
ordinal_dvd_times_div: fixes y :: "ordinal"   and x :: "ordinal" assumes "y dvd x" shows "y * (x div y) = x"
next_commutes_2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>(F \<or> G) \<equiv> \<circle>F \<or> \<circle>G"
am_d_def: fixes x :: "'a" shows "am_d x = ad (ad x)"
prems_3: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
ABind_eq: fixes v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABind v e\<cdot>ae = \<A>\<^sup>\<bottom>\<^bsub>ae v\<^esub> e"
tl_nth: fixes l :: "'b list"   and n :: "nat" assumes "l \<noteq> []" shows "tl l ! n = l ! (n + 1)"
prems: shows "s1 = t1 on L (WHILE b DO c) X"
kD_but_not_vD_is_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt"   and "ip \<notin> vD rt" shows "ip \<in> iD rt"
dagger_slide: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<dagger> = (x \<cdot> y)\<^sup>\<dagger> \<cdot> x"
less_imp_encode_less: fixes base :: "nat"   and n :: "nat"   and p :: "nat"   and e :: "nat" assumes "2 \<le> base"   and "n < p" shows "encode base e n < encode base e p"
vwU: shows "adjuster n w us + w \<notin> local.span (set us)"
exists_ctran_Basic_None: fixes x :: "('b com option \<times> 'b) list"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and i :: "nat" assumes "x \<in> cptn"   and "x ! 0 = (Some (Basic f), s)"   and "i < length x"   and "fst (x ! i) = None" shows "\<exists>j<i. x ! j -c\<rightarrow> x ! Suc j"
prefix_Suc: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "prefix f (Suc n) = prefix f n @ [the (f (Suc n))]"
indaut_resplabels: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and B :: "'c set"   and \<phi> :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "ChamberComplex.label_wrt X B \<phi>"   and "v \<in> \<Union> X" shows "\<phi> (OpposedThinChamberComplexFoldings.induced_automorphism X f g v) = \<phi> v"
assms: shows "f x \<le> \<Down>\<^sub>E E R M'"
ord_\<Gamma>_def: shows "ord_\<Gamma> = {uu_. \<exists>CAs DA AAs As E. uu_ = Infer (mset CAs) DA E \<and> ord_resolve CAs DA AAs As E}"
CFG_Construct_axioms: shows "CFG_Construct \<alpha>e \<alpha>n invar inEdges' Entry defs uses"
get_disconnected_nodes_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "h \<turnstile> ok get_disconnected_nodes document_ptr" shows "document_ptr |\<in>| document_ptr_kinds h"
prj_in_hom_5: shows "\<guillemotleft>f_src_src.Prj\<^sub>1\<^sub>0 : Chn (f \<star> src f \<star> src f) \<rightarrow>\<^sub>C src.apex\<guillemotright>"
induct: fixes P :: "'b dual \<Rightarrow> bool"   and dual :: "'b dual" assumes "\<And>xa. P (\<partial> xa)" shows "P dual"
Sup_eq_PInfty: fixes S :: "ereal set" assumes "\<infinity> \<in> S" shows "Sup S = \<infinity>"
rep_eq: fixes x :: "float" shows "is_float_pos x = (0 < real_of_float x)"
weakSimTauChain: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and Q :: "('d, 'e, 'f) psi"   and Q' :: "('d, 'e, 'f) psi"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Rel :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(Q, Q') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>*"   and "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> env.weakSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi>' R Rel S" obtains P' :: "('d, 'e, 'f) psi" where "(P, P') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>*" and "(\<Psi>, P', Q') \<in> Rel"
tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes s :: "('fun, 'var) stateful_strand_step"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p s"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p s) \<inter> range_vars \<theta> = {}" shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"
distinct_41: fixes x31 :: "nat"   and x32 :: "int"   and x33 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "CN x31 x32 x33 \<noteq> CF x91 x92 x93"
fresh_perm: fixes a :: "atom"   and p :: "perm" shows "a \<sharp> p = (p \<bullet> a = a)"
prime_aprimedivisor': fixes n :: "'a" assumes "n \<noteq> (0::'a)"   and "\<not> is_unit n" shows "prime (aprimedivisor n)"
finite_FvarsT: fixes t :: "'trm" assumes "t \<in> trm" shows "finite (FvarsT t)"
snd_snd_foldl_echelon_form_of_column_k_iarrays: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and k :: "nat" shows "snd (snd (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<k])) = bezout"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "size (x # xs) = max (size x) (size xs) + 1"
pivot_unsat_core_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<U>\<^sub>c (pivot x\<^sub>i x\<^sub>j s) = \<U>\<^sub>c s"
\<mu>_bound_rowD: fixes fs :: "int vec list"   and bnd :: "rat"   and i :: "nat"   and j :: "nat" assumes "\<mu>_bound_row fs bnd i"   and "j \<le> i" shows "(gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)\<^sup>2 \<le> bnd"
dom_map_graph: fixes f :: "'c \<Rightarrow> 'd option" shows "dom f = Domain (map_graph f)"
apartment_chamber_distance_def: fixes X :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "X \<in> \<A>" shows "ChamberComplex.chamber_distance X C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D]))))"
convergent_imp_Bseq: fixes f :: "nat \<Rightarrow> 'b" assumes "convergent f" shows "Bseq f"
assms_6: shows "orient_lit_inst L1 t s polarity \<sigma>"
1: shows "sqrt 2 powr sqrt 2 \<notin> \<rat>"
IH: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" assumes "s\<^sub>1 = (h, l(V := None))" shows "P,E \<turnstile> \<langle>{V:T; e\<^sub>1},(h, l)\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l V))\<rangle>"
iprefixI: fixes f :: "nat \<Rightarrow> 'b"   and xs :: "'b list"   and g :: "nat \<Rightarrow> 'b" assumes "f = xs \<frown> g" shows "xs \<sqsubseteq> f"
vsv_vdomain_atD: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "vsv r" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b"
in_nlists_UNIV: fixes xs :: "'b list"   and n :: "nat" shows "(xs \<in> nlists UNIV n) = (length xs = n)"
that: shows "fx \<in> aform_err e (fst X, snd X + err)"
has_density_def: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "has_density M N f = (f \<in> borel_measurable N \<and> space N \<noteq> {} \<and> M = density N f)"
simps_3: fixes x3 :: "val"   and y3 :: "val" shows "(Val x3 = Val y3) = (x3 = y3)"
last_bin_last: fixes xs :: "bool list" assumes "0 < length xs" shows "last xs = odd (bl_to_bin xs)"
abs_rep2_ID': fixes pat :: "[:'r:]"   and usvs :: "[:'r:] \<times> [:'r:]" shows "abs2\<cdot>(rep2\<cdot>pat\<cdot>usvs) = usvs"
normalizability_I': fixes a :: "'c"   and b :: "'c"   and A :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>!" shows "(a, c) \<in> A\<^sup>!"
half_bounded_equal: fixes x :: "real" assumes "1 \<le> x * 2" shows "(x * 2 \<le> 1) = (x = 1 / 2)"
prems_2: shows "set (w # ws) \<subseteq> carrier_vec n"
prems: shows "\<forall>g\<in>set (take (Suc n) gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
degree_pdevs_of_list_append: fixes xs :: "'b list"   and ys :: "'b list" shows "degree (pdevs_of_list (xs @ ys)) \<le> length xs + degree (pdevs_of_list ys)"
product_nn_integral_prod: fixes I :: "'b set"   and f :: "'b \<Rightarrow> real \<Rightarrow> ennreal" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (interval_measure (\<lambda>x. x))" shows "\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I (\<lambda>_. interval_measure (\<lambda>x. x)) = (\<Prod>i\<in>I. integral\<^sup>N (interval_measure (\<lambda>x. x)) (f i))"
scale_sum_right: fixes a :: "'a"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "a *b sum f A = (\<Sum>x\<in>A. a *b f x)"