hcju/isabelleps huggingface dataset:
f: shows "\<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)"
hyps_3: assumes "r \<in> I" shows "(x, y) \<in> (\<Union>r\<in>I. r)\<^sup>*"
UnI1: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A" shows "c \<in> A \<union> B"
intro: fixes g_ops :: "('a, 'b, 'c, 'd) graph_ops_scheme"   and mr_ops :: "('a, ('a \<times> 'b \<times> 'a) list \<times> 'b, 'e, 'f) map_ops_scheme"   and qw_ops :: "('a, 'b infty, 'g, 'h) uprio_ops_scheme"   and ga :: "('a, 'b) graph"   and v0 :: "'a"   and g :: "'c" assumes "dijkstraC g_ops mr_ops qw_ops"   and "Dijkstra ga v0"   and "dijkstraC_fixg_axioms g_ops ga g" shows "dijkstraC_fixg g_ops mr_ops qw_ops ga v0 g"
id: shows "set us = insert (us ! i) (set us - {us ! i})"
wbalanced_bal_list: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "wbalanced (bal_list n xs)"
le_le_imp_sum_mult_le_sum_mult: fixes i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat" assumes "i \<le> j"   and "m \<le> n" shows "i * n + j * m \<le> i * m + j * n"
diff_ev_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "local.diff_ev A i j \<equiv> i < j \<longrightarrow> A $$ (i, i) \<noteq> A $$ (j, j) \<longrightarrow> A $$ (i, j) = (0::'a)"
0: fixes ps :: "(int \<times> int) list" shows "real (time (closest_pair_tm ps)) \<le> (closest_pair_time \<circ> length) ps"
unat_1_0: fixes x :: "'b word" shows "(1 \<le> x) = (0 < unat x)"
sim: shows "strong_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)"
trss_valid_preserve_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "valid fg (add_mset s c)"   and "((s, c), e, s', c') \<in> trss fg" shows "valid fg (add_mset s' c')"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P') A\<^sub>P \<Psi>\<^sub>P"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> N"   and "x \<sharp> A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' \<lparr>\<nu>x\<rparr>P') (x # A\<^sub>P) \<Psi>\<^sub>P"
real_of_int_div_aux: fixes x :: "int"   and d :: "int" shows "real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d"
\<pi>_induct: fixes i :: "nat"   and T :: "Unary_PCF.type"   and j :: "nat"   and P :: "Unary_PCF.type \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "i < ar T"   and "j \<le> ar (T !- i)"   and "\<And>T i. i < ar T \<Longrightarrow> P T i 0"   and "\<And>T i j. \<lbrakk>i < ar T; j < ar (T !- i); P (T !- i) j 0; \<forall>x<ar (T !- i !- j). P (T !- i) j (x + 1)\<rbrakk> \<Longrightarrow> P T i (j + 1)" shows "P T i j"
dghm_cn_comp_ArrMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ArrMap\<rparr>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<rparr>"
real_seq_to_hypreal_approx: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<forall>n. norm (X n - x) < inverse (real (Suc n))" shows "star_n X \<approx> star_of x"
conjugate_comp_ext: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f"   and "residuated g" shows "conjugate (\<lambda>x. f (g x)) x = conjugate g (conjugate f x)"
sup_mono: fixes a :: "'d"   and c :: "'d"   and b :: "'d"   and d :: "'d" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
impl_of_inject: fixes x :: "('b, 'c) RBT.rbt"   and y :: "('b, 'c) RBT.rbt" shows "(rbt.impl_of x = rbt.impl_of y) = (x = y)"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "x + y = y + x"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
less_literal_def: shows "(<) \<equiv> map_fun literal.explode (map_fun literal.explode id) (ord.lexordp (\<lambda>c d. of_char c < of_char d))"
card: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "type_definition Rep Abs A" shows "CARD('c) = card A"
coboundedI1: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># c" shows "a \<inter># b \<subseteq># c"
dim_map: fixes C :: "'a set" assumes "folding.chamber C" shows "card (f ` C) = card C"
f_analytic: shows "f analytic_on {s. \<sigma> \<le> Re s}"
wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>" shows "typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<sigma> \<equiv> \<forall>v. \<Gamma> (Var v) = \<Gamma> (\<sigma> v)"
bi_uniqueDr: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "bi_unique A"   and "A x y"   and "A x z" shows "y = z"
a: shows "AE x in MA. (\<lambda>n. A.birkhoff_sum fA n x / real n) \<longlonglongrightarrow> real_cond_exp MA A.Invariants fA x"
assms_1: shows "(a, b) \<in> A\<^sup>\<down>"
RCOSETS_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" shows "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})"
semiring_axioms_def: fixes R :: "('a, 'b) ring_scheme" shows "semiring_axioms R \<equiv> ((\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z) \<and> (\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y)) \<and> (\<forall>x. x \<in> carrier R \<longrightarrow> \<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>) \<and> (\<forall>x. x \<in> carrier R \<longrightarrow> x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>)"
type_less_eq_x_Real_intro_2: fixes \<tau> :: "'a basic_type" assumes "\<tau> = Integer" shows "\<tau> \<le> Real"
semiring_normalization_rules_16: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lx :: "'a"   and ly :: "'a"   and rx :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times (times lx ly) rx = times (times lx rx) ly"
REC_rule: fixes body :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and pre :: "'a \<Rightarrow> bool"   and x :: "'a"   and M :: "'a \<Rightarrow> 'b" assumes "trimono body"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x. pre x \<Longrightarrow> f x \<le> M x; pre x; f \<le> REC body\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC body x \<le> M x"
tensor_mat_scale2: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and a :: "'a" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "tensor_mat m1 (a \<cdot>\<^sub>m m2) = a \<cdot>\<^sub>m tensor_mat m1 m2"
match_build: fixes v :: "'v" shows "match (build v) = Some v"
elims_2: fixes inPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and x :: "'c \<times> 'b" assumes "Vertex_Graph.valid_in_port inPorts vertices nodeOf x" obtains v :: "'c" and p :: "'b" where "x = (v, p)" and "v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v)"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a"   and SRel :: "('b \<times> 'b) set" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelRSTPO Enc SRel TRel"
T_2: fixes t :: "real" assumes "T \<le> t" shows "\<bar>Si t\<bar> \<le> pi / 2 + 1"
cases_scheme: fixes r :: "'a t_context_scheme" obtains types_t :: "tf list" and func_t :: "tf list" and global :: "tg list" and table :: "nat option" and memory :: "nat option" and local :: "t list" and label :: "t list list" and return :: "t list option" and more :: "'a" where "r = \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr>"
atLeastLessThan_eq_atLeastAtMost_diff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order less_eq less" shows "ord.atLeastLessThan less_eq less a b = ord.atLeastAtMost less_eq a b - {b}"
exhaust: fixes y :: "'a \<times> 'b" obtains a :: "'a" and b :: "'b" where "y = (a, b)"
unfold': fixes E :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> 'res) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state"   and self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3)" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self \<and> (case a1 of (a1, a2, a3) \<Rightarrow> \<lambda>\<tau>. \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2 \<and> \<tau> \<Turnstile> \<upsilon> a3) \<tau>"   and "\<tau> \<Turnstile> case a1 of (a, b, c) \<Rightarrow> PRE self a b c"   and "\<exists>res. \<tau> \<Turnstile> (case a1 of (a, b, c) \<Rightarrow> POST self a b c) (\<lambda>_. res)"   and "\<And>res. \<tau> \<Turnstile> (case a1 of (a, b, c) \<Rightarrow> POST self a b c) (\<lambda>_. res) \<Longrightarrow> \<tau> \<Turnstile> E (\<lambda>_. res)" shows "\<tau> \<Turnstile> E (case a1 of (a, b, c) \<Rightarrow> f self a b c)"
case_sum_expand_Inr_pointfree: fixes f :: "'d + 'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "f \<circ> Inl = g" shows "case_sum g (f \<circ> Inr) = f"
boolean_binop_type_det: fixes op :: "boolean_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "boolean_binop_type op \<tau> \<sigma> \<rho>\<^sub>1"   and "boolean_binop_type op \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
fresh_var_def: fixes Vs :: "String.literal list" shows "fresh_var Vs = sum_list (STR ''V'' # Vs)"
Int_stable_def: fixes M :: "'d set set" shows "Int_stable M = (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M)"
has_codomain_char: fixes \<mu> :: "'a arrow_of_spans_data" shows "(V.codomains \<mu> \<noteq> {}) = arrow_of_spans (\<cdot>) \<mu>"
cycle_rotate_2_aux: fixes i :: "nat"   and j :: "nat"   and a :: "nat"   and b :: "nat"   and xs :: "nat list"   and c :: "nat" assumes "(i, j) \<in> set (arcs a b (xs @ [c]))"   and "(i, j) \<noteq> (c, b)" shows "(i, j) \<in> set (arcs a c xs)"
Quotient_linear_poly: shows "Quotient (eq_onp Abstract_Linear_Poly.inv) Abs_linear_poly Rep_linear_poly cr_linear_poly"
graph_eq_intro: fixes G :: "'b graph"   and G' :: "'b graph" assumes "nodes G = nodes G'"   and "edges G = edges G'" shows "G = G'"
square_part_dvd: fixes n :: "nat" shows "(square_part n)\<^sup>2 dvd n"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "partial_magma.ide V \<mu>" shows "identity_arrow_of_tabulations_in_maps_axioms V \<mu>"
assms: shows "fm2 \<subseteq>\<^sub>f fm1"
that: fixes N :: "nat" assumes "\<And>n. N < n \<Longrightarrow> eint k < val_Zp (a \<ominus> s n)" shows "thesis"
set_to_map_simp: fixes S :: "('a \<times> 'b) set"   and k :: "'a"   and v :: "'b" assumes "inj_on fst S" shows "(set_to_map S k = Some v) = ((k, v) \<in> S)"
ctxt_3: shows "\<forall>A\<in>\<Gamma>. typ_of A = Some propT"
simps: fixes v :: "tm"   and x :: "tm" shows "VarNonOccFormP v x = FormP x AND SubstFormP v Zero x x"
kruskal_forest_components_inf: fixes f :: "'a"   and w :: "'a" assumes "f \<le> w \<squnion> id w"   and "w \<sqinter> id w \<le> top" shows "top \<sqinter> top \<le> top \<sqinter> top"
1: shows "\<alpha> = x @ [N] @ y \<and> \<beta> = x @ \<delta> @ y \<and> is_sentence x \<and> is_sentence y \<and> (N, \<delta>) \<in> \<RR>"
turn_true_simps_1: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" assumes "\<A> \<Turnstile> F" shows "turn_true \<A> F = F"
cases: fixes a1 :: "com"   and a2 :: "state" assumes "a1 \<down> a2" obtains f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = Do f__" and "a2 = s__" and "f__ s__ \<noteq> {}"   | c1 :: "com" and s0 :: "state" and c2 :: "com" where "a1 = (c1__; c2__)" and "a2 = s0" and "c1__ \<down> s0" and "\<And>s1. s0 -c1__\<rightarrow> s1 \<Longrightarrow> c2__ \<down> s1"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" where "a1 = IF b__ THEN c1a__ ELSE c2a__" and "a2 = sa__" and "b__ sa__" and "c1a__ \<down> sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and c1 :: "com" where "a1 = IF ba__ THEN c1 ELSE c2" and "a2 = sb__" and "\<not> ba__ sb__" and "c2 \<down> sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE bb__ DO c__" and "a2 = sc__" and "\<not> bb__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE b DO ca__" and "a2 = sd__" and "b sd__" and "ca__ \<down> sd__" and "\<And>t. sd__ -ca__\<rightarrow> t \<Longrightarrow> WHILE b DO ca__ \<down> t"   | s :: "state" where "a1 = CALL" and "a2 = se__" and "body \<down> se__"   | c :: "com" and f :: "state \<Rightarrow> state" and s :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = LOCAL f; c; g" and "a2 = s" and "c \<down> f s"
field_one_plus_frac3: fixes x :: "'a" assumes "x \<in> carrier K"   and "x \<noteq> 1\<^sub>r"   and "1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x) \<noteq> \<zero>" shows "1\<^sub>r \<plusminus> -\<^sub>a x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x))\<^bsup>\<hyphen> K\<^esup> = (1\<^sub>r \<plusminus> -\<^sub>a x^\<^bsup>K Suc (Suc 0)\<^esup>) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x))\<^bsup>\<hyphen> K\<^esup>"
intro: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h new_element_ptr h' r ptr'. \<lbrakk>h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr; h \<turnstile> new_element \<rightarrow>\<^sub>h h'; r \<in> get_disconnected_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_new_element_get_disconnected_nodes get_disconnected_nodes_locs"
encode_problem_parallel_complete_ii: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"   and "\<forall>v l. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l \<and> l < length \<pi> + 1 \<longrightarrow> \<A> (State l (index (\<Pi>\<^sub>\<V>) v)) = \<A> (State (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1) (index (\<Pi>\<^sub>\<V>) v))" shows "\<A> \<Turnstile> (\<Phi>\<^sub>G \<Pi>) (length \<pi>)"
mult_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec"   and k :: "'b" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "A *\<^sub>v (k \<cdot>\<^sub>v v) = k \<cdot>\<^sub>v (A *\<^sub>v v)"
simps_2: fixes L :: "'b language"   and x :: "'b"   and xs :: "'b list" shows "in_language L (x # xs) = in_language (\<dd> L x) xs"
round_down: fixes prec :: "int"   and x :: "real" shows "round_down prec x \<le> x"
select_convs_2: fixes lens_get :: "'a \<Rightarrow> 'b"   and lens_put :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and more :: "'c" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
simps_2: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and t :: "'a"   and b :: "bool"   and ta :: "('a, 'd, 'e) new_thread_action" shows "nta_bisim bisim (ThreadExists t b) ta = (ta = ThreadExists t b)"
h0: fixes x :: "'a" shows "h (0, x) = f x"
lt_rep_list_spair: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and a :: "'a"   and b :: "'a" assumes "rep_list p \<noteq> 0"   and "rep_list q \<noteq> 0"   and "rep_list (spair p q) \<noteq> 0"   and "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)" shows "punit.lt (rep_list (spair p q)) \<prec> a - gcs a b + punit.lt (rep_list p)"
can_take_def: fixes a :: "nat"   and g :: "vname gexp list"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "can_take a g i r = (length i = a \<and> apply_guards g (join_ir i r))"
of_nat_power_eq_of_nat_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero x = power.power one times (semiring_1.of_nat one plus zero b) w) = (x = b ^ w)"
tendsto_mult_filterlim_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> (0::'b)"   and "filterlim g at_infinity F" shows "LIM x F. f x * g x :> at_infinity"
opt_bst2_opt_bst: fixes i :: "int"   and j :: "int" shows "opt_bst2 i j = local.opt_bst i j"
converse_rtranclE3: fixes xa :: "'a"   and xb :: "'b"   and xc :: "'c"   and za :: "'a"   and zb :: "'b"   and zc :: "'c"   and r :: "(('a \<times> 'b \<times> 'c) \<times> 'a \<times> 'b \<times> 'c) set" assumes "((xa, xb, xc), za, zb, zc) \<in> r\<^sup>*" obtains "(xa, xb, xc) = (za, zb, zc)"   | a :: "'a" and aa :: "'b" and b :: "'c" where "((xa, xb, xc), a, aa, b) \<in> r" and "((a, aa, b), za, zb, zc) \<in> r\<^sup>*"
assms_2: shows "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"
bounded_bilinear_def: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "bounded_bilinear prod \<equiv> ((\<forall>a a' b. prod (a + a') b = prod a b + prod a' b) \<and> (\<forall>a b b'. prod a (b + b') = prod a b + prod a b')) \<and> (\<forall>r a b. prod (r *\<^sub>R a) b = r *\<^sub>R prod a b) \<and> (\<forall>a r b. prod a (r *\<^sub>R b) = r *\<^sub>R prod a b) \<and> (\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K)"
is_float_def: fixes r :: "real" shows "is_float r = (r \<in> float)"
has_integral_cmul: fixes f :: "'b \<Rightarrow> 'c"   and k :: "'c"   and S :: "'b set"   and c :: "real" assumes "(f has_integral k) S" shows "((\<lambda>x. c *\<^sub>R f x) has_integral c *\<^sub>R k) S"
psubstEnv_updEnv_idEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "idEnv [x \<leftarrow> X]_xs &[rho] = rho [x \<leftarrow> X #[rho]]_xs"
next_in_basics_3: fixes A :: "'b set"   and a :: "'b"   and b :: "'b"   and u :: "'b" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a..<b}" shows "A \<inter> {u<..<next_in A u} = {}"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list"   and a2 :: "'a list list" assumes "\<And>i. P i [] []"   and "\<And>i a as bs bss. P i as bss \<Longrightarrow> P i (a # as) (bs # bss)"   and "\<And>a v va. P a (v # va) []"   and "\<And>a v va. P a [] (v # va)" shows "P a0 a1 a2"
sub_distrib_pconj_def: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "sub_distrib_pconj prog \<equiv> \<forall>P Q. unitary P \<longrightarrow> unitary Q \<longrightarrow> wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)"
error_def: fixes n :: "nat" shows "error n = map (\<lambda>x. (x, Err)) [0..<n]"
axioms_1: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "Category C"
n_add_idem: fixes x :: "'a" shows "n x \<oplus> n x = n x"
invar_children: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e a r ts)" shows "BinomialHeapStruc.queue_invar ts"
igSubstInpIPresIGWlsInp_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstInpIPresIGWlsInp MOD \<equiv> \<forall>ys y Y delta inp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<longrightarrow> igWlsInp MOD delta (igSubstInp MOD ys Y y inp)"
wls_Abs_swap_fresh: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "wls s X"   and "fresh xs x' X" shows "Abs xs x' (X #[x' \<and> x]_xs) = Abs xs x X"
subset_inext_closed: fixes n :: "nat"   and B :: "nat set"   and A :: "nat set" assumes "n \<in> B"   and "A \<subseteq> B" shows "inext n A \<in> B"
c_top_welldef: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "N \<bullet>\<^bsub>G\<^esub> (a \<cdot>\<^bsub>G\<^esub> b) = c_top G N (N \<bullet>\<^bsub>G\<^esub> a) (N \<bullet>\<^bsub>G\<^esub> b)"
simps_6: fixes V :: "'c set"   and uu :: "'c list"   and uv :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and S :: "('d, 'c) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<forall>uu\<langle>\<or>\<noteq>: uv\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t V S"
bind_lunit: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b option" shows "Option.bind (Some x) f = f x"